Monte Carlo Methods: A Hands-On Computational Introduction Utilizing Excel 1636390714, 9781636390710

Table of contents :
Introduction
Random Variable
Continuous Random Variable
Uniform Random Variable
Normal or Gaussian Random Variable
Transformation or Modeling of Random Variable
Examples of Transformation or Modeling of Random Variable
Variance Reduction and Importance Sampling
Evaluation of Definite Integrals Using the Monte Carlo Method
Evaluation of Definite Integrals Using the Monte Carlo Method: Example I
Evaluation of Definite Integrals Using the Monte Carlo Method: Example II
Evaluation of Definite Integrals Using the Monte Carlo Method: Example III
Evaluation of Definite Integrals Using the Monte Carlo Method: Example IV
Evaluation of Definite Integrals Using the Monte Carlo Method: Example V
Variational Monte Carlo Method Applied to the Ground State of a Simple Harmonic Oscillator
The Variational Method of Quantum Mechanics Applied to the Ground State of Any Quantum Mechanical System
The Variational Method of Quantum Mechanics Applied to the Ground State of a Simple Harmonic Oscillator
Ground State Energy of a Simple Harmonic Oscillator Using the Monte Carlo Method
Variational Monte Carlo Method Applied to the Ground State of a Hydrogen Atom
The Variational Method of Quantum Mechanics Applied to the Ground State of Any Quantum Mechanical System (again)
The Variational Method of Quantum Mechanics Applied to the Ground State of a Hydrogen Atom
Ground State Energy of a Hydrogen Atom Using the Variational Monte Carlo Method
Concluding Remarks
Bibliography
Author's Biography
Blank Page

Citation preview

ℝ

Series ISSN: 1938-1743

Sujaul Chowdhury, Shahjalal University of Science and Technology

MONTE CARLO METHODS

This book is intended for undergraduate students of Mathematics, Statistics, and Physics who know nothing about Monte Carlo Methods but wish to know how they work. All treatments have been done as much manually as is practicable. The treatments are deliberately manual to let the readers get the real feel of how Monte Carlo Methods work. Definite integrals of a total of five functions F(x), namely Sin(x), Cos(x), ex, loge(x), and 1/(1+x2), have been evaluated using constant, linear, Gaussian, and exponential probability density functions p(x). It is shown that results agree with known exact values better if p(x) is proportional to F(x). Deviation from the proportionality results in worse agreement. Two separate chapters have been dedicated to Variational Quantum Monte Carlo method applied to ground state of simple harmonic oscillator and of Hydrogen atom. The book is intended to aid hands-on learning of Monte Carlo method.



Monte Carlo Methods A Hands-On Computational Introduction Utilizing Excel

CHOWDHURY

Series Editor: Steven G. Krantz, Washington University in St. Louis



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Monte Carlo Methods A Hands-On Computational Introduction Utilizing Excel Sujaul Chowdhury

Monte Carlo Methods A Hands-On Computational Introduction Utilizing Excel

Synthesis Lectures on Mathematics and Statistics Editor Steven G. Krantz, Washington University, St. Louis

Monte Carlo Methods: A Hands-On Computational Introduction Utilizing Excel Sujaul Chowdhury 2021

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iv

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Statistics is Easy! Dennis Shasha and Manda Wilson 2008

A Gyrovector Space Approach to Hyperbolic Geometry Abraham Albert Ungar 2008

Copyright © 2021 by Morgan & Claypool

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means—electronic, mechanical, photocopy, recording, or any other except for brief quotations in printed reviews, without the prior permission of the publisher. Monte Carlo Methods: A Hands-On Computational Introduction Utilizing Excel Sujaul Chowdhury www.morganclaypool.com ISBN: 9781636390710 ISBN: 9781636390727 ISBN: 9781636390734

paperback ebook hardcover

DOI 10.2200/S01073ED1V01Y202101MAS037

A Publication in the Morgan & Claypool Publishers series SYNTHESIS LECTURES ON MATHEMATICS AND STATISTICS Lecture #37 Series Editor: Steven G. Krantz, Washington University, St. Louis Series ISSN Print 1938-1743 Electronic 1938-1751

Monte Carlo Methods A Hands-On Computational Introduction Utilizing Excel

Sujaul Chowdhury Shahjalal University of Science and Technology

SYNTHESIS LECTURES ON MATHEMATICS AND STATISTICS #37

M &C

Morgan

& cLaypool publishers

ABSTRACT This book is intended for undergraduate students of Mathematics, Statistics, and Physics who know nothing about Monte Carlo Methods but wish to know how they work. All treatments have been done as much manually as is practicable. The treatments are deliberately manual to let the readers get the real feel of how Monte Carlo Methods work. Definite integrals of a total of five functions F .x/, namely Sin.x/, Cos.x/, ex , loge .x/, and 1=.1 C x 2 /, have been evaluated using constant, linear, Gaussian, and exponential probability density functions p.x/. It is shown that results agree with known exact values better if p.x/ is proportional to F .x/. Deviation from the proportionality results in worse agreement. This book is on Monte Carlo Methods which are numerical methods for Computational Physics. These are parts of a syllabus for undergraduate students of Mathematics and Physics for the course titled “Computational Physics.” Need for the book: Besides the three referenced books, this is the only book that teaches how basic Monte Carlo methods work. This book is much more explicit and easier to follow than the three referenced books. The two chapters on the Variational Quantum Monte Carlo method are additional contributions of the book. Pedagogical features: After a thorough acquaintance with background knowledge in Chapter 1, five thoroughly worked out examples on how to carry out Monte Carlo integration is included in Chapter 2. Moreover, the book contains two chapters on the Variational Quantum Monte Carlo method applied to a simple harmonic oscillator and a hydrogen atom. The book is a good read; it is intended to make readers adept at using the method. The book is intended to aid in hands-on learning of the Monte Carlo methods.

KEYWORDS Monte Carlo methods, basic Monte Carlo integration, variational quantum Monte Carlo method, simple harmonic oscillator, hydrogen atom

ix

Contents 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7

2

Evaluation of Definite Integrals Using the Monte Carlo Method . . . . . . . . . . . 37 2.1 2.2 2.3 2.4 2.5

3

Evaluation of Definite Integrals Using the Monte Carlo Method: Example I Evaluation of Definite Integrals Using the Monte Carlo Method: Example II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evaluation of Definite Integrals Using the Monte Carlo Method: Example III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evaluation of Definite Integrals Using the Monte Carlo Method: Example IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evaluation of Definite Integrals Using the Monte Carlo Method: Example V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 44 66 70 75

Variational Monte Carlo Method Applied to the Ground State of a Simple Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.1 3.2 3.3

4

Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Continuous Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Uniform Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Normal or Gaussian Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Transformation or Modeling of Random Variable . . . . . . . . . . . . . . . . . . . . . . . 2 Examples of Transformation or Modeling of Random Variable . . . . . . . . . . . . 5 Variance Reduction and Importance Sampling . . . . . . . . . . . . . . . . . . . . . . . . . 32

The Variational Method of Quantum Mechanics Applied to the Ground State of Any Quantum Mechanical System . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 The Variational Method of Quantum Mechanics Applied to the Ground State of a Simple Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Ground State Energy of a Simple Harmonic Oscillator Using the Monte Carlo Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Variational Monte Carlo Method Applied to the Ground State of a Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

x

4.1 4.2 4.3

5

The Variational Method of Quantum Mechanics Applied to the Ground State of Any Quantum Mechanical System (again) . . . . . . . . . . . . . . . . . . . . 107 The Variational Method of Quantum Mechanics Applied to the Ground State of a Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Ground State Energy of a Hydrogen Atom Using the Variational Monte Carlo Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Author’s Biography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

1

CHAPTER

1

Introduction 1.1

RANDOM VARIABLE

In layman terms, a random variable refers to a variable the value of which is not predictable. We do not know what value the variable can or will assume. In connection with the Monte Carlo method, the term “random variable” has a precise meaning. We do not know the value of the variable in any given case, but we do know the values that the variable can assume and the probabilities of these values. The result of a single trial cannot be precisely predicted, but the result of a large number of trials can be predicted very reliably. To define a random variable, we must indicate the values that the variable can assume and the probabilities of occurrence of these values.

1.2

CONTINUOUS RANDOM VARIABLE

A random variable x is called “continuous” if it can assume any (fractional) value in a certain interval, say a to b . Besides specifying the interval containing all its possible values, we need to state a function p.x/ called probability density function or probability distribution function. We have: 1. p.x/  0 for a  x  b , 2. the product p.x/dx is probability that x  x  x C dx , R b0 3. a0 p.x/dx is probability that a0  x  b 0 where a < a0 and b 0 < b , and Rb 4. a p.x/dx D 1 indicating the surety that x lies in the interval a to b . As such, the average or so-called expectation value of x is given by Ax D Rb and average value of any function of x , say f .x/, is Af D a f .x/p.x/dx .

1.3

Rb a

xp.x/dx ,

UNIFORM RANDOM VARIABLE

A random variable u in the interval 0  x  1 having constant, say C , probability density funcR1 tion p.x/ D C is said to be a uniform random variable. The requirement that 0 p.x/dx D 1 R1 gives 0 Cdx D 1 which gives C D 1. Thus, probability density function of uniform random variable is p.x/ D 1.

2

1. INTRODUCTION

Here is a manual way of obtaining values of a uniform random variable. Take ten table tennis balls; write on each of them one of the ten digits 0, 1, 2, 3, …, 9; mingle them well in a box; take one ball out and note the digit; place the ball back to the box and mingle the ten balls well again; then take one ball out and note the digit. Continue the process until we get a table of 300 digits. From this table, take a sequence of three digits, say 1, 6, 9; construct the number 169; then divide the number by 1000 and get 0.169. Repeat the process until we get a list of 100 numbers such as 0.169. Thus, we have a table of 100 numbers in the interval 0 to 1. Since each of the digits 0, 1, 2, 3, …, 9 was equally likely to be picked up during the trials, the said 100 numbers are uniformly distributed in the interval 0 to 1. Thus, we have 100 values of uniform random variable u. See Table 1.1.

1.4

NORMAL OR GAUSSIAN RANDOM VARIABLE

Probability density function p.x/ for normal random variable x is given by p.x/ D

1 p Exp  2



 1  x a 2 ; 2 

(1.1)

defined for /< x < C /. The average value of x is a and variance of x is  2 . According to R aC3 the so-called rule of 3 sigma, a 3 p.x/dx  0:997.

1.5

TRANSFORMATION OR MODELING OF RANDOM VARIABLE

Values of random variable  corresponding to a given probability density function p.x/ defined for a  x  b can be obtained by transformation of values of one “standard” random variable which is usually taken as the uniform random variable u (distributed in the interval 0  x  1). See Table 1.1. We already know the values of u. The process of finding the values of  by transforming the values of u is called modeling or transformation of the random variable. With every available value of u, the corresponding value of  is obtainable by solving the equation Z a



p.x/dx D u:

(1.2)

This is verified below. Here values of  are distributed in the interval a  x  b with probability density function p.x/. Let Z x y.x/ D p.x/dx: (1.3) a

1.5. TRANSFORMATION OR MODELING OF RANDOM VARIABLE

Table 1.1: Showing 100 uniform random numbers ui (in the interval 0–1) i

ui

i

ui

i

ui

i

ui

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

0.169 0.050 0.844 0.182 0.599 0.985 0.726 0.148 0.788 0.698 0.010 0.397 0.852 0.761 0.309 0.113 0.212 0.809 0.084 0.276 0.243 0.989 0.508 0.817 0.174

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

0.225 0.687 0.060 0.070 0.835 0.466 0.271 0.813 0.376 0.983 0.181 0.583 0.612 0.468 0.614 0.768 0.811 0.461 0.985 0.735 0.559 0.884 0.229 0.920 0.524

51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75

0.393 0.306 0.819 0.620 0.875 0.773 0.492 0.332 0.185 0.565 0.434 0.665 0.516 0.659 0.709 0.515 0.683 0.092 0.417 0.598 0.682 0.948 0.125 0.149 0.246

76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

0.231 0.836 0.167 0.545 0.748 0.440 0.674 0.287 0.052 0.940 0.679 0.910 0.254 0.236 0.854 0.877 0.483 0.117 0.847 0.201 0.294 0.071 0.065 0.740 0.954

3

4

1. INTRODUCTION

y 1 y(b′) y(a′) Q

u

(0, 0)

a

a′ b′

η

b

x

Figure 1.1: Figure to help explain transformation or modeling of random variable.

Since p.x/ > 0, and

Rb a

p.x/dx D 1, we have y.a/ D y.b/ D

Z

a

p.x/dx D 0

a

Z

a

b

p.x/dx D 1

emphasizing that p.x/ is normalized, and y 0 .x/ D p.x/ > 0:

This means that y monotonically increases from 0 to 1 for the interval a < x < b . See Figure 1.1. Straight line y D u where 0 < R u < 1 intersects the curve y.x/ at Q giving us the value of  which satisfies Equation (1.2): a p.x/dx D u. p.a0 <  < b 0 / D p.y.a0 / < u < y.b 0 // D y.b 0 / y.a0 / Z b0 Z a0 Z b0 D p.x/dx p.x/dx D p.x/dx: a

a0

a

Thus, random variable  which obeys Equation (1.2): function p.x/.

R a

p.x/dx D u has the probability density

1.6. EXAMPLES OF TRANSFORMATION OR MODELING OF RANDOM VARIABLE

1.6

EXAMPLES OF TRANSFORMATION OR MODELING OF RANDOM VARIABLE

Values of random variable  obeying normalized, arbitrary probability density function p.x/ in R the interval a < x < b are obtainable from Equation (1.2): a p.x/dx D u where u is uniform random variable in the interval 0 to 1. Values of random variable  uniformly distributed in the interval a < x < b R Rb are obtainable from a C dx D u where p.x/ should be normalized, i.e., a p.x/dx D 1, Rb R a/. Thus, a b 1 a dx D u. Hence, a Cdx D 1, C D 1=.b

Example 1.1

 D a C .b

a/u:

(1.4)

This is a linear transformation of u. Use of tabulated values of u (see Table 1.1) in Equation (1.4) gives us values of random variable  obeying uniform probability density function p.x/ D 1=.b a/ in the interval a < x < b . See Table 1.2 which shows 100 uniform random numbers i in the interval =2 to C=2 obtained using Equation (1.4):  D a C .b a/u which gives i ’s as a linear transformation of ui ’s. We now discuss modeling of random variable g obeying Gaussian or normal probability density function given by   1 1  x a 2 pg .xI a D 0;  D 1/ D p Exp : (1.5) 2   2 R with p given by Equation (1.5) Values of g are obtainable from Equation (1.2): a hp.x/dx D u
2 i Rg 1 1 x a with a D 0 and  D 1, i.e., from a 3  p2 Exp 2  dx D u with a D 0 and  D 1, i.e., from   Z g 1 2 1 x dx D u: (1.6) p Exp 2 2 3 But Equation (1.6) is not analytically solvable  such, we have used Program 1.3 in R  for g . As Mathematica 6.0 and obtained values of 3 p12 Exp 12 x 2 dx for  ranging from 3 to in step of 0.005. According to Equation (1.6), values of g are those values of  for which RC3  1 2 p1 x dx D u. Hence, we have literally and manually hunted for each value of u 3 2 Exp 2 of Table 1.1 in the 3rd and 6th columns of Table 1.3 and wrote down the corresponding value of  from the 2nd and 5th columns of Table 1.3 in Table 1.4. Example 1.2

5

6

1. INTRODUCTION

Table 1.2: Showing 100 uniform random numbers i in the interval =2 to C=2 obtained using Equation (1.4):  D a C .b a/u which gives i ’s as linear transformation of ui ’s i

ui

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

0.169 0.050 0.844 0.182 0.599 0.985 0.726 0.148 0.788 0.698 0.010 0.397 0.852 0.761 0.309 0.113 0.212 0.809 0.084 0.276 0.243 0.989 0.508 0.817 0.174

ηi = −π/2 + πui i –1.040 –1.414 1.081 –0.999 0.311 1.524 0.710 –1.106 0.905 0.622 –1.539 –0.324 1.106 0.820 –0.600 –1.216 –0.905 0.971 –1.307 –0.704 –0.807 1.536 0.025 0.996 –1.024

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

ui 0.225 0.687 0.060 0.070 0.835 0.466 0.271 0.813 0.376 0.983 0.181 0.583 0.612 0.468 0.614 0.768 0.811 0.461 0.985 0.735 0.559 0.884 0.229 0.920 0.524

ηi = −π/2 + πui i –0.864 0.587 –1.382 –1.351 1.052 –0.107 –0.719 0.983 –0.390 1.517 –1.002 0.261 0.352 –0.101 0.358 0.842 0.977 –0.123 1.524 0.738 0.185 1.206 –0.851 1.319 0.075

51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75

ui

ηi = −π/2 + πui

i

ui

ηi = −π/2 + πui

0.393 0.306 0.819 0.620 0.875 0.773 0.492 0.332 0.185 0.565 0.434 0.665 0.516 0.659 0.709 0.515 0.683 0.092 0.417 0.598 0.682 0.948 0.125 0.149 0.246

–0.336 –0.609 1.002 0.377 1.178 0.858 –0.025 –0.528 –0.990 0.204 –0.207 0.518 0.050 0.500 0.657 0.047 0.575 –1.282 –0.261 0.308 0.572 1.407 –1.178 –1.103 –0.798

76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

0.231 0.836 0.167 0.545 0.748 0.440 0.674 0.287 0.052 0.940 0.679 0.910 0.254 0.236 0.854 0.877 0.483 0.117 0.847 0.201 0.294 0.071 0.065 0.740 0.954

–0.845 1.056 –1.046 0.141 0.779 –0.188 0.547 –0.669 –1.407 1.382 0.562 1.288 –0.773 –0.829 1.112 1.184 –0.053 –1.203 1.090 –0.939 –0.647 –1.348 –1.367 0.754 1.426

1.6. EXAMPLES OF TRANSFORMATION OR MODELING OF RANDOM VARIABLE

  R Table 1.3: Showing values of 3 p12 Exp 12 x 2 dx for  ranging from 3 to C3 in step of 0.005. Tabulated using Program 1.3 via Microsoft Excel. η

i

η

1

η

1 – exp3– – x24dx ∫ Ï· 2 2π

i

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.001 0.001 0.001 0.001

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

η

–3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

–3.000 –2.995 –2.990 –2.985 –2.980 –2.975 –2.970 –2.965 –2.960 –2.955 –2.950 –2.945 –2.940 –2.935 –2.930 –2.925 –2.920 –2.915 –2.910 –2.905 –2.900 –2.895 –2.890 –2.885 –2.880

1

1 – exp3– – x24dx ∫ Ï· 2 2π

–3

–2.875 –2.870 –2.865 –2.860 –2.855 –2.850 –2.845 –2.840 –2.835 –2.830 –2.825 –2.820 –2.815 –2.810 –2.805 –2.800 –2.795 –2.790 –2.785 –2.780 –2.775 –2.770 –2.765 –2.760 –2.755

0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.002 0.002

7

8

1. INTRODUCTION

η

i

η

1

η

1 – exp3– – x24dx ∫ Ï· 2 2π

i

0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.003 0.003 0.003 0.003 0.003 0.003 0.003

76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

η

–3

51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75

–2.750 –2.745 –2.740 –2.735 –2.730 –2.725 –2.720 –2.715 –2.710 –2.705 –2.700 –2.695 –2.690 –2.685 –2.680 –2.675 –2.670 –2.665 –2.660 –2.655 –2.650 –2.645 –2.640 –2.635 –2.630

1

1 – exp3– – x24dx ∫ Ï· 2 2π

–3

–2.625 –2.620 –2.615 –2.610 –2.605 –2.600 –2.595 –2.590 –2.585 –2.580 –2.575 –2.570 –2.565 –2.560 –2.555 –2.550 –2.545 –2.540 –2.535 –2.530 –2.525 –2.520 –2.515 –2.510 –2.505

0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.005 0.005 0.005 0.005

1.6. EXAMPLES OF TRANSFORMATION OR MODELING OF RANDOM VARIABLE

η

i

η

1

η

1 – exp3– – x24dx ∫ Ï· 2 2π

i

0.005 0.005 0.005 0.005 0.005 0.005 0.005 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.006 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007

126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150

η

–3

101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125

–2.500 –2.495 –2.490 –2.485 –2.480 –2.475 –2.470 –2.465 –2.460 –2.455 –2.450 –2.445 –2.440 –2.435 –2.430 –2.425 –2.420 –2.415 –2.410 –2.405 –2.400 –2.395 –2.390 –2.385 –2.380

1

1 – exp3– – x24dx ∫ Ï· 2 2π

–3

–2.375 –2.370 –2.365 –2.360 –2.355 –2.350 –2.345 –2.340 –2.335 –2.330 –2.325 –2.320 –2.315 –2.310 –2.305 –2.300 –2.295 –2.290 –2.285 –2.280 –2.275 –2.270 –2.265 –2.260 –2.255

0.007 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.009 0.009 0.009 0.009 0.009 0.009 0.009 0.010 0.010 0.010 0.010 0.010 0.010 0.010 0.011 0.011

9

10

1. INTRODUCTION

η

i

η

1

η

1 – exp3– – x24dx ∫ Ï· 2 2π

i

0.011 0.011 0.011 0.011 0.012 0.012 0.012 0.012 0.012 0.012 0.013 0.013 0.013 0.013 0.013 0.013 0.014 0.014 0.014 0.014 0.014 0.015 0.015 0.015 0.015

176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200

η

–3

151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175

–2.250 –2.245 –2.240 –2.235 –2.230 –2.225 –2.220 –2.215 –2.210 –2.205 –2.200 –2.195 –2.190 –2.185 –2.180 –2.175 –2.170 –2.165 –2.160 –2.155 –2.150 –2.145 –2.140 –2.135 –2.130

1

1 – exp3– – x24dx ∫ Ï· 2 2π

–3

–2.125 –2.120 –2.115 –2.110 –2.105 –2.100 –2.095 –2.090 –2.085 –2.080 –2.075 –2.070 –2.065 –2.060 –2.055 –2.050 –2.045 –2.040 –2.035 –2.030 –2.025 –2.020 –2.015 –2.010 –2.005

0.015 0.016 0.016 0.016 0.016 0.017 0.017 0.017 0.017 0.017 0.018 0.018 0.018 0.018 0.019 0.019 0.019 0.019 0.020 0.020 0.020 0.020 0.021 0.021 0.021

1.6. EXAMPLES OF TRANSFORMATION OR MODELING OF RANDOM VARIABLE

η

i

η

1

η

1 – exp3– – x24dx ∫ Ï· 2 2π

i

0.021 0.022 0.022 0.022 0.023 0.023 0.023 0.023 0.024 0.024 0.024 0.025 0.025 0.025 0.025 0.026 0.026 0.026 0.027 0.027 0.027 0.028 0.028 0.028 0.029

226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250

η

–3

201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225

–2.000 –1.995 –1.990 –1.985 –1.980 –1.975 –1.970 –1.965 –1.960 –1.955 –1.950 –1.945 –1.940 –1.935 –1.930 –1.925 –1.920 –1.915 –1.910 –1.905 –1.900 –1.895 –1.890 –1.885 –1.880

1

1 – exp3– – x24dx ∫ Ï· 2 2π

–3

–1.875 –1.870 –1.865 –1.860 –1.855 –1.850 –1.845 –1.840 –1.835 –1.830 –1.825 –1.820 –1.815 –1.810 –1.805 –1.800 –1.795 –1.790 –1.785 –1.780 –1.775 –1.770 –1.765 –1.760 –1.755

0.029 0.029 0.030 0.030 0.030 0.031 0.031 0.032 0.032 0.032 0.033 0.033 0.033 0.034 0.034 0.035 0.035 0.035 0.036 0.036 0.037 0.037 0.037 0.038 0.038

11

12

1. INTRODUCTION

η

i

η

1

η

1 – exp3– – x24dx ∫ Ï· 2 2π

i

0.039 0.039 0.040 0.040 0.040 0.041 0.041 0.042 0.042 0.043 0.043 0.044 0.044 0.045 0.045 0.046 0.046 0.047 0.047 0.048 0.048 0.049 0.049 0.050 0.050

276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300

η

–3

251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275

–1.750 –1.745 –1.740 –1.735 –1.730 –1.725 –1.720 –1.715 –1.710 –1.705 –1.700 –1.695 –1.690 –1.685 –1.680 –1.675 –1.670 –1.665 –1.660 –1.655 –1.650 –1.645 –1.640 –1.635 –1.630

1

1 – exp3– – x24dx ∫ Ï· 2 2π

–3

–1.625 –1.620 –1.615 –1.610 –1.605 –1.600 –1.595 –1.590 –1.585 –1.580 –1.575 –1.570 –1.565 –1.560 –1.555 –1.550 –1.545 –1.540 –1.535 –1.530 –1.525 –1.520 –1.515 –1.510 –1.505

0.051 0.051 0.052 0.052 0.053 0.053 0.054 0.055 0.055 0.056 0.056 0.057 0.057 0.058 0.059 0.059 0.060 0.060 0.061 0.062 0.062 0.063 0.064 0.064 0.065

1.6. EXAMPLES OF TRANSFORMATION OR MODELING OF RANDOM VARIABLE

η

i

η

1

η

1 – exp3– – x24dx ∫ Ï· 2 2π

i

0.065 0.066 0.067 0.067 0.068 0.069 0.069 0.070 0.071 0.071 0.072 0.073 0.074 0.074 0.075 0.076 0.076 0.077 0.078 0.079 0.079 0.080 0.081 0.082 0.082

326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350

η

–3

301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325

–1.500 –1.495 –1.490 –1.485 –1.480 –1.475 –1.470 –1.465 –1.460 –1.455 –1.450 –1.445 –1.440 –1.435 –1.430 –1.425 –1.420 –1.415 –1.410 –1.405 –1.400 –1.395 –1.390 –1.385 –1.380

1

1 – exp3– – x24dx ∫ Ï· 2 2π

–3

–1.375 –1.370 –1.365 –1.360 –1.355 –1.350 –1.345 –1.340 –1.335 –1.330 –1.325 –1.320 –1.315 –1.310 –1.305 –1.300 –1.295 –1.290 –1.285 –1.280 –1.275 –1.270 –1.265 –1.260 –1.255

0.083 0.084 0.085 0.086 0.086 0.087 0.088 0.089 0.090 0.090 0.091 0.092 0.093 0.094 0.095 0.095 0.096 0.097 0.098 0.099 0.100 0.101 0.102 0.102 0.103

13

14

1. INTRODUCTION

η

i

η

1

η

1 – exp3– – x24dx ∫ Ï· 2 2π

i

0.104 0.105 0.106 0.107 0.108 0.109 0.110 0.111 0.112 0.113 0.114 0.115 0.116 0.117 0.118 0.119 0.120 0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.128

376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400

η

–3

351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375

–1.250 –1.245 –1.240 –1.235 –1.230 –1.225 –1.220 –1.215 –1.210 –1.205 –1.200 –1.195 –1.190 –1.185 –1.180 –1.175 –1.170 –1.165 –1.160 –1.155 –1.150 –1.145 –1.140 –1.135 –1.130

1

1 – exp3– – x24dx ∫ Ï· 2 2π

–3

–1.125 –1.120 –1.115 –1.110 –1.105 –1.100 –1.095 –1.090 –1.085 –1.080 –1.075 –1.070 –1.065 –1.060 –1.055 –1.050 –1.045 –1.040 –1.035 –1.030 –1.025 –1.020 –1.015 –1.010 –1.005

0.129 0.130 0.131 0.132 0.133 0.134 0.135 0.137 0.138 0.139 0.140 0.141 0.142 0.143 0.144 0.146 0.147 0.148 0.149 0.150 0.151 0.153 0.154 0.155 0.156

1.6. EXAMPLES OF TRANSFORMATION OR MODELING OF RANDOM VARIABLE

η

i

η

1

η

1 – exp3– – x24dx ∫ Ï· 2 2π

i

0.157 0.159 0.160 0.161 0.162 0.163 0.165 0.166 0.167 0.168 0.170 0.171 0.172 0.174 0.175 0.176 0.177 0.179 0.180 0.181 0.183 0.184 0.185 0.187 0.188

426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450

η

–3

401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425

–1.000 –0.995 –0.990 –0.985 –0.980 –0.975 –0.970 –0.965 –0.960 –0.955 –0.950 –0.945 –0.940 –0.935 –0.930 –0.925 –0.920 –0.915 –0.910 –0.905 –0.900 –0.895 –0.890 –0.885 –0.880

1

1 – exp3– – x24dx ∫ Ï· 2 2π

–3

–0.875 –0.870 –0.865 –0.860 –0.855 –0.850 –0.845 –0.840 –0.835 –0.830 –0.825 –0.820 –0.815 –0.810 –0.805 –0.800 –0.795 –0.790 –0.785 –0.780 –0.775 –0.770 –0.765 –0.760 –0.755

0.189 0.191 0.192 0.194 0.195 0.196 0.198 0.199 0.201 0.202 0.203 0.205 0.206 0.208 0.209 0.211 0.212 0.213 0.215 0.216 0.218 0.219 0.221 0.222 0.224

15

16

1. INTRODUCTION

η

i

η

1

η

1 – exp3– – x24dx ∫ Ï· 2 2π

i

0.225 0.227 0.228 0.230 0.231 0.233 0.234 0.236 0.238 0.239 0.241 0.242 0.244 0.245 0.247 0.248 0.250 0.252 0.253 0.255 0.256 0.258 0.260 0.261 0.263

476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500

η

–3

451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475

–0.750 –0.745 –0.740 –0.735 –0.730 –0.725 –0.720 –0.715 –0.710 –0.705 –0.700 –0.695 –0.690 –0.685 –0.680 –0.675 –0.670 –0.665 –0.660 –0.655 –0.650 –0.645 –0.640 –0.635 –0.630

1

1 – exp3– – x24dx ∫ Ï· 2 2π

–3

–0.625 –0.620 –0.615 –0.610 –0.605 –0.600 –0.595 –0.590 –0.585 –0.580 –0.575 –0.570 –0.565 –0.560 –0.555 –0.550 –0.545 –0.540 –0.535 –0.530 –0.525 –0.520 –0.515 –0.510 –0.505

0.265 0.266 0.268 0.270 0.271 0.273 0.275 0.276 0.278 0.280 0.281 0.283 0.285 0.286 0.288 0.290 0.292 0.293 0.295 0.297 0.298 0.300 0.302 0.304 0.305

1.6. EXAMPLES OF TRANSFORMATION OR MODELING OF RANDOM VARIABLE

η

i

η

1

η

1 – exp3– – x24dx ∫ Ï· 2 2π

i

0.307 0.309 0.311 0.312 0.314 0.316 0.318 0.320 0.321 0.323 0.325 0.327 0.329 0.330 0.332 0.334 0.336 0.338 0.340 0.341 0.343 0.345 0.347 0.349 0.351

526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550

η

–3

501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525

–0.500 –0.495 –0.490 –0.485 –0.480 –0.475 –0.470 –0.465 –0.460 –0.455 –0.450 –0.445 –0.440 –0.435 –0.430 –0.425 –0.420 –0.415 –0.410 –0.405 –0.400 –0.395 –0.390 –0.385 –0.380

1

1 – exp3– – x24dx ∫ Ï· 2 2π

–3

–0.375 –0.370 –0.365 –0.360 –0.355 –0.350 –0.345 –0.340 –0.335 –0.330 –0.325 –0.320 –0.315 –0.310 –0.305 –0.300 –0.295 –0.290 –0.285 –0.280 –0.275 –0.270 –0.265 –0.260 –0.255

0.352 0.354 0.356 0.358 0.360 0.362 0.364 0.366 0.367 0.369 0.371 0.373 0.375 0.377 0.379 0.381 0.383 0.385 0.386 0.388 0.390 0.392 0.394 0.396 0.398

17

18

1. INTRODUCTION

η

i

η

1

η

1 – exp3– – x24dx ∫ Ï· 2 2π

i

0.400 0.402 0.404 0.406 0.408 0.410 0.412 0.414 0.415 0.417 0.419 0.421 0.423 0.425 0.427 0.429 0.431 0.433 0.435 0.437 0.439 0.441 0.443 0.445 0.447

576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600

η

–3

551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575

–0.250 –0.245 –0.240 –0.235 –0.230 –0.225 –0.220 –0.215 –0.210 –0.205 –0.200 –0.195 –0.190 –0.185 –0.180 –0.175 –0.170 –0.165 –0.160 –0.155 –0.150 –0.145 –0.140 –0.135 –0.130

1

1 – exp3– – x24dx ∫ Ï· 2 2π

–3

–0.125 –0.120 –0.115 –0.110 –0.105 –0.100 –0.095 –0.090 –0.085 –0.080 –0.075 –0.070 –0.065 –0.060 –0.055 –0.050 –0.045 –0.040 –0.035 –0.030 –0.025 –0.020 –0.015 –0.010 –0.005

0.449 0.451 0.453 0.455 0.457 0.459 0.461 0.463 0.465 0.467 0.469 0.471 0.473 0.475 0.477 0.479 0.481 0.483 0.485 0.487 0.489 0.491 0.493 0.495 0.497

1.6. EXAMPLES OF TRANSFORMATION OR MODELING OF RANDOM VARIABLE

η

i

η

1

η

1 – exp3– – x24dx ∫ Ï· 2 2π

i

0.499 0.501 0.503 0.505 0.507 0.509 0.511 0.513 0.515 0.517 0.519 0.521 0.523 0.525 0.527 0.529 0.531 0.533 0.535 0.536 0.538 0.540 0.542 0.544 0.546

626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650

η

–3

601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 0.055 0.060 0.065 0.070 0.075 0.080 0.085 0.090 0.095 0.100 0.105 0.110 0.115 0.120

1

1 – exp3– – x24dx ∫ Ï· 2 2π

–3

0.125 0.130 0.135 0.140 0.145 0.150 0.155 0.160 0.165 0.170 0.175 0.180 0.185 0.190 0.195 0.200 0.205 0.210 0.215 0.220 0.225 0.230 0.235 0.240 0.245

0.548 0.550 0.552 0.554 0.556 0.558 0.560 0.562 0.564 0.566 0.568 0.570 0.572 0.574 0.576 0.578 0.580 0.582 0.584 0.586 0.588 0.590 0.592 0.593 0.595

19

20

1. INTRODUCTION

η

i

η

1

η

1 – exp3– – x24dx ∫ Ï· 2 2π

i

0.597 0.599 0.601 0.603 0.605 0.607 0.609 0.611 0.613 0.615 0.617 0.618 0.620 0.622 0.624 0.626 0.628 0.630 0.632 0.634 0.635 0.637 0.639 0.641 0.643

676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700

η

–3

651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675

0.250 0.255 0.260 0.265 0.270 0.275 0.280 0.285 0.290 0.295 0.300 0.305 0.310 0.315 0.320 0.325 0.330 0.335 0.340 0.345 0.350 0.355 0.360 0.365 0.370

1

1 – exp3– – x24dx ∫ Ï· 2 2π

–3

0.375 0.380 0.385 0.390 0.395 0.400 0.405 0.410 0.415 0.420 0.425 0.430 0.435 0.440 0.445 0.450 0.455 0.460 0.465 0.470 0.475 0.480 0.485 0.490 0.495

0.645 0.647 0.649 0.650 0.652 0.654 0.656 0.658 0.660 0.661 0.663 0.665 0.667 0.669 0.670 0.672 0.674 0.676 0.678 0.679 0.681 0.683 0.685 0.687 0.688

1.6. EXAMPLES OF TRANSFORMATION OR MODELING OF RANDOM VARIABLE

η

i

η

1

η

1 – exp3– – x24dx ∫ Ï· 2 2π

i

0.690 0.692 0.694 0.695 0.697 0.699 0.701 0.702 0.704 0.706 0.707 0.709 0.711 0.713 0.714 0.716 0.718 0.719 0.721 0.723 0.724 0.726 0.728 0.729 0.731

726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750

η

–3

701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725

0.500 0.505 0.510 0.515 0.520 0.525 0.530 0.535 0.540 0.545 0.550 0.555 0.560 0.565 0.570 0.575 0.580 0.585 0.590 0.595 0.600 0.605 0.610 0.615 0.620

1

1 – exp3– – x24dx ∫ Ï· 2 2π

–3

0.625 0.630 0.635 0.640 0.645 0.650 0.655 0.660 0.665 0.670 0.675 0.680 0.685 0.690 0.695 0.700 0.705 0.710 0.715 0.720 0.725 0.730 0.735 0.740 0.745

0.733 0.734 0.736 0.738 0.739 0.741 0.742 0.744 0.746 0.747 0.749 0.750 0.752 0.754 0.755 0.757 0.758 0.760 0.761 0.763 0.764 0.766 0.767 0.769 0.771

21

22

1. INTRODUCTION

η

i

η

1

η

1 – exp3– – x24dx ∫ Ï· 2 2π

i

0.772 0.774 0.775 0.777 0.778 0.779 0.781 0.782 0.784 0.785 0.787 0.788 0.790 0.791 0.793 0.794 0.795 0.797 0.798 0.800 0.801 0.802 0.804 0.805 0.806

776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800

η

–3

751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775

0.750 0.755 0.760 0.765 0.770 0.775 0.780 0.785 0.790 0.795 0.800 0.805 0.810 0.815 0.820 0.825 0.830 0.835 0.840 0.845 0.850 0.855 0.860 0.865 0.870

1

1 – exp3– – x24dx ∫ Ï· 2 2π

–3

0.875 0.880 0.885 0.890 0.895 0.900 0.905 0.910 0.915 0.920 0.925 0.930 0.935 0.940 0.945 0.950 0.955 0.960 0.965 0.970 0.975 0.980 0.985 0.990 0.995

0.808 0.809 0.811 0.812 0.813 0.815 0.816 0.817 0.819 0.820 0.821 0.822 0.824 0.825 0.826 0.828 0.829 0.830 0.831 0.833 0.834 0.835 0.836 0.838 0.839

1.6. EXAMPLES OF TRANSFORMATION OR MODELING OF RANDOM VARIABLE

η

i

η

1

η

1 – exp3– – x24dx ∫ Ï· 2 2π

i

0.840 0.841 0.842 0.844 0.845 0.846 0.847 0.848 0.849 0.851 0.852 0.853 0.854 0.855 0.856 0.857 0.859 0.860 0.861 0.862 0.863 0.864 0.865 0.866 0.867

826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850

η

–3

801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825

1.000 1.005 1.010 1.015 1.020 1.025 1.030 1.035 1.040 1.045 1.050 1.055 1.060 1.065 1.070 1.075 1.080 1.085 1.090 1.095 1.100 1.105 1.110 1.115 1.120

1

1 – exp3– – x24dx ∫ Ï· 2 2π

–3

1.125 1.130 1.135 1.140 1.145 1.150 1.155 1.160 1.165 1.170 1.175 1.180 1.185 1.190 1.195 1.200 1.205 1.210 1.215 1.220 1.225 1.230 1.235 1.240 1.245

0.868 0.869 0.870 0.872 0.873 0.874 0.875 0.876 0.877 0.878 0.879 0.880 0.881 0.882 0.883 0.884 0.885 0.886 0.886 0.887 0.888 0.889 0.890 0.891 0.892

23

24

1. INTRODUCTION

η

i

η

1

η

1 – exp3– – x24dx ∫ Ï· 2 2π

i

0.893 0.894 0.895 0.896 0.897 0.897 0.898 0.899 0.900 0.901 0.902 0.903 0.904 0.904 0.905 0.906 0.907 0.908 0.909 0.909 0.910 0.911 0.912 0.913 0.913

876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900

η

–3

851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875

1.250 1.255 1.260 1.265 1.270 1.275 1.280 1.285 1.290 1.295 1.300 1.305 1.310 1.315 1.320 1.325 1.330 1.335 1.340 1.345 1.350 1.355 1.360 1.365 1.370

1

1 – exp3– – x24dx ∫ Ï· 2 2π

–3

1.375 1.380 1.385 1.390 1.395 1.400 1.405 1.410 1.415 1.420 1.425 1.430 1.435 1.440 1.445 1.450 1.455 1.460 1.465 1.470 1.475 1.480 1.485 1.490 1.495

0.914 0.915 0.916 0.916 0.917 0.918 0.919 0.919 0.920 0.921 0.922 0.922 0.923 0.924 0.924 0.925 0.926 0.927 0.927 0.928 0.929 0.929 0.930 0.931 0.931

1.6. EXAMPLES OF TRANSFORMATION OR MODELING OF RANDOM VARIABLE

η

i

η

1

η

1 – exp3– – x24dx ∫ Ï· 2 2π

i

0.932 0.932 0.933 0.934 0.934 0.935 0.936 0.936 0.937 0.937 0.938 0.939 0.939 0.940 0.940 0.941 0.942 0.942 0.943 0.943 0.944 0.944 0.945 0.945 0.946

926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950

η

–3

901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925

1.500 1.505 1.510 1.515 1.520 1.525 1.530 1.535 1.540 1.545 1.550 1.555 1.560 1.565 1.570 1.575 1.580 1.585 1.590 1.595 1.600 1.605 1.610 1.615 1.620

1

1 – exp3– – x24dx ∫ Ï· 2 2π

–3

1.625 1.630 1.635 1.640 1.645 1.650 1.655 1.660 1.665 1.670 1.675 1.680 1.685 1.690 1.695 1.700 1.705 1.710 1.715 1.720 1.725 1.730 1.735 1.740 1.745

0.947 0.947 0.948 0.948 0.949 0.949 0.950 0.950 0.951 0.951 0.952 0.952 0.953 0.953 0.954 0.954 0.955 0.955 0.955 0.956 0.956 0.957 0.957 0.958 0.958

25

26

1. INTRODUCTION

η

i

η

1

η

1 – exp3– – x24dx ∫ Ï· 2 2π

i

0.959 0.959 0.959 0.960 0.960 0.961 0.961 0.962 0.962 0.962 0.963 0.963 0.964 0.964 0.964 0.965 0.965 0.965 0.966 0.966 0.966 0.967 0.967 0.968 0.968

976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000

η

–3

951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975

1.750 1.755 1.760 1.765 1.770 1.775 1.780 1.785 1.790 1.795 1.800 1.805 1.810 1.815 1.820 1.825 1.830 1.835 1.840 1.845 1.850 1.855 1.860 1.865 1.870

1

1 – exp3– – x24dx ∫ Ï· 2 2π

–3

1.875 1.880 1.885 1.890 1.895 1.900 1.905 1.910 1.915 1.920 1.925 1.930 1.935 1.940 1.945 1.950 1.955 1.960 1.965 1.970 1.975 1.980 1.985 1.990 1.995

0.968 0.969 0.969 0.969 0.970 0.970 0.970 0.971 0.971 0.971 0.972 0.972 0.972 0.972 0.973 0.973 0.973 0.974 0.974 0.974 0.975 0.975 0.975 0.975 0.976

1.6. EXAMPLES OF TRANSFORMATION OR MODELING OF RANDOM VARIABLE

η

i

η

1

η

1 – exp3– – x24dx ∫ Ï· 2 2π

i

0.976 0.976 0.976 0.977 0.977 0.977 0.977 0.978 0.978 0.978 0.978 0.979 0.979 0.979 0.979 0.980 0.980 0.980 0.980 0.981 0.981 0.981 0.981 0.981 0.982

1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050

η

–3

1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025

2.000 2.005 2.010 2.015 2.020 2.025 2.030 2.035 2.040 2.045 2.050 2.055 2.060 2.065 2.070 2.075 2.080 2.085 2.090 2.095 2.100 2.105 2.110 2.115 2.120

1

1 – exp3– – x24dx ∫ Ï· 2 2π

–3

2.125 2.130 2.135 2.140 2.145 2.150 2.155 2.160 2.165 2.170 2.175 2.180 2.185 2.190 2.195 2.200 2.205 2.210 2.215 2.220 2.225 2.230 2.235 2.240 2.245

0.982 0.982 0.982 0.982 0.983 0.983 0.983 0.983 0.983 0.984 0.984 0.984 0.984 0.984 0.985 0.985 0.985 0.985 0.985 0.985 0.986 0.986 0.986 0.986 0.986

27

28

1. INTRODUCTION

η

i

η

1

η

1 – exp3– – x24dx ∫ Ï· 2 2π

i

0.986 0.987 0.987 0.987 0.987 0.987 0.987 0.987 0.988 0.988 0.988 0.988 0.988 0.988 0.988 0.989 0.989 0.989 0.989 0.989 0.989 0.989 0.990 0.990 0.990

1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100

η

–3

1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075

2.250 2.255 2.260 2.265 2.270 2.275 2.280 2.285 2.290 2.295 2.300 2.305 2.310 2.315 2.320 2.325 2.330 2.335 2.340 2.345 2.350 2.355 2.360 2.365 2.370

1

1 – exp3– – x24dx ∫ Ï· 2 2π

–3

2.375 2.380 2.385 2.390 2.395 2.400 2.405 2.410 2.415 2.420 2.425 2.430 2.435 2.440 2.445 2.450 2.455 2.460 2.465 2.470 2.475 2.480 2.485 2.490 2.495

0.990 0.990 0.990 0.990 0.990 0.990 0.991 0.991 0.991 0.991 0.991 0.991 0.991 0.991 0.991 0.992 0.992 0.992 0.992 0.992 0.992 0.992 0.992 0.992 0.992

1.6. EXAMPLES OF TRANSFORMATION OR MODELING OF RANDOM VARIABLE

η

i

η

1

η

1 – exp3– – x24dx ∫ Ï· 2 2π

i

0.992 0.993 0.993 0.993 0.993 0.993 0.993 0.993 0.993 0.993 0.993 0.993 0.993 0.993 0.994 0.994 0.994 0.994 0.994 0.994 0.994 0.994 0.994 0.994 0.994

1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150

η

–3

1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125

2.500 2.505 2.510 2.515 2.520 2.525 2.530 2.535 2.540 2.545 2.550 2.555 2.560 2.565 2.570 2.575 2.580 2.585 2.590 2.595 2.600 2.605 2.610 2.615 2.620

1

1 – exp3– – x24dx ∫ Ï· 2 2π

–3

2.625 2.630 2.635 2.640 2.645 2.650 2.655 2.660 2.665 2.670 2.675 2.680 2.685 2.690 2.695 2.700 2.705 2.710 2.715 2.720 2.725 2.730 2.735 2.740 2.745

0.994 0.994 0.994 0.995 0.995 0.995 0.995 0.995 0.995 0.995 0.995 0.995 0.995 0.995 0.995 0.995 0.995 0.995 0.995 0.995 0.995 0.995 0.996 0.996 0.996

29

30

1. INTRODUCTION

η

i

η

1

1 – exp3– – x24dx ∫ Ï· 2 2π

η

i

η

–3

1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 1172 1173 1174 1175

2.750 2.755 2.760 2.765 2.770 2.775 2.780 2.785 2.790 2.795 2.800 2.805 2.810 2.815 2.820 2.825 2.830 2.835 2.840 2.845 2.850 2.855 2.860 2.865 2.870

0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.996 0.997 0.997 0.997

1

1 – exp3– – x24dx ∫ Ï· 2 2π

–3

1176 1177 1178 1179 1180 1181 1182 1183 1184 1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200 1201

2.875 2.880 2.885 2.890 2.895 2.900 2.905 2.910 2.915 2.920 2.925 2.930 2.935 2.940 2.945 2.950 2.955 2.960 2.965 2.970 2.975 2.980 2.985 2.990 2.995 3.000

0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.997 0.997

1.6. EXAMPLES OF TRANSFORMATION OR MODELING OF RANDOM VARIABLE

31

Table 1.4: Showing 100 random numbers g obeying Gaussian or Normal probability density h
2 i 1 x a 1 function pg .x/ D  p2 Exp 2  with a D 0 and  D 1. ui ’s are corresponding values of uniform random numbers in the interval 0 to 1 (of Table 1.1). i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

ui 0.169 0.050 0.844 0.182 0.599 0.985 0.726 0.148 0.788 0.698 0.010 0.397 0.852 0.761 0.309 0.113 0.212 0.809 0.084 0.276 0.243 0.989 0.508 0.817 0.174

gi –0.955 –1.635 1.015 –0.905 0.255 2.195 0.605 –1.040 0.805 0.520 –2.295 –0.255 1.050 0.715 –0.495 –1.205 –0.795 0.880 –1.370 –0.590 –0.695 2.325 0.020 0.910 –0.935

i 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

ui 0.225 0.687 0.060 0.070 0.835 0.466 0.271 0.813 0.376 0.983 0.181 0.583 0.612 0.468 0.614 0.768 0.811 0.461 0.985 0.735 0.559 0.884 0.229 0.920 0.524

gi –0.750 0.490 –1.540 –1.465 0.980 –0.085 –0.605 0.895 –0.315 2.145 –0.905 0.210 0.285 –0.080 0.290 0.735 0.885 –0.095 2.195 0.630 0.150 1.200 –0.740 1.415 0.060

i 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75

ui 0.393 0.306 0.819 0.620 0.875 0.773 0.492 0.332 0.185 0.565 0.434 0.665 0.516 0.659 0.709 0.515 0.683 0.092 0.417 0.598 0.682 0.948 0.125 0.149 0.246

gi i –0.270 76 –0.505 77 0.915 78 0.310 79 1.155 80 0.750 81 –0.020 82 –0.430 83 –0.890 84 0.165 85 –0.165 86 0.430 87 0.040 88 0.410 89 0.555 90 0.040 91 0.480 92 –1.320 93 –0.205 94 0.250 95 0.475 96 1.635 97 –1.145 98 –1.035 99 –0.685 100

ui 0.231 0.836 0.167 0.545 0.748 0.440 0.674 0.287 0.052 0.940 0.679 0.910 0.254 0.236 0.854 0.877 0.483 0.117 0.847 0.201 0.294 0.071 0.065 0.740 0.954

gi –0.730 0.985 –0.960 0.115 0.670 –0.150 0.455 –0.560 –1.615 1.565 0.470 1.350 –0.660 –0.715 1.060 1.165 –0.040 –1.185 1.030 –0.835 –0.540 –1.460 –1.505 0.645 1.695

32

1. INTRODUCTION

Program 1.3

h=0.005; eta=-3.005; i=0; Table[{i=i+1,eta=eta+h,int=(1/Sqrt[2*3.1416])*Integrate [Exp[-0.5*x^2], {x,-3,eta}]},{eta,-3.005,2.995,h}]; TableForm[%,TableSpacing->{2,2}, TableHeadings->{None,{"i","eta","integral"}}]

We now discuss modeling of random variable G obeying Gaussian or Normal probability density function given by   1  x a 2 1 PG .x/ D p Exp (1.7) 2   2

Example 1.4

with non-zero value of a and  ¤ 1. It is well known that values of G are related to those of g via G D a C g; (1.8) where values of g are random numbers obeying Equation (1.7) with a D 0 and  D 1.

1.7

VARIANCE REDUCTION AND IMPORTANCE SAMPLING

Let us consider a definite integral I D

Z

b

F .x/dx:

(1.9)

Z

(1.10)

a

Let us re-write Equation (1.9) as I D

Z a

b

F .x/ p.x/dx D p.x/

b

f .x/p.x/dx

a

where f .x/ D F .x/=p.x/. As discussed in Section 1.2, average value of any function of x , say f .x/, is given by Rb Af D a f .x/p.x/dx where p.x/ is a normalized probability density function. Thus, according to Equation (1.10), value I of the integral is average value of F=p .

1.7. VARIANCE REDUCTION AND IMPORTANCE SAMPLING

33

Table 1.5: Showing 100 random G obeying Gaussian or Normal probability density h numbers
2 i x a 1 1 function PG .x/ D  p2 Exp 2  with a D =2 and  D 0:5. ui ’s are corresponding values of uniform random numbers in the interval 0 to 1i and g ’s are corresponding values of h
2 1 1 x random numbers obeying pg .x/ D  p2 Exp 2  a with a D 0 and  D 1. Values of G have been obtained using G D a C g D =2 C 0:5g . i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

ui gi 0.169 –0.955 0.050 –1.635 0.844 1.015 0.182 –0.905 0.599 0.255 0.985 2.195 0.726 0.605 0.148 –1.040 0.788 0.805 0.698 0.520 0.010 –2.295 0.397 –0.255 0.852 1.050 0.761 0.715 0.309 –0.495 0.113 –1.205 0.212 –0.795 0.809 0.880 0.084 –1.370 0.276 –0.590 0.243 –0.695 0.989 2.325 0.508 0.020 0.817 0.910 0.174 –0.935

Gi 1.093 0.753 2.078 1.118 1.698 2.668 1.873 1.051 1.973 1.831 0.423 1.443 2.096 1.928 1.323 0.968 1.173 2.011 0.886 1.276 1.223 2.733 1.581 2.026 1.103

i 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

ui gi Gi 0.225 –0.750 1.196 0.687 0.490 1.816 0.060 –1.540 0.801 0.070 –1.465 0.838 0.835 0.980 2.061 0.466 –0.085 1.528 0.271 –0.605 1.268 0.813 0.895 2.018 0.376 –0.315 1.413 0.983 2.145 2.643 0.181 –0.905 1.118 0.583 0.210 1.676 0.612 0.285 1.713 0.468 –0.080 1.531 0.614 0.290 1.716 0.768 0.735 1.938 0.811 0.885 2.013 0.461 –0.095 1.523 0.985 2.195 2.668 0.735 0.630 1.886 0.559 0.150 1.646 0.884 1.200 2.171 0.229 –0.740 1.201 0.920 1.415 2.278 0.524 0.060 1.601

34

1. INTRODUCTION

i 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75

ui gi 0.393 –0.270 0.306 –0.505 0.819 0.915 0.620 0.310 0.875 1.155 0.773 0.750 0.492 –0.020 0.332 –0.430 0.185 –0.890 0.565 0.165 0.434 –0.165 0.665 0.430 0.516 0.040 0.659 0.410 0.709 0.555 0.515 0.040 0.683 0.480 0.092 –1.320 0.417 –0.205 0.598 0.250 0.682 0.475 0.948 1.635 0.125 –1.145 0.149 –1.035 0.246 –0.685

Gi 1.436 1.318 2.028 1.726 2.148 1.946 1.561 1.356 1.126 1.653 1.488 1.786 1.591 1.776 1.848 1.591 1.811 0.911 1.468 1.696 1.808 2.388 0.998 1.053 1.228

i 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

ui gi Gi 0.231 –0.730 1.206 0.836 0.985 2.063 0.167 –0.960 1.091 0.545 0.115 1.628 0.748 0.670 1.906 0.440 –0.150 1.496 0.674 0.455 1.798 0.287 –0.560 1.291 0.052 –1.615 0.763 0.940 1.565 2.353 0.679 0.470 1.806 0.910 1.350 2.246 0.254 –0.660 1.241 0.236 –0.715 1.213 0.854 1.060 2.101 0.877 1.165 2.153 0.483 –0.040 1.551 0.117 –1.185 0.978 0.847 1.030 2.086 0.201 –0.835 1.153 0.294 –0.540 1.301 0.071 –1.460 0.841 0.065 –1.505 0.818 0.740 0.645 1.893 0.954 1.695 2.418

1.7. VARIANCE REDUCTION AND IMPORTANCE SAMPLING

35

We can choose any functional form for p.x/, but as proved below, choosing p.x/ as proportional to F .x/ ensures that the variance of f .x/ D F=p is small. This will ensure a better Rb value of the average and hence a better value of the integral I D a F .x/dx . Variance of f .x/ is square of standard deviation. Variance of f .x/ is given by h
2 i V Œf .x/ D A f Af ; (1.11) where A stands for average value. Thus, h i
Vf .x/ D A f 2 2fAf C Af2 D A f 2 D A.f 2 /

2Af Af C Af2

Af2

or, Vf .x/ D D

Z

Z

b 2

f .x/p.x/dx

a

Z a

b

f .x/p.x/dx

!2

a b

Z

F2 pdx p2

b

F p.x/dx p

a

Thus, the variance is Vf .x/ D

Z

b

a

!2

D

F2 dx p

Z

b

F2 dx p

a

Z

b

F dx

!2

:

a

I 2:

(1.12)

If p.x/ is taken as p.x/ D R b a

jF .x/j

(1.13)

jF .x/j dx

the variance given by Equation (1.12) becomes Vf .x/ D

Z a

b

Rb a

F2 jF .x/j

I2

dx

jF .x/j dx

or, Vf .x/ D

Z

b a

jF .x/j dx

! Z a

b

jF .x/j dx

!

I2 D

Z

b a

jF .x/j dx

!2

I 2:

(1.14)

Rb Equation (1.14) in conjunction with Equation (1.9): I D a F .x/dx reveals that the variance vanishes if the integrand F .x/ does not change sign. If it does, the variance will be small if

36

1. INTRODUCTION

condition laid by Equation (1.13) is met. Thus, the probability density function p.x/ should be proportional to the integrand jF .x/j. This is the so-called importance sampling. Looking back at Equation (1.10), we find that value of the integral is approximately determined by the value of the average N 1 X F .xi / I D : N p.xi / i D1

As recommended by importance sampling, we need to take p.x/ D C F .x/ where C is a conRb Rb stant; we need to normalize p.x/ first; a p.x/dx D 1 gives a C F .x/dx D 1 or, C D Rb a

1

:

F .x/dx

Thus, p.x/ D R b a

The sum

F .x/

:

F .x/dx

N 1 X F .xi / N p.xi / i D1

becomes

N 1 X N i D1

Rb a

F .xi / F .xi /

D

Z

b

F .x/dx:

a

F .x/dx

Thus, the sum N 1 X F .xi / N p.xi / i D1

equals the integral if probability density function p.x/ is taken proportional to the integrand jF .x/j. Slight variation of p.x/ from proportionality with jF .x/j will result in slight difference of N 1 X F .xi / N p.xi / i D1 Rb from the actual value of the integral I D a F .x/dx .

37

CHAPTER

2

Evaluation of Definite Integrals Using the Monte Carlo Method 2.1

EVALUATION OF DEFINITE INTEGRALS USING THE MONTE CARLO METHOD: EXAMPLE I

As Example I, we now take up the integral Z b Z I D F .x/dx D a



Sin.x/dx;

(2.1)

0

where F .x/ D Sin.x/. We re-write Equation (2.1) as Z b F .x/ I D p.x/dx a p.x/

(2.2)

which, as discussed in Section 1.7, implies that average value of F=p is the value of the integral, i.e., N 1 X F .xi / I D ; (2.3) N p.xi / iD1 where xi ’s are random values of x in the interval a < x < b obeying probability density function p.x/. Here p.x/ is a suitable probability density function which, as demonstrated in Section 1.7, at least should follow F .x/, if not be proportional to F .x/. See Figure 2.1 in which we have plotted F .x/ D Sin.x/ along with normalized probability density function of a uniform random variable in the R interval 0 to  given by p.x/ D 1= . Let p.x/ D C , a constant. Normalization requires that 0 Cdx D 1 which gives C D 1= . Uniform p.x/ D 1= is not too different from a slowly varying function of x like Sin.x/. Hence, p.x/ D 1= is a good choice. We now generate random variable  obeying p.x/ D 1= in the interval 0 < x <  using Equation (1.4):  D a C .b a/u where u is random variable (of Table 1.1) of uniform probability density function in the interval 0 < x R< 1. Thus, we have  D u. That  D u can be R  calculated using a p.x/dx D u which gives 0 .1=/dx D u or,  D u.

38

2. EVALUATION OF DEFINITE INTEGRALS USING THE MONTE CARLO METHOD

F(x) = Sin(x) or p(x) = 1/π

1.0 0.8 0.6 0.4 0.2 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

x

Figure 2.1: Showing F .x/ D Sin.x/ as undashed curve and p.x/ D 1= as dashed curve obtained using the command Plot[{Sin[x],1/Pi},{x,0,Pi},Frame->True,FrameLabel-> {"x","F(x) = Sin(x) or p(x) = 1/\[Pi]"},PlotStyle-> {{Black},{Dashed,Black}}] in Mathematica 6.0.

Using Table 2.1 in Microsoft Excel, we have evaluated the integral with the result 1.973 rather than 2. The difference is attributable to the fact that F .x/ and p.x/ used are not proportional to each other. We now take up a linear variation for p.x/ given by p.x/ D C x where C is a conR stant. Normalization requires that 0 C xdx D 1 which gives C D 2= 2 . Thus, normalized p.x/ D 2x= 2 . See Figure 2.2 in which we have plotted F .x/ D Sin.x/ along with the normalized linear probability density function p.x/ D 2x= 2 . The linear probability density function p.x/ D 2x= 2 is not too different from a slowly varying function of x like Sin.x/. Hence, it is a good choice. 2 We now generate R  random variable R  obeying p.x/ D 2x= in the interval 0 < x <  using Equation (1.2): a p.x/dx D u or, 0 p.x/dx D u where u is random variable (of Table 1.1) of uniform probability density function in the interval 0 < x < 1. Thus, we have  D u1=2 . Using Table 2.2 in Microsoft Excel, we have evaluated the integral with the result 1.947 rather than 2. The difference is attributable to the fact that F .x/ and p.x/ used are not proportional to each other. We h now take i up a normalized Gaussian variation for p.x/ given by p.x/ D
2 1 p Exp 21 x  a where a is average and  is variance of x . See Figure 2.3 in which we  2 have plotted F .x/ D Sin.x/ along with the normalized Gaussian probability density function

2.1. EVALUATION OF DEFINITE INTEGRALS: EXAMPLE I

39

Table 2.1: Showing tabulated values of ui , i D ui , Sin.i / from which we get I D F .xi / F .i / 1 PN 1 PN  PN i D1 p.xi / D N i D1 1= D N i D1 Sin.i / as an approximate value of the integral N Rb R I D a F .x/dx D 0 Sin.x/dx D 1:973 whereas the exact value is 2. Using p.x/ D 1= . i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

ui ηi = π ui Sin(ηi) 0.169 0.531 0.506 0.050 0.157 0.156 0.844 2.652 0.471 0.182 0.572 0.541 0.599 1.882 0.952 0.985 3.094 0.047 0.726 2.281 0.758 0.148 0.465 0.448 0.788 2.476 0.618 0.698 2.193 0.813 0.010 0.031 0.031 0.397 1.247 0.948 0.852 2.677 0.448 0.761 2.391 0.682 0.309 0.971 0.825 0.113 0.355 0.348 0.212 0.666 0.618 0.809 2.542 0.565 0.084 0.264 0.261 0.276 0.867 0.762 0.243 0.763 0.691 0.989 3.107 0.035 0.508 1.596 1.000 0.817 2.567 0.544 0.174 0.547 0.520

i 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

ui ηi = π ui Sin(ηi) 0.225 0.707 0.649 0.687 2.158 0.832 0.060 0.188 0.187 0.070 0.220 0.218 0.835 2.623 0.495 0.466 1.464 0.994 0.271 0.851 0.752 0.813 2.554 0.554 0.376 1.181 0.925 0.983 3.088 0.053 0.181 0.569 0.538 0.583 1.832 0.966 0.612 1.923 0.939 0.468 1.470 0.995 0.614 1.929 0.937 0.768 2.413 0.666 0.811 2.548 0.559 0.461 1.448 0.993 0.985 3.094 0.047 0.735 2.309 0.740 0.559 1.756 0.983 0.884 2.777 0.356 0.229 0.719 0.659 0.920 2.890 0.249 0.524 1.646 0.997

40

2. EVALUATION OF DEFINITE INTEGRALS USING THE MONTE CARLO METHOD

i 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75

ui ηi = π ui Sin(ηi) i 0.393 1.235 0.944 76 0.306 0.961 0.820 77 0.819 2.573 0.538 78 0.620 1.948 0.930 79 0.875 2.749 0.383 80 0.773 2.428 0.654 81 0.492 1.546 1.000 82 0.332 1.043 0.864 83 0.185 0.581 0.549 84 0.565 1.775 0.979 85 0.434 1.363 0.979 86 0.665 2.089 0.869 87 0.516 1.621 0.999 88 0.659 2.070 0.878 89 0.709 2.227 0.792 90 0.515 1.618 0.999 91 0.683 2.146 0.839 92 0.092 0.289 0.285 93 0.417 1.310 0.966 94 0.598 1.879 0.953 95 0.682 2.143 0.841 96 0.948 2.978 0.163 97 0.125 0.393 0.383 98 0.149 0.468 0.451 99 0.246 0.773 0.698 100

ui ηi = π ui Sin(ηi) 0.231 0.726 0.664 0.836 2.626 0.493 0.167 0.525 0.501 0.545 1.712 0.990 0.748 2.350 0.712 0.440 1.382 0.982 0.674 2.117 0.854 0.287 0.902 0.784 0.052 0.163 0.163 0.940 2.953 0.187 0.679 2.133 0.846 0.910 2.859 0.279 0.254 0.798 0.716 0.236 0.741 0.675 0.854 2.683 0.443 0.877 2.755 0.377 0.483 1.517 0.999 0.117 0.368 0.359 0.847 2.661 0.462 0.201 0.631 0.590 0.294 0.924 0.798 0.071 0.223 0.221 0.065 0.204 0.203 0.740 2.325 0.729 0.954 2.997 0.144 Sum = 62.800 I = 1.973

2.1. EVALUATION OF DEFINITE INTEGRALS: EXAMPLE I

41

F(x) = S in(x) or p(x)

1.0 0.8 0.6 0.4 0.2 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

x

Figure 2.2: Showing F .x/ D Sin.x/ as undashed curve and p.x/ D 2x= 2 as dashed curve obtained using the command Plot[{Sin[x],(2*x)/(Pi^2)},{x,0,Pi},Frame->True,FrameLabel-> {"x","F(x) = Sin(x) or p(x)"},PlotStyle-> {{Black},{Dashed,Black}}] in Mathematica 6.0. F(x) = Sin(x) or Gaussian(x)

1.0 0.8 0.6 0.4 0.2 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

x

Figure 2.3: Showing F .x/ D Sin.x/ as undashed curve and Gaussian p.x/ D h
2 i Exp 21 x  a as dashed curve obtained using the command sig=0.5;a=Pi/2; p1=Plot[Sin[x],{x,0,Pi},PlotStyle->{Black},Frame->True, FrameLabel->{"x","F(x) = Sin(x) or Gaussian (x)"}]; p2=Plot[(1/(sig*Sqrt[2*Pi]))*Exp[-0.5*((x-a)/sig)^2],{x,0,Pi}, PlotStyle->{Dashed,Black}];Show[p1,p2] in Mathematica 6.0.



p1 2

42

2. EVALUATION OF DEFINITE INTEGRALS USING THE MONTE CARLO METHOD

Table 2.2: Showing tabulated values of ui , i D u1=2 i , Sin.i /=i from which we get I D F .xi / Sin.i / Sin.i / 1 PN 1 PN  2 PN as an approximate value of the integral i D1 p.xi / D N i D1 2i = 2 D 2N i D1 N i Rb R I D a F .x/dx D 0 Sin.x/dx D 1:947 whereas the exact value is 2. Using p.x/ D 2x= 2 . i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

ui ηi = π ui1/2 Sin(ηi)/ηi 0.169 1.291 0.744 0.050 0.702 0.920 0.844 2.886 0.088 0.182 1.340 0.726 0.599 2.431 0.268 0.985 3.118 0.008 0.726 2.677 0.167 0.148 1.209 0.774 0.788 2.789 0.124 0.698 2.625 0.188 0.010 0.314 0.984 0.397 1.979 0.464 0.852 2.900 0.083 0.761 2.741 0.142 0.309 1.746 0.564 0.113 1.056 0.824 0.212 1.447 0.686 0.809 2.826 0.110 0.084 0.911 0.867 0.276 1.650 0.604 0.243 1.549 0.646 0.989 3.124 0.006 0.508 2.239 0.351 0.817 2.840 0.105 0.174 1.310 0.737

i 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

ui ηi = π ui1/2 Sin(ηi)/ηi 0.225 1.490 0.669 0.687 2.604 0.197 0.060 0.770 0.904 0.070 0.831 0.889 0.835 2.871 0.093 0.466 2.145 0.392 0.271 1.635 0.610 0.813 2.833 0.107 0.376 1.926 0.487 0.983 3.115 0.009 0.181 1.337 0.728 0.583 2.399 0.282 0.612 2.458 0.257 0.468 2.149 0.390 0.614 2.462 0.255 0.768 2.753 0.138 0.811 2.829 0.109 0.461 2.133 0.397 0.985 3.118 0.008 0.735 2.693 0.161 0.559 2.349 0.303 0.884 2.954 0.063 0.229 1.503 0.664 0.920 3.013 0.042 0.524 2.274 0.335

2.1. EVALUATION OF DEFINITE INTEGRALS: EXAMPLE I

i 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75

ui ηi = π ui1/2 Sin(ηi)/ηi i 0.393 1.969 0.468 76 0.306 1.738 0.567 77 0.819 2.843 0.103 78 0.620 2.474 0.250 79 0.875 2.939 0.069 80 0.773 2.762 0.134 81 0.492 2.204 0.366 82 0.332 1.810 0.537 83 0.185 1.351 0.722 84 0.565 2.361 0.298 85 0.434 2.070 0.424 86 0.665 2.562 0.214 87 0.516 2.257 0.343 88 0.659 2.550 0.219 89 0.709 2.645 0.180 90 0.515 2.255 0.344 91 0.683 2.596 0.200 92 0.092 0.953 0.855 93 0.417 2.029 0.442 94 0.598 2.429 0.269 95 0.682 2.594 0.201 96 0.948 3.059 0.027 97 0.125 1.111 0.807 98 0.149 1.213 0.772 99 0.246 1.558 0.642 100

ui ηi = π ui1/2 Sin(ηi)/ηi 0.231 1.510 0.661 0.836 2.872 0.093 0.167 1.284 0.747 0.545 2.319 0.316 0.748 2.717 0.152 0.440 2.084 0.418 0.674 2.579 0.207 0.287 1.683 0.590 0.052 0.716 0.917 0.940 3.046 0.031 0.679 2.589 0.203 0.910 2.997 0.048 0.254 1.583 0.632 0.236 1.526 0.655 0.854 2.903 0.081 0.877 2.942 0.067 0.483 2.183 0.375 0.117 1.075 0.818 0.847 2.891 0.086 0.201 1.408 0.701 0.294 1.703 0.582 0.071 0.837 0.887 0.065 0.801 0.896 0.740 2.703 0.157 0.954 3.068 0.024 Sum = 39.461 I = 1.947

43

44

2. EVALUATION OF DEFINITE INTEGRALS USING THE MONTE CARLO METHOD

p.x/. As Figure 2.3 shows, the Gaussian probability density function p.x/ with a D =2 and  D 0:5 is not too different from the function Sin.x/. Hence, it is ideal for use as p.x/. We now generate random variable G obeying "   # 1 1 G a 2 p.G/ D p Exp 2   2

with a D =2 and  D 0:5 using the tabulated values of random variable g of Table 1.4 obeying   1  g a 2 1 p.g/ D p Exp 2   2 for a D 0 and  D 1. The connection is G D a C g D =2 C 0:5g . Using Table 2.3 in Microsoft Excel, we have evaluated the integral with the result 1.950 rather than 2. The difference is attributable to the fact that F .x/ and p.x/ used are not proportional to each other.

2.2

EVALUATION OF DEFINITE INTEGRALS USING THE MONTE CARLO METHOD: EXAMPLE II

As Example II, we now take up the integral Z b Z I D F .x/dx D a

C=2

Cos.x/dx;

(2.4)

=2

where F .x/ D Cos.x/. We re-write Equation (2.4) as Z b F .x/ I D p.x/dx a p.x/

(2.5)

which, as discussed in Section 1.7, implies that average value of F=p is the value of the integral, i.e., N 1 X F .xi / ; (2.6) I D N p.xi / i D1

where xi ’s are random values of x in the interval a < x < b obeying probability density function p.x/. Here, p.x/ is a suitable probability density function which, as demonstrated in Section 1.7, at least should follow F .x/, if not be proportional to F .x/. See Figure 2.4 in which we have plotted F .x/ D Cos.x/ along with normalized probability density function of a uniform random variable in the interval =2 to C=2 given by p.x/ D 1= . Let p.x/ D C , a constant. NorR C=2 malization requires that =2 Cdx D 1 which gives C D 1= . Uniform p.x/ D 1= is not too different from a slowly varying function of x like Cos.x/. Hence, p.x/ D 1= is a good choice.

2.2. EVALUATION OF DEFINITE INTEGRALS: EXAMPLE II

45

Table 2.3: Showing tabulated values of ui , gi , Gi , Sin.Gi /=pi from which we get I D P F .xi / Sin.Gi / Sin 1 PN 1 PN  .Gi/ D N1 N   as an approximate value of i D1 p.xi / D N i D1 i D1 N pi G a 2 the integralhI D p1  2

Exp

1 2

Rb

a F .x/dx
i x a 2 . 

i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

D

R 0



1 p Exp 2

1 2

i 

Sin.x/dx D 1:950 whereas the exact value is 2. Using p.x/ D

ui gi 0.169 –0.955 0.050 –1.635 0.844 1.015 0.182 –0.905 0.599 0.255 0.985 2.195 0.726 0.605 0.148 –1.040 0.788 0.805 0.698 0.520 0.010 –2.295 0.397 –0.255 0.852 1.050 0.761 0.715 0.309 –0.495 0.113 –1.205 0.212 –0.795 0.809 0.880 0.084 –1.370 0.276 –0.590 0.243 –0.695 0.989 2.325 0.508 0.020 0.817 0.910 0.174 –0.935

Gi 1.093 0.753 2.078 1.118 1.698 2.668 1.873 1.051 1.973 1.831 0.423 1.443 2.096 1.928 1.323 0.968 1.173 2.011 0.886 1.276 1.223 2.733 1.581 2.026 1.103

SinGi 0.888 0.684 0.874 0.899 0.992 0.456 0.955 0.868 0.920 0.966 0.411 0.992 0.865 0.937 0.970 0.824 0.922 0.905 0.774 0.957 0.940 0.397 1.000 0.898 0.893

pi (SinGi)/pi 0.506 1.756 0.210 3.263 0.477 1.833 0.530 1.698 0.772 1.284 0.072 6.354 0.664 1.437 0.465 1.868 0.577 1.594 0.697 1.387 0.057 7.168 0.772 1.284 0.460 1.882 0.618 1.516 0.706 1.373 0.386 2.134 0.582 1.585 0.542 1.670 0.312 2.481 0.670 1.427 0.627 1.500 0.053 7.425 0.798 1.254 0.527 1.703 0.515 1.732

46

2. EVALUATION OF DEFINITE INTEGRALS USING THE MONTE CARLO METHOD

i 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

ui gi 0.225 –0.750 0.687 0.490 0.060 –1.540 0.070 –1.465 0.835 0.980 0.466 –0.085 0.271 –0.605 0.813 0.895 0.376 –0.315 0.983 2.145 0.181 –0.905 0.583 0.210 0.612 0.285 0.468 –0.080 0.614 0.290 0.768 0.735 0.811 0.885 0.461 –0.095 0.985 2.195 0.735 0.630 0.559 0.150 0.884 1.200 0.229 –0.740 0.920 1.415 0.524 0.060

Gi 1.196 1.816 0.801 0.838 2.061 1.528 1.268 2.018 1.413 2.643 1.118 1.676 1.713 1.531 1.716 1.938 2.013 1.523 2.668 1.886 1.646 2.171 1.201 2.278 1.601

SinGi 0.931 0.970 0.718 0.744 0.882 0.999 0.955 0.902 0.988 0.478 0.899 0.994 0.990 0.999 0.990 0.933 0.904 0.999 0.456 0.951 0.997 0.825 0.932 0.760 1.000

pi (SinGi)/pi 0.602 1.545 0.708 1.371 0.244 2.945 0.273 2.725 0.494 1.787 0.795 1.257 0.664 1.437 0.535 1.686 0.759 1.301 0.080 5.978 0.530 1.698 0.780 1.274 0.766 1.292 0.795 1.256 0.765 1.293 0.609 1.532 0.539 1.676 0.794 1.258 0.072 6.354 0.654 1.453 0.789 1.264 0.388 2.125 0.607 1.537 0.293 2.592 0.796 1.255

2.2. EVALUATION OF DEFINITE INTEGRALS: EXAMPLE II

i 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75

ui gi 0.393 –0.270 0.306 –0.505 0.819 0.915 0.620 0.310 0.875 1.155 0.773 0.750 0.492 –0.020 0.332 –0.430 0.185 –0.890 0.565 0.165 0.434 –0.165 0.665 0.430 0.516 0.040 0.659 0.410 0.709 0.555 0.515 0.040 0.683 0.480 0.092 –1.320 0.417 –0.205 0.598 0.250 0.682 0.475 0.948 1.635 0.125 –1.145 0.149 –1.035 0.246 –0.685

Gi 1.436 1.318 2.028 1.726 2.148 1.946 1.561 1.356 1.126 1.653 1.488 1.786 1.591 1.776 1.848 1.591 1.811 0.911 1.468 1.696 1.808 2.388 0.998 1.053 1.228

SinGi 0.991 0.968 0.897 0.988 0.838 0.931 1.000 0.977 0.903 0.997 0.997 0.977 1.000 0.979 0.962 1.000 0.971 0.790 0.995 0.992 0.972 0.684 0.841 0.869 0.942

pi (SinGi)/pi 0.769 1.288 0.702 1.379 0.525 1.709 0.760 1.299 0.410 2.046 0.602 1.545 0.798 1.254 0.727 1.343 0.537 1.681 0.787 1.266 0.787 1.266 0.727 1.343 0.797 1.254 0.734 1.335 0.684 1.406 0.797 1.254 0.711 1.366 0.334 2.366 0.781 1.273 0.773 1.283 0.713 1.364 0.210 3.263 0.414 2.029 0.467 1.861 0.631 1.493

47

48

2. EVALUATION OF DEFINITE INTEGRALS USING THE MONTE CARLO METHOD

i 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

ui gi 0.231 –0.730 0.836 0.985 0.167 –0.960 0.545 0.115 0.748 0.670 0.440 –0.150 0.674 0.455 0.287 –0.560 0.052 –1.615 0.940 1.565 0.679 0.470 0.910 1.350 0.254 –0.660 0.236 –0.715 0.854 1.060 0.877 1.165 0.483 –0.040 0.117 –1.185 0.847 1.030 0.201 –0.835 0.294 –0.540 0.071 –1.460 0.065 –1.505 0.740 0.645 0.954 1.695

Gi 1.206 2.063 1.091 1.628 1.906 1.496 1.798 1.291 0.763 2.353 1.806 2.246 1.241 1.213 2.101 2.153 1.551 0.978 2.086 1.153 1.301 0.841 0.818 1.893 2.418

SinGi 0.934 0.881 0.887 0.998 0.944 0.997 0.974 0.961 0.691 0.709 0.973 0.781 0.946 0.937 0.863 0.835 1.000 0.830 0.870 0.914 0.964 0.745 0.730 0.948 0.662

pi (SinGi)/pi 0.611 1.528 0.491 1.794 0.503 1.762 0.793 1.260 0.637 1.481 0.789 1.264 0.719 1.354 0.682 1.409 0.217 3.192 0.234 3.024 0.714 1.361 0.321 2.434 0.642 1.474 0.618 1.516 0.455 1.897 0.405 2.063 0.797 1.254 0.395 2.098 0.469 1.854 0.563 1.624 0.690 1.398 0.275 2.711 0.257 2.839 0.648 1.464 0.190 3.489 Sum = 195.012 I = 1.950

2.2. EVALUATION OF DEFINITE INTEGRALS: EXAMPLE II

49

F(x) = Cos(x) or p(x)

1.0 0.8 0.6 0.4 0.2 0.0 –1.5

–1.0

–0.5

0.0

0.5

1.0

1.5

x

Figure 2.4: Showing F .x/ D Cos.x/ as undashed curve and p.x/ D 1= as dashed curve obtained using the command Plot[{Cos[x],1/Pi},{x,-Pi/2,Pi/2}, Frame->True,FrameLabel->{"x","F(x) = Cos(x) or p(x)"}, PlotStyle->{{Black},{Dashed,Black}}] in Mathematica 6.0.

We now generate random variable  obeying p.x/ D 1= in the interval =2 < x < C=2 using Equation (1.4):  D a C .b a/u where u is random variable (of Table 1.1) of uniform probability density function in the Rinterval 0 < x < 1. Thus, we have  D u =2.  That  D u =2 can be calculated using a p.x/dx D u which gives Z

 =2

.1=/dx D u

or,  D u

=2:

Using Table 2.4 in Microsoft Excel, we have evaluated the integral with the result 1.973 rather than 2. The difference is attributable to the fact that F .x/ and p.x/ used are not proportional to each other. We now take up a linear variation for p.x/ given by p.x/ D C x C d where C and d are parameters such that slope C D intercept d on y axis =.=2/; hence, d D C =2. As such, R C=2 p.x/ D C.x C =2/. Normalization of p.x/ requires =2 p.x/dx D 1 or C D 2= 2 . As such, normalized probability density function is p.x/ D 2x= 2 C 1= . See Figure 2.5 in which we have plotted F .x/ D Cos.x/ along with the normalized linear probability density function p.x/ D 2x= 2 C 1= . The linear probability density function p.x/ D 2x= 2 C 1= is not too different from a slowly varying function of x like Cos.x/. Hence, it is a good choice.

50

2. EVALUATION OF DEFINITE INTEGRALS USING THE MONTE CARLO METHOD

Table 2.4: Showing tabulated values of ui , i D ui =2, Cos.i / from which we get I D F .xi / F .i / 1 PN  PN 1 PN i D1 p.xi / D N i D1 1= D N i D1 Cos.i / as an approximate value of the integral N Rb R C=2 I D a F .x/dx D =2 Cos.x/dx D 1:973 whereas the exact value is 2. Using p.x/ D 1= . i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

ui ηi = π ui –π/2 Cos(ηi) 0.169 –1.040 0.506 0.050 –1.414 0.156 0.844 1.081 0.471 0.182 –0.999 0.541 0.599 0.311 0.952 0.985 1.524 0.047 0.726 0.710 0.758 0.148 –1.106 0.448 0.788 0.905 0.618 0.698 0.622 0.813 0.010 –1.539 0.031 0.397 –0.324 0.948 0.852 1.106 0.448 0.761 0.820 0.682 0.309 –0.600 0.825 0.113 –1.216 0.348 0.212 –0.905 0.618 0.809 0.971 0.565 0.084 –1.307 0.261 0.276 –0.704 0.762 0.243 –0.807 0.691 0.989 1.536 0.035 0.508 0.025 1.000 0.817 0.996 0.544 0.174 –1.024 0.520

i 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

ui ηi = π ui –π/2 Cos(ηi) 0.225 –0.864 0.649 0.687 0.587 0.832 0.060 –1.382 0.187 0.070 –1.351 0.218 0.835 1.052 0.495 0.466 –0.107 0.994 0.271 –0.719 0.752 0.813 0.983 0.554 0.376 –0.390 0.925 0.983 1.517 0.053 0.181 –1.002 0.538 0.583 0.261 0.966 0.612 0.352 0.939 0.468 –0.101 0.995 0.614 0.358 0.937 0.768 0.842 0.666 0.811 0.977 0.559 0.461 –0.123 0.993 0.985 1.524 0.047 0.735 0.738 0.740 0.559 0.185 0.983 0.884 1.206 0.356 0.229 –0.851 0.659 0.920 1.319 0.249 0.524 0.075 0.997

2.2. EVALUATION OF DEFINITE INTEGRALS: EXAMPLE II

i 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75

ui ηi = π ui –π/2 Cos(ηi) i 0.393 –0.336 0.944 76 0.306 –0.609 0.820 77 0.819 1.002 0.538 78 0.620 0.377 0.930 79 0.875 1.178 0.383 80 0.773 0.858 0.654 81 0.492 –0.025 1.000 82 0.332 –0.528 0.864 83 0.185 –0.990 0.549 84 0.565 0.204 0.979 85 0.434 –0.207 0.979 86 0.665 0.518 0.869 87 0.516 0.050 0.999 88 0.659 0.500 0.878 89 0.709 0.657 0.792 90 0.515 0.047 0.999 91 0.683 0.575 0.839 92 0.092 –1.282 0.285 93 0.417 –0.261 0.966 94 0.598 0.308 0.953 95 0.682 0.572 0.841 96 0.948 1.407 0.163 97 0.125 –1.178 0.383 98 0.149 –1.103 0.451 99 0.246 –0.798 0.698 100

ui ηi = π ui –π/2 Cos(ηi) 0.231 –0.845 0.664 0.836 1.056 0.493 0.167 –1.046 0.501 0.545 0.141 0.990 0.748 0.779 0.712 0.440 –0.188 0.982 0.674 0.547 0.854 0.287 –0.669 0.784 0.052 –1.407 0.163 0.940 1.382 0.187 0.679 0.562 0.846 0.910 1.288 0.279 0.254 –0.773 0.716 0.236 –0.829 0.675 0.854 1.112 0.443 0.877 1.184 0.377 0.483 –0.053 0.999 0.117 –1.203 0.359 0.847 1.090 0.462 0.201 –0.939 0.590 0.294 –0.647 0.798 0.071 –1.348 0.221 0.065 –1.367 0.203 0.740 0.754 0.729 0.954 1.426 0.144 Sum = 62.800 I = 1.973

51

52

2. EVALUATION OF DEFINITE INTEGRALS USING THE MONTE CARLO METHOD

F(x) = Cos(x) or p(x)

1.0 0.8 0.6 0.4 0.2 0.0 –1.5

–1.0

–0.5

0.0

0.5

1.0

1.5

x

Figure 2.5: Showing F .x/ D Cos.x/ as undashed curve and p.x/ D 2x= 2 C 1= as dashed curve obtained using the command Plot[{Cos[x],2*x/Pi^2+1/Pi},{x,-Pi/2,Pi/2},Frame->True, FrameLabel->{"x","F(x) = Cos(x) or p(x)"}, PlotStyle->{{Black},{Dashed,Black}}] in Mathematica 6.0. We now generate random variable  obeying p.x/ D 2x= 2 C 1= in the interval =2 < x < C=2 using Equation (1.2): Z  Z  p.x/dx D u or, p.x/dx D u; a

=2

where u is random variable (of Table 1.1) of uniform probability density function in the interval 0 < x < 1. Thus, we have  given by roots of the quadratic equation 2 C  C . 2 =4

 2 u/ D 0

which are D

=2 ˙ u1=2 :

Using  D =2 C u1=2 in Microsoft Excel we have Table 2.5 in which we have evaluated the integral with the result 1.947 rather than 2. The difference is attributable to the fact that F .x/ and p.x/ used are not proportional to each other. Table 2.6 uses  D =2 u1=2 . We h now take i up a normalized Gaussian variation for p.x/ given by p.x/ D
2 1 p Exp 21 x  a where a is average and  is variance of x . See Figure 2.6 in which we have  2 plotted F .x/ D Cos.x/ along with the normalized Gaussian probability density function p.x/.

2.2. EVALUATION OF DEFINITE INTEGRALS: EXAMPLE II

53

Table 2.5: Showing tabulated values of ui , i D =2 C u1=2 i , Cos.i /=pi from which we F .xi / Cos.i / Cos.i / 1 PN 1 PN 1 PN get I D N i D1 p.xi / D N i D1 pi D N i D1 1=C2 = 2 as an approximate value of the i Rb R C=2 integral I D a F .x/dx D =2 Cos.x/dx D 1:947 whereas the exact value is 2. Using p.x/ D 2x= 2 C 1= . i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

ui ηi = –π/2 + πui1/2 Cos(ηi)/pi = Cos(ηi)/(1/π + 2ηi /π2) 0.169 –0.279 3.673 0.050 –0.868 4.539 0.844 1.315 0.432 0.182 –0.231 3.585 0.599 0.861 1.323 0.985 1.547 0.037 0.726 1.106 0.826 0.148 –0.362 3.818 0.788 1.218 0.611 0.698 1.054 0.929 0.010 –1.257 4.854 0.397 0.409 2.288 0.852 1.329 0.407 0.761 1.170 0.703 0.309 0.176 2.782 0.113 –0.515 4.067 0.212 –0.124 3.385 0.809 1.255 0.543 0.084 –0.660 4.281 0.276 0.080 2.980 0.243 –0.022 3.186 0.989 1.553 0.027 0.508 0.668 1.730 0.817 1.269 0.517 0.174 –0.260 3.639

54

2. EVALUATION OF DEFINITE INTEGRALS USING THE MONTE CARLO METHOD

i 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

ui ηi = –π/2 + πui1/2 Cos(ηi)/pi = Cos(ηi)/(1/π + 2ηi /π2) 0.225 –0.081 3.301 0.687 1.033 0.971 0.060 –0.801 4.462 0.070 –0.740 4.386 0.835 1.300 0.460 0.466 0.574 1.933 0.271 0.065 3.011 0.813 1.262 0.530 0.376 0.356 2.401 0.983 1.544 0.042 0.181 –0.234 3.591 0.583 0.828 1.391 0.612 0.887 1.269 0.468 0.578 1.923 0.614 0.891 1.260 0.768 1.182 0.679 0.811 1.258 0.536 0.461 0.562 1.957 0.985 1.547 0.037 0.735 1.123 0.794 0.559 0.778 1.496 0.884 1.383 0.312 0.229 –0.067 3.275 0.920 1.443 0.210 0.524 0.703 1.655

2.2. EVALUATION OF DEFINITE INTEGRALS: EXAMPLE II

i 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75

ui ηi = –π/2 + πui1/2 Cos(ηi)/pi = Cos(ηi)/(1/π + 2ηi /π2) 0.393 0.399 2.309 0.306 0.167 2.800 0.819 1.272 0.510 0.620 0.903 1.236 0.875 1.368 0.338 0.773 1.191 0.662 0.492 0.633 1.806 0.332 0.239 2.648 0.185 –0.220 3.564 0.565 0.791 1.470 0.434 0.499 2.094 0.665 0.991 1.055 0.516 0.686 1.692 0.659 0.980 1.079 0.709 1.074 0.888 0.515 0.684 1.697 0.683 1.026 0.986 0.092 –0.618 4.221 0.417 0.458 2.182 0.598 0.859 1.327 0.682 1.024 0.990 0.948 1.488 0.133 0.125 –0.460 3.981 0.149 –0.358 3.811 0.246 –0.013 3.167

55

56

2. EVALUATION OF DEFINITE INTEGRALS USING THE MONTE CARLO METHOD

i 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

ui ηi = –π/2 + πui1/2 Cos(ηi)/pi = Cos(ηi)/(1/π + 2ηi /π2) 0.231 –0.061 3.262 0.836 1.302 0.457 0.167 –0.287 3.687 0.545 0.748 1.559 0.748 1.146 0.748 0.440 0.513 2.063 0.674 1.008 1.020 0.287 0.112 2.914 0.052 –0.854 4.523 0.940 1.475 0.155 0.679 1.018 1.001 0.910 1.426 0.237 0.254 0.013 3.117 0.236 –0.045 3.230 0.854 1.332 0.401 0.877 1.371 0.332 0.483 0.613 1.849 0.117 –0.496 4.038 0.847 1.320 0.423 0.201 –0.162 3.458 0.294 0.133 2.872 0.071 –0.734 4.378 0.065 –0.770 4.424 0.740 1.132 0.776 0.954 1.498 0.117 Sum = 194.736 I = 1.947

2.2. EVALUATION OF DEFINITE INTEGRALS: EXAMPLE II

57

Table 2.6: Showing tabulated values of ui , i D =2 u1=2 i , Cos.i /=pi from which we F .xi / Cos.i / Cos.i / 1 PN 1 PN 1 PN get I D N i D1 p.xi / D N i D1 pi D N i D1 1=C2 = 2 as an approximate value of the i Rb R C=2 integral I D a F .x/dx D =2 Cos.x/dx D 1:947 whereas the exact value is 2. Using p.x/ D 1= C 2x= 2 . i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

ui ηi = –π/2 – πui1/2 Cos(ηi)/pi = Cos(ηi)/(1/π + 2ηi /π2) 3.673 –2.862 0.169 4.539 –2.273 0.050 0.432 –4.457 0.844 3.585 –2.911 0.182 1.323 –4.002 0.599 0.037 –4.689 0.985 0.826 –4.248 0.726 3.818 –2.779 0.148 0.611 –4.360 0.788 0.929 –4.195 0.698 4.854 –1.885 0.010 2.288 –3.550 0.397 0.407 –4.471 0.852 0.703 –4.311 0.761 2.782 –3.317 0.309 4.067 –2.627 0.113 3.385 –3.017 0.212 0.543 –4.396 0.809 4.281 –2.481 0.084 2.980 –3.221 0.276 3.186 –3.119 0.243 0.027 –4.695 0.989 1.730 –3.810 0.508 0.517 –4.410 0.817 3.639 –2.881 0.174

58

2. EVALUATION OF DEFINITE INTEGRALS USING THE MONTE CARLO METHOD

i 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

ui ηi = –π/2 – πui1/2 Cos(ηi)/pi = Cos(ηi)/(1/π + 2ηi /π2) 0.225 –3.061 3.301 0.687 –4.175 0.971 0.060 –2.340 4.462 0.070 –2.402 4.386 0.835 –4.442 0.460 0.466 –3.715 1.933 0.271 –3.206 3.011 0.813 –4.403 0.530 0.376 –3.497 2.401 0.983 –4.686 0.042 0.181 –2.907 3.591 0.583 –3.970 1.391 0.612 –4.028 1.269 0.468 –3.720 1.923 0.614 –4.032 1.260 0.768 –4.324 0.679 0.811 –4.400 0.536 0.461 –3.704 1.957 0.985 –4.689 0.037 0.735 –4.264 0.794 0.559 –3.920 1.496 0.884 –4.525 0.312 0.229 –3.074 3.275 0.920 –4.584 0.209 0.524 –3.845 1.655

2.2. EVALUATION OF DEFINITE INTEGRALS: EXAMPLE II

i 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75

ui ηi = –π/2 – πui1/2 Cos(ηi)/pi = Cos(ηi)/(1/π + 2ηi /π2) 2.309 –3.540 0.393 2.800 –3.309 0.306 0.510 –4.414 0.819 1.236 –4.044 0.620 0.338 –4.509 0.875 0.662 –4.333 0.773 1.806 –3.774 0.492 2.648 –3.381 0.332 3.564 –2.922 0.185 1.470 –3.932 0.565 2.094 –3.640 0.434 1.055 –4.133 0.665 1.692 –3.828 0.516 1.079 –4.121 0.659 0.888 –4.216 0.709 1.697 –3.825 0.515 0.986 –4.167 0.683 4.221 –2.524 0.092 2.182 –3.600 0.417 1.327 –4.000 0.598 0.990 –4.165 0.682 0.133 –4.630 0.948 3.981 –2.682 0.125 3.811 –2.783 0.149 3.167 –3.129 0.246

59

60

2. EVALUATION OF DEFINITE INTEGRALS USING THE MONTE CARLO METHOD

i 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

ui ηi = –π/2 – πui1/2 Cos(ηi)/pi = Cos(ηi)/(1/π + 2ηi /π2) 3.262 –3.081 0.231 0.457 –4.443 0.836 3.687 –2.855 0.167 1.559 –3.890 0.545 0.748 –4.288 0.748 2.063 –3.655 0.440 1.020 –4.150 0.674 2.914 –3.254 0.287 4.523 –2.287 0.052 0.155 –4.617 0.940 1.001 –4.160 0.679 0.237 –4.568 0.910 3.117 –3.154 0.254 3.230 –3.097 0.236 0.401 –4.474 0.854 0.332 –4.513 0.877 1.849 –3.754 0.483 4.038 –2.645 0.117 0.423 –4.462 0.847 3.458 –2.979 0.201 2.872 –3.274 0.294 4.378 –2.408 0.071 4.424 –2.372 0.065 0.776 –4.273 0.740 0.117 –4.639 0.954 Sum = 194.735 I = 1.947

2.2. EVALUATION OF DEFINITE INTEGRALS: EXAMPLE II

61

F(x) = Cos(x) or Gaussian(x)

1.0 0.8 0.6 0.4 0.2 0.0 –1.5

–1.0

–0.5

0.0

0.5

1.0

1.5

x

Figure 2.6: Showing F .x/ D Cos.x/ as undashed curve and Gaussian p.x/ D h
2 i Exp 21 x  a as dashed curve obtained using the command sig=0.5; a=0; p1=Plot[Cos[x],{x,-Pi/2,Pi/2},PlotStyle->{Black},Frame->True, FrameLabel->{"x","F(x)= Cos (x) or Gaussian (x)"}]; p2=Plot[(1/(sig*Sqrt[2*Pi]))*Exp[-0.5*((x-a)/sig)^2], {x,-Pi/2,Pi/2},PlotStyle->{Dashed,Black}]; Show[p1,p2] in Mathematica 6.0.



p1 2

As Figure 2.6 shows, the Gaussian probability density function p.x/ with a D 0 and  D 0:5 is not too different from the function Cos.x/. Hence, it is ideal for use as p.x/. We now generate random variable G obeying "   # 1 G a 2 1 p.G/ D p Exp 2   2 with a D 0 and  D 0:5 using the tabulated values of random variable g of Table 1.4 obeying   1 1  g a 2 p.g/ D p Exp 2   2 for a D 0 and  D 1. The connection is G D a C g D 0 C 0:5g . Using Table 2.7 in Microsoft Excel, we have evaluated the integral with the result 1.950 rather than 2. The difference is attributable to the fact that F .x/ and p.x/ used are not proportional to each other.

62

2. EVALUATION OF DEFINITE INTEGRALS USING THE MONTE CARLO METHOD

P Table 2.7: Showing tabulated values of ui , gi , Gi , Cos.Gi /=pi from which we get I D N1 N i D1 F .xi / Cos.Gi / Cos 1 PN 1 PN  as an approximate value of the inte .Gi / D D  i D1 i D1 p.xi / N pi N G a 2 

1 p Exp 2

1 2

i 

Rb R C=2 gral I D a F .x/dx D =2 Cos.x/dx D 1:950 whereas the exact value is 2. Using p.x/ D h
2 i p1 Exp 21 x  a . 

2

i

ui

gi

Gi

Cos Gi

pi

(Cos Gi)/pi

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

0.169 0.050 0.844 0.182 0.599 0.985 0.726 0.148 0.788 0.698 0.010 0.397 0.852 0.761 0.309 0.113 0.212 0.809 0.084 0.276 0.243 0.989 0.508 0.817 0.174

–0.955 –1.635 1.015 –0.905 0.255 2.195 0.605 –1.040 0.805 0.520 –2.295 –0.255 1.050 0.715 –0.495 –1.205 –0.795 0.880 –1.370 –0.590 –0.695 2.325 0.020 0.910 –0.935

–0.478 –0.818 0.508 –0.453 0.128 1.098 0.303 –0.520 0.403 0.260 –1.148 –0.128 0.525 0.358 –0.248 –0.603 –0.398 0.440 –0.685 –0.295 –0.348 1.163 0.010 0.455 –0.468

0.888 0.684 0.874 0.899 0.992 0.456 0.955 0.868 0.920 0.966 0.411 0.992 0.865 0.937 0.970 0.824 0.922 0.905 0.774 0.957 0.940 0.397 1.000 0.898 0.893

0.506 0.210 0.477 0.530 0.772 0.072 0.664 0.465 0.577 0.697 0.057 0.772 0.460 0.618 0.706 0.386 0.582 0.542 0.312 0.670 0.627 0.053 0.798 0.527 0.515

1.756 3.263 1.833 1.698 1.284 6.354 1.437 1.868 1.594 1.387 7.168 1.284 1.882 1.516 1.373 2.134 1.585 1.670 2.481 1.427 1.500 7.425 1.254 1.703 1.732

2.2. EVALUATION OF DEFINITE INTEGRALS: EXAMPLE II

i

ui

gi

Gi

Cos Gi

pi

(Cos Gi)/pi

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

0.225 0.687 0.060 0.070 0.835 0.466 0.271 0.813 0.376 0.983 0.181 0.583 0.612 0.468 0.614 0.768 0.811 0.461 0.985 0.735 0.559 0.884 0.229 0.920 0.524

–0.750 0.490 –1.540 –1.465 0.980 –0.085 –0.605 0.895 –0.315 2.145 –0.905 0.210 0.285 –0.080 0.290 0.735 0.885 –0.095 2.195 0.630 0.150 1.200 –0.740 1.415 0.060

–0.375 0.245 –0.770 –0.733 0.490 –0.043 –0.303 0.448 –0.158 1.073 –0.453 0.105 0.143 –0.040 0.145 0.368 0.443 –0.048 1.098 0.315 0.075 0.600 –0.370 0.708 0.030

0.931 0.970 0.718 0.744 0.882 0.999 0.955 0.902 0.988 0.478 0.899 0.994 0.990 0.999 0.990 0.933 0.904 0.999 0.456 0.951 0.997 0.825 0.932 0.760 1.000

0.602 0.708 0.244 0.273 0.494 0.795 0.664 0.535 0.759 0.080 0.530 0.780 0.766 0.795 0.765 0.609 0.539 0.794 0.072 0.654 0.789 0.388 0.607 0.293 0.796

1.545 1.371 2.945 2.725 1.787 1.257 1.437 1.686 1.301 5.978 1.698 1.274 1.292 1.256 1.293 1.532 1.676 1.258 6.354 1.453 1.264 2.125 1.537 2.592 1.255

63

64

2. EVALUATION OF DEFINITE INTEGRALS USING THE MONTE CARLO METHOD

i

ui

gi

Gi

Cos Gi

pi

(Cos Gi)/pi

51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75

0.393 0.306 0.819 0.620 0.875 0.773 0.492 0.332 0.185 0.565 0.434 0.665 0.516 0.659 0.709 0.515 0.683 0.092 0.417 0.598 0.682 0.948 0.125 0.149 0.246

–0.270 –0.505 0.915 0.310 1.155 0.750 –0.020 –0.430 –0.890 0.165 –0.165 0.430 0.040 0.410 0.555 0.040 0.480 –1.320 –0.205 0.250 0.475 1.635 –1.145 –1.035 –0.685

–0.135 –0.253 0.458 0.155 0.578 0.375 –0.010 –0.215 –0.445 0.083 –0.083 0.215 0.020 0.205 0.278 0.020 0.240 –0.660 –0.103 0.125 0.238 0.818 –0.573 –0.518 –0.343

0.991 0.968 0.897 0.988 0.838 0.931 1.000 0.977 0.903 0.997 0.997 0.977 1.000 0.979 0.962 1.000 0.971 0.790 0.995 0.992 0.972 0.684 0.841 0.869 0.942

0.769 0.702 0.525 0.760 0.410 0.602 0.798 0.727 0.537 0.787 0.787 0.727 0.797 0.734 0.684 0.797 0.711 0.334 0.781 0.773 0.713 0.210 0.414 0.467 0.631

1.288 1.379 1.709 1.299 2.046 1.545 1.254 1.343 1.681 1.266 1.266 1.343 1.254 1.335 1.406 1.254 1.366 2.366 1.273 1.283 1.364 3.263 2.029 1.861 1.493

2.2. EVALUATION OF DEFINITE INTEGRALS: EXAMPLE II

i

ui

gi

Gi

Cos Gi

76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

0.231 0.836 0.167 0.545 0.748 0.440 0.674 0.287 0.052 0.940 0.679 0.910 0.254 0.236 0.854 0.877 0.483 0.117 0.847 0.201 0.294 0.071 0.065 0.740 0.954

–0.730 0.985 –0.960 0.115 0.670 –0.150 0.455 –0.560 –1.615 1.565 0.470 1.350 –0.660 –0.715 1.060 1.165 –0.040 –1.185 1.030 –0.835 –0.540 –1.460 –1.505 0.645 1.695

–0.365 0.493 –0.480 0.058 0.335 –0.075 0.228 –0.280 –0.808 0.783 0.235 0.675 –0.330 –0.358 0.530 0.583 –0.020 –0.593 0.515 –0.418 –0.270 –0.730 –0.753 0.323 0.848

0.934 0.881 0.887 0.998 0.944 0.997 0.974 0.961 0.691 0.709 0.973 0.781 0.946 0.937 0.863 0.835 1.000 0.830 0.870 0.914 0.964 0.745 0.730 0.948 0.662

pi

(Cos Gi)/pi

0.611 1.528 0.491 1.794 0.503 1.762 0.793 1.260 0.637 1.481 0.789 1.264 0.719 1.354 0.682 1.409 0.217 3.192 0.234 3.024 0.714 1.361 0.321 2.434 0.642 1.474 0.618 1.516 0.455 1.897 0.405 2.063 0.797 1.254 0.395 2.098 0.469 1.854 0.563 1.624 0.690 1.398 0.275 2.711 0.257 2.839 0.648 1.464 0.190 3.489 Sum = 195.012 I = 1.950

65

66

2. EVALUATION OF DEFINITE INTEGRALS USING THE MONTE CARLO METHOD

2.3

EVALUATION OF DEFINITE INTEGRALS USING THE MONTE CARLO METHOD: EXAMPLE III

As Example III, we now take up the integral I D

Z

b

F .x/dx D

a

Z

2

e x dx;

(2.7)

0

where F .x/ D e x . We re-write Equation (2.7) as I D

Z

b

a

F .x/ p.x/dx p.x/

(2.8)

which, as discussed in Section 1.7, implies that average value of F=p is the value of the integral, i.e., N 1 X F .xi / I D (2.9) ; N p.xi / i D1 where xi ’s are random values of x in the interval a < x < b obeying probability density function p.x/. We now take up a linear variation for p.x/ given by p.x/ D C x where C is a constant. R2 Normalization of p.x/ requires 0 p.x/dx D 1 or, C D 1=2. As such, normalized probability density function is p.x/ D x=2. See Figure 2.7 in which we have plotted F .x/ D e x along with the normalized linear probability density function p.x/ D x=2. The linear probability density function p.x/ D x=2 follows the function e x to some extent. Hence, it is not a bad choice. We now generate random variable  obeying p.x/ D x=2 in the interval 0 < x < 2 using Equation (1.2): Z 

a

p.x/dx D u;

where u is random variable (of Table 1.1) of uniform probability density function in the interval 0 < x < 1. Thus, we have  D 2u1=2 . Using  D 2u1=2 in Microsoft Excel we have Table 2.8 in which we have evaluated the integral with the result 6.234 rather than 6.39. The difference is attributable to the fact that F .x/ and p.x/ used are not proportional to each other. We now take up a normalized Gaussian variation for p.x/ given by   1 1  x a 2 p.x/ D p Exp ; 2   2 where a is average and  is variance of x . See Figure 2.8 in which we have plotted F .x/ D e x along with the normalized Gaussian probability density function p.x/. As Figure 2.8 shows, the Gaussian probability density function p.x/ with a D 1 and  D 0:3 is different from the

2.3. EVALUATION OF DEFINITE INTEGRALS: EXAMPLE III

67

7

F(x) = e x or p(x)

6 5 4 3 2 1 0 0.0

0.5

1.0

x

1.5

2.0

Figure 2.7: Showing F .x/ D e x as undashed curve and p.x/ D x=2 as dashed curve obtained using the command Plot[{Exp[x],x/2},{x,0,2},Frame->True,FrameLabel-> {"x","F(x) = e^x or p(x)"},PlotStyle->{{Black},{Dashed, Black}}] in Mathematica 6.0.

F(x) = e x or Gaussian(x)

7 6 5 4 3 2 1 0

0.0

0.5

1.0

x

1.5

2.0

Figure 2.8: Showing F .x/ D e x as undashed curve and Gaussian p.x/ D h
2 i Exp 21 x  a as dashed curve obtained using the command sig=0.3; a=1; p1=Plot[Exp[x],{x,0,2},PlotStyle->{Black},Frame->True, FrameLabel->{"x","F(x) = e^x or Gaussian (x)"},PlotRange->{0,7}]; p2=Plot[(1/(sig*Sqrt[2*Pi]))*Exp[-0.5*((x-a)/sig)^2], {x,0,7},PlotStyle->{Dashed,Black},PlotRange->{0,7}]; Show[p1,p2] in Mathematica 6.0.



p1 2

68

2. EVALUATION OF DEFINITE INTEGRALS USING THE MONTE CARLO METHOD

i Table 2.8: Showing tabulated values of ui , i D 2u1=2 i , e =i from which we get I D P P P   N N N F .xi / 1 1 e i 2 e i i D1 p.xi / D N i D1 i =2 D N i D1 i as an approximate value of the integral I D N Rb R2 x a F .x/dx D 0 e dx D 6:234 whereas the exact value is 6.389. Using p.x/ D x=2.

i

ui

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

0.169 0.050 0.844 0.182 0.599 0.985 0.726 0.148 0.788 0.698 0.010 0.397 0.852 0.761 0.309 0.113 0.212 0.809 0.084 0.276 0.243 0.989 0.508 0.817 0.174

ηi = 2ui1/2 eηi /ηi 0.822 0.447 1.837 0.853 1.548 1.985 1.704 0.769 1.775 1.671 0.200 1.260 1.846 1.745 1.112 0.672 0.921 1.799 0.580 1.051 0.986 1.989 1.425 1.808 0.834

2.768 3.497 3.418 2.751 3.037 3.667 3.225 2.805 3.325 3.182 6.107 2.798 3.432 3.281 2.734 2.913 2.727 3.359 3.080 2.722 2.719 3.674 2.918 3.373 2.761

i

ui

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

0.225 0.687 0.060 0.070 0.835 0.466 0.271 0.813 0.376 0.983 0.181 0.583 0.612 0.468 0.614 0.768 0.811 0.461 0.985 0.735 0.559 0.884 0.229 0.920 0.524

ηi = 2ui1/2 eηi /ηi 0.949 1.658 0.490 0.529 1.828 1.365 1.041 1.803 1.226 1.983 0.851 1.527 1.565 1.368 1.567 1.753 1.801 1.358 1.985 1.715 1.495 1.880 0.957 1.918 1.448

2.722 3.165 3.332 3.208 3.403 2.869 2.721 3.366 2.780 3.663 2.752 3.015 3.056 2.871 3.058 3.292 3.363 2.863 3.667 3.240 2.983 3.487 2.721 3.550 2.938

2.3. EVALUATION OF DEFINITE INTEGRALS: EXAMPLE III

i

ui

51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75

0.393 0.306 0.819 0.620 0.875 0.773 0.492 0.332 0.185 0.565 0.434 0.665 0.516 0.659 0.709 0.515 0.683 0.092 0.417 0.598 0.682 0.948 0.125 0.149 0.246

ηi = 2ui1/2 eηi /ηi 1.254 1.106 1.810 1.575 1.871 1.758 1.403 1.152 0.860 1.503 1.318 1.631 1.437 1.624 1.684 1.435 1.653 0.607 1.292 1.547 1.652 1.947 0.707 0.772 0.992

i

2.794 76 2.733 77 3.376 78 3.067 79 3.471 80 3.300 81 2.899 82 2.747 83 2.748 84 2.991 85 2.834 86 3.132 87 2.928 88 3.123 89 3.199 90 2.927 91 3.159 92 3.024 93 2.817 94 3.036 95 3.158 96 3.600 97 2.868 98 2.803 99 2.718 100

ui 0.231 0.836 0.167 0.545 0.748 0.440 0.674 0.287 0.052 0.940 0.679 0.910 0.254 0.236 0.854 0.877 0.483 0.117 0.847 0.201 0.294 0.071 0.065 0.740 0.954

ηi = 2ui1/2 eηi /ηi 2.720 0.961 3.404 1.829 2.771 0.817 2.965 1.476 3.260 1.730 2.841 1.327 3.146 1.642 2.725 1.071 3.460 0.456 3.585 1.939 3.153 1.648 3.532 1.908 2.718 1.008 2.719 0.972 3.435 1.848 3.474 1.873 2.888 1.390 2.897 0.684 3.423 1.841 2.734 0.897 2.727 1.084 3.197 0.533 3.266 0.510 3.247 1.720 3.611 1.953 Sum = 311.710 I = 6.234

69

70

2. EVALUATION OF DEFINITE INTEGRALS USING THE MONTE CARLO METHOD

function e x in that it does not follow e x for x > 1. Hence, it is not expected to yield a good estimate of the value of the definite integral. Other values of  yield worse results. We now generate random variable G obeying "   # 1 1 G a 2 p.G/ D p Exp 2   2 with a D 1 and  D 0:3 using the tabulated values of random variable g of Table 1.4 obeying   1 1  g a 2 p.g/ D p Exp 2   2 for a D 0 and  D 1. The connection is G D a C g D 1 C 0:3g . Using Table 2.9 in Microsoft Excel, we have evaluated the integral with the result 5.154 rather than the exact value 6.389. The difference is attributable to the fact that F .x/ and p.x/ used are not proportional to each other for x > 1.

2.4

EVALUATION OF DEFINITE INTEGRALS USING THE MONTE CARLO METHOD: EXAMPLE IV

As Example IV, we now take up the integral I D

Z a

b

F .x/dx D

Z

5

1

loge xdx;

(2.10)

where F .x/ D loge x . We re-write Equation (2.10) as I D

Z

a

b

F .x/ p.x/dx p.x/

(2.11)

which, as discussed in Section 1.7, implies that average value of F=p is the value of the integral, i.e., N 1 X F .xi / I D ; (2.12) N p.xi / i D1 where xi ’s are random values of x in the interval a < x < b obeying probability density function p.x/. We now take up an exponential variation for p.x/ given by p.x/ D C e x where C is a R5 constant. Normalization of p.x/ requires 1 p.x/dx D 1 or, C D 1=145. As such, normalized probability density function is p.x/ D e x =145. See Figure 2.9 in which we have plotted F .x/ D loge x along with the normalized probability density function p.x/ D e x =145. The probability density function p.x/ D e x =145 follows the function loge x well. Hence, it is a good choice.

2.4. EVALUATION OF DEFINITE INTEGRALS: EXAMPLE IV

71

Table 2.9: Showing tabulated values of ui , gi , Gi , e Gi , pi , e Gi =pi from which we get I D G F .xi / 1 PN 1 PN 1 PN e Gi e i    as an approximate value of the i D1 p.xi / D N i D1 pi D N i D1 N G a 2 integral I D h p1 Exp  2

Rb

a F .x/dx
i 1 x a 2 , 2 

D

R2 0



1 p Exp 2

1 2

i 

e x dx D 5:154 whereas the exact value is 6.389. Using p.x/ D

a D 1 and  D 0:3.

i

ui

gi

Gi

eGi

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

0.169 0.050 0.844 0.182 0.599 0.985 0.726 0.148 0.788 0.698 0.010 0.397 0.852 0.761 0.309 0.113 0.212 0.809 0.084 0.276 0.243 0.989 0.508 0.817 0.174

–0.955 –1.635 1.015 –0.905 0.255 2.195 0.605 –1.040 0.805 0.520 –2.295 –0.255 1.050 0.715 –0.495 –1.205 –0.795 0.880 –1.370 –0.590 –0.695 2.325 0.020 0.910 –0.935

0.714 0.510 1.305 0.729 1.077 1.659 1.182 0.688 1.242 1.156 0.312 0.924 1.315 1.215 0.852 0.639 0.762 1.264 0.589 0.823 0.792 1.698 1.006 1.273 0.720

2.041 1.664 3.686 2.072 2.934 5.251 3.259 1.990 3.461 3.177 1.365 2.518 3.725 3.369 2.343 1.894 2.141 3.540 1.802 2.277 2.207 5.460 2.735 3.572 2.053

pi

eGi /pi

0.843 2.422 0.349 4.764 0.794 4.639 0.883 2.347 1.287 2.280 0.120 43.925 1.107 2.943 0.774 2.570 0.962 3.598 1.162 2.735 0.096 14.296 1.287 1.956 0.766 4.861 1.030 3.271 1.176 1.992 0.643 2.943 0.969 2.209 0.903 3.920 0.520 3.464 1.117 2.038 1.044 2.113 0.089 61.269 1.330 2.057 0.879 4.063 0.859 2.391

72

2. EVALUATION OF DEFINITE INTEGRALS USING THE MONTE CARLO METHOD

i

ui

gi

Gi

eGi

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

0.225 0.687 0.060 0.070 0.835 0.466 0.271 0.813 0.376 0.983 0.181 0.583 0.612 0.468 0.614 0.768 0.811 0.461 0.985 0.735 0.559 0.884 0.229 0.920 0.524

–0.750 0.490 –1.540 –1.465 0.980 –0.085 –0.605 0.895 –0.315 2.145 –0.905 0.210 0.285 –0.080 0.290 0.735 0.885 –0.095 2.195 0.630 0.150 1.200 –0.740 1.415 0.060

0.775 1.147 0.538 0.561 1.294 0.975 0.819 1.269 0.906 1.644 0.729 1.063 1.086 0.976 1.087 1.221 1.266 0.972 1.659 1.189 1.045 1.360 0.778 1.425 1.018

2.171 3.149 1.713 1.752 3.647 2.650 2.267 3.556 2.473 5.173 2.072 2.895 2.961 2.654 2.965 3.389 3.545 2.642 5.251 3.284 2.843 3.896 2.177 4.156 2.768

pi

eGi /pi

1.004 2.162 1.179 2.670 0.406 4.215 0.455 3.852 0.823 4.433 1.325 2.000 1.107 2.047 0.891 3.991 1.265 1.954 0.133 38.822 0.883 2.347 1.301 2.226 1.277 2.319 1.326 2.002 1.275 2.326 1.015 3.339 0.899 3.944 1.324 1.996 0.120 43.925 1.090 3.011 1.315 2.162 0.647 6.019 1.011 2.153 0.489 8.504 1.327 2.085

2.4. EVALUATION OF DEFINITE INTEGRALS: EXAMPLE IV

i

ui

gi

Gi

eGi

51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75

0.393 0.306 0.819 0.620 0.875 0.773 0.492 0.332 0.185 0.565 0.434 0.665 0.516 0.659 0.709 0.515 0.683 0.092 0.417 0.598 0.682 0.948 0.125 0.149 0.246

–0.270 –0.505 0.915 0.310 1.155 0.750 –0.020 –0.430 –0.890 0.165 –0.165 0.430 0.040 0.410 0.555 0.040 0.480 –1.320 –0.205 0.250 0.475 1.635 –1.145 –1.035 –0.685

0.919 0.849 1.275 1.093 1.347 1.225 0.994 0.871 0.733 1.050 0.951 1.129 1.012 1.123 1.167 1.012 1.144 0.604 0.939 1.075 1.143 1.491 0.657 0.690 0.795

2.507 2.336 3.577 2.983 3.844 3.404 2.702 2.389 2.081 2.856 2.587 3.093 2.751 3.074 3.211 2.751 3.139 1.829 2.556 2.930 3.135 4.439 1.928 1.993 2.213

pi

eGi /pi

1.282 1.955 1.171 1.996 0.875 4.088 1.267 2.354 0.683 5.632 1.004 3.391 1.330 2.032 1.212 1.971 0.895 2.326 1.312 2.177 1.312 1.972 1.212 2.551 1.329 2.070 1.223 2.514 1.140 2.816 1.329 2.070 1.185 2.649 0.556 3.288 1.302 1.963 1.289 2.273 1.188 2.639 0.349 12.706 0.690 2.793 0.778 2.560 1.052 2.105

73

74

2. EVALUATION OF DEFINITE INTEGRALS USING THE MONTE CARLO METHOD

i

ui

gi

Gi

eGi

76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

0.231 0.836 0.167 0.545 0.748 0.440 0.674 0.287 0.052 0.940 0.679 0.910 0.254 0.236 0.854 0.877 0.483 0.117 0.847 0.201 0.294 0.071 0.065 0.740 0.954

–0.730 0.985 –0.960 0.115 0.670 –0.150 0.455 –0.560 –1.615 1.565 0.470 1.350 –0.660 –0.715 1.060 1.165 –0.040 –1.185 1.030 –0.835 –0.540 –1.460 –1.505 0.645 1.695

0.781 1.296 0.712 1.035 1.201 0.955 1.137 0.832 0.516 1.470 1.141 1.405 0.802 0.786 1.318 1.350 0.988 0.645 1.309 0.750 0.838 0.562 0.549 1.194 1.509

2.184 3.653 2.038 2.814 3.323 2.599 3.116 2.298 1.674 4.347 3.130 4.076 2.230 2.194 3.736 3.855 2.686 1.905 3.702 2.116 2.312 1.754 1.731 3.299 4.520

pi

eGi /pi

1.019 2.143 0.819 4.462 0.839 2.430 1.321 2.130 1.062 3.128 1.315 1.976 1.199 2.599 1.137 2.021 0.361 4.639 0.391 11.124 1.191 2.629 0.535 7.623 1.070 2.085 1.030 2.130 0.758 4.927 0.675 5.715 1.329 2.021 0.659 2.891 0.782 4.732 0.938 2.255 1.149 2.011 0.458 3.830 0.428 4.039 1.080 3.054 0.316 14.296 Sum = 515.351 I = 5.154

2.5. EVALUATION OF DEFINITE INTEGRALS: EXAMPLE V

75

F(x) = Log(x) or p(x)

1.5

1.0

0.5

0.0 1

2

3

4

x

5

Figure 2.9: Showing F .x/ D loge x as undashed curve and p.x/ D e x =145 as dashed curve obtained using the command a=1/145; p1=Plot[Log[E,x],{x,1,5},Frame->True, FrameLabel->{"x","F(x) = Log(x) or p(x)"},PlotStyle->{Black}]; p2=Plot[a*Exp[x],{x,1,5},PlotStyle->{Dashed,Black},Frame->True, FrameLabel->{"x","F(x) = e^x or Gaussian (x)"}];Show[p1,p2] in Mathematica 6.0. We now generate random variable  obeying p.x/ D e x =145 in the interval 1 < x < 5 using Equation (1.2): Z  Z  p.x/dx D u or, p.x/dx D u; a

1

where u is random variable (of Table 1.1) of uniform probability density function in the interval 0 < x < 1. Thus, we have  given by  D loge .145u C 2:72/. Using  D loge .145u C 2:72/ in Microsoft Excel we have Table 2.10 in which we have evaluated the integral with the result 3.915 rather than 4.05. The difference is attributable to the fact that F .x/ and p.x/ used are not proportional to each other.

2.5

EVALUATION OF DEFINITE INTEGRALS USING THE MONTE CARLO METHOD: EXAMPLE V

As Example V, we now take up the integral I D

Z a

b

F .x/dx D

Z 0

3

1 dx; 1 C x2

(2.13)

76

2. EVALUATION OF DEFINITE INTEGRALS USING THE MONTE CARLO METHOD

Table 2.10: Showing tabulated values of ui , i D loge .145ui C 2:72/, loge i =e  i from which P loge i loge i F .xi / 1 PN 145 PN we get I D N1 N as an approximate value of iD1 p.xi / D N i D1 e i =145 D N i D1 e i R5 Rb the integral I D a F .x/dx D 1 loge xdx D 3:915 whereas the exact value is 4.045. Using p.x/ D e x =145. i

ui

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

0.169 0.050 0.844 0.182 0.599 0.985 0.726 0.148 0.788 0.698 0.010 0.397 0.852 0.761 0.309 0.113 0.212 0.809 0.084 0.276 0.243 0.989 0.508 0.817 0.174

Loge ηi ηi = Loge (145ui + 2.72) — e ηi 0.046 3.237 0.091 2.109 0.013 4.812 0.044 3.308 0.017 4.473 0.011 4.965 0.015 4.663 0.051 3.110 0.014 4.744 0.015 4.624 –0.045 0.896 0.024 4.068 0.013 4.822 0.014 4.710 0.029 3.822 0.060 2.854 0.039 3.455 0.013 4.770 0.072 2.577 0.032 3.712 0.035 3.588 0.011 4.969 0.020 4.310 0.013 4.780 0.045 3.265

i

ui

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

0.225 0.687 0.060 0.070 0.835 0.466 0.271 0.813 0.376 0.983 0.181 0.583 0.612 0.468 0.614 0.768 0.811 0.461 0.985 0.735 0.559 0.884 0.229 0.920 0.524

Loge ηi ηi = Loge (145ui + 2.72) — e ηi 0.037 3.513 0.015 4.608 0.085 2.271 0.079 2.410 0.013 4.802 0.021 4.225 0.032 3.694 0.013 4.775 0.025 4.014 0.011 4.963 0.044 3.303 0.017 4.446 0.017 4.494 0.021 4.229 0.017 4.497 0.014 4.719 0.013 4.773 0.021 4.215 0.011 4.965 0.014 4.675 0.018 4.405 0.012 4.858 0.037 3.530 0.012 4.898 0.019 4.341

2.5. EVALUATION OF DEFINITE INTEGRALS: EXAMPLE V

i

ui

51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75

0.393 0.306 0.819 0.620 0.875 0.773 0.492 0.332 0.185 0.565 0.434 0.665 0.516 0.659 0.709 0.515 0.683 0.092 0.417 0.598 0.682 0.948 0.125 0.149 0.246

Loge ηi i ηi = Loge (145ui + 2.72) — e ηi 76 0.024 4.058 77 0.030 3.812 78 0.013 4.782 79 0.017 4.507 80 0.012 4.848 81 0.014 4.725 82 0.020 4.279 83 0.028 3.892 84 0.043 3.324 85 0.018 4.415 86 0.022 4.155 87 0.016 4.576 88 0.019 4.326 89 0.016 4.567 90 0.015 4.640 91 0.019 4.324 92 0.015 4.603 93 0.068 2.661 94 0.023 4.116 95 0.017 4.471 96 0.015 4.601 97 0.012 4.927 98 0.057 2.949 99 0.050 3.116 0.035 100 3.600

ui 0.231 0.836 0.167 0.545 0.748 0.440 0.674 0.287 0.052 0.940 0.679 0.910 0.254 0.236 0.854 0.877 0.483 0.117 0.847 0.201 0.294 0.071 0.065 0.740 0.954

Loge ηi ηi = Loge (145ui + 2.72) — e ηi 0.037 3.539 0.013 4.803 0.047 3.225 0.018 4.380 0.014 4.693 0.022 4.169 0.015 4.589 0.031 3.750 0.089 2.143 0.012 4.919 0.015 4.597 0.012 4.887 0.034 3.631 0.036 3.559 0.013 4.824 0.012 4.850 0.020 4.260 0.059 2.887 0.013 4.816 0.041 3.404 0.030 3.773 0.078 2.423 0.082 2.343 0.014 4.682 0.011 4.934 Sum = 2.700 I = 3.915

77

78

2. EVALUATION OF DEFINITE INTEGRALS USING THE MONTE CARLO METHOD

F(x) = 1/(1 + x 2) or p(x)

1.0 0.8 0.6 0.4 0.2 0.0 0.0

0.5

1.0

1.5

x

2.0

2.5

3.0

1 x Figure 2.10: Showing F .x/ D 1Cx as dashed curve 2 as undashed curve and p.x/ D 1:05e obtained using the command p1=Plot[1/(1+x^2),{x,0,3},PlotRange->{0,1},Frame->True, FrameLabel-> {"x","F(x) = 1/(1+x^2) or p(x)"},PlotStyle->{Black}]; p2=Plot[1.05*Exp[-x],{x,0,3},PlotStyle->{Dashed,Black}]; Show[p1,p2] in Mathematica 6.0.

where F .x/ D

1 . 1Cx 2

We re-write Equation (2.13) as I D

Z a

b

F .x/ p.x/dx p.x/

(2.14)

which, as discussed in Section 1.7, implies that average value of F=p is the value of the integral, i.e., N 1 X F .xi / ; I D N p.xi /

(2.15)

i D1

where xi ’s are random values of x in the interval a < x < b obeying probability density function p.x/. We now take up an exponential variation for p.x/ given by p.x/ D C e x where C is R3 a constant. Normalization of p.x/ requires 0 p.x/dx D 1 or, C D 1:05. As such normalized probability density function is p.x/ D 1:05e x . See Figure 2.10 in which we have plot1 x . The ted F .x/ D 1Cx 2 along with the normalized probability density function p.x/ D 1:05e 1 x probability density function p.x/ D 1:05e follows the function 1Cx 2 quite well. Hence, it is a very good choice.

2.5. EVALUATION OF DEFINITE INTEGRALS: EXAMPLE V

79

x

We now generate random variable  obeying p.x/ D 1:05e in the interval 0 < x < 3 using Equation (1.2): Z  Z  p.x/dx D u or, p.x/dx D u; a

0

where u is random variable (of Table 1.1) of uniform probability density function in the interval 0 < x < 1. Thus, we have  given by  D loge .1 0:95u/. Using  D loge .1 0:95u/ in Microsoft Excel we have Table 2.11 in which we have evaluated the integral with the result 1.251 while the exact value is also 1.249. The absence of difference is attributable to the fact that F .x/ and p.x/ used are almost proportional to each other.

80

2. EVALUATION OF DEFINITE INTEGRALS USING THE MONTE CARLO METHOD

Table 2.11: Showing tabulated values of ui , i D loge .1 0:95ui /, e i =.1 C 2i / from which P 1=.1Ci2 / F .xi / 1 PN 0:95 PN e i we get I D N1 N i D1 p.xi / D N i D1 1:05e i D N i D1 .1C 2 / as an approximate value i Rb R3 1 of the integral I D a F .x/dx D 0 1Cx 2 dx D 1:251 whereas the exact value is 1.249. Using p.x/ D 1:05e x . i

ui

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

0.169 0.050 0.844 0.182 0.599 0.985 0.726 0.148 0.788 0.698 0.010 0.397 0.852 0.761 0.309 0.113 0.212 0.809 0.084 0.276 0.243 0.989 0.508 0.817 0.174

e ηi ηi = –Loge (1 – 0.95ui ) — i (1 + ηi 2) 0.175 0.049 1.617 0.190 0.841 2.743 1.169 0.151 1.380 1.087 0.010 0.473 1.657 1.283 0.347 0.113 0.225 1.462 0.083 0.304 0.262 2.804 0.659 1.496 0.181

1.156 1.047 1.394 1.167 1.358 1.822 1.360 1.137 1.369 1.359 1.009 1.311 1.400 1.363 1.263 1.106 1.192 1.375 1.079 1.241 1.216 1.863 1.348 1.379 1.160

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

ui 0.225 0.687 0.060 0.070 0.835 0.466 0.271 0.813 0.376 0.983 0.181 0.583 0.612 0.468 0.614 0.768 0.811 0.461 0.985 0.735 0.559 0.884 0.229 0.920 0.524

e ηi ηi = –Loge (1 – 0.95ui ) — (1 + ηi 2) 0.240 1.057 0.059 0.069 1.575 0.584 0.297 1.479 0.442 2.714 0.189 0.807 0.870 0.588 0.875 1.307 1.471 0.576 2.743 1.197 0.757 1.830 0.245 2.070 0.688

1.202 1.359 1.057 1.066 1.388 1.337 1.237 1.377 1.301 1.804 1.166 1.357 1.359 1.338 1.359 1.364 1.376 1.336 1.822 1.361 1.355 1.433 1.205 1.500 1.351

2.5. EVALUATION OF DEFINITE INTEGRALS: EXAMPLE V

i

ui

51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75

0.393 0.306 0.819 0.620 0.875 0.773 0.492 0.332 0.185 0.565 0.434 0.665 0.516 0.659 0.709 0.515 0.683 0.092 0.417 0.598 0.682 0.948 0.125 0.149 0.246

e ηi ηi = –Loge (1 – 0.95ui ) — (1 + ηi 2) 0.467 0.343 1.504 0.889 1.778 1.325 0.630 0.379 0.193 0.769 0.531 0.998 0.673 0.983 1.119 0.671 1.046 0.091 0.504 0.839 1.043 2.307 0.126 0.153 0.266

1.310 1.261 1.379 1.359 1.422 1.365 1.344 1.277 1.169 1.356 1.327 1.359 1.349 1.359 1.359 1.349 1.359 1.087 1.320 1.358 1.359 1.589 1.117 1.138 1.219

i

ui

76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

0.231 0.836 0.167 0.545 0.748 0.440 0.674 0.287 0.052 0.940 0.679 0.910 0.254 0.236 0.854 0.877 0.483 0.117 0.847 0.201 0.294 0.071 0.065 0.740 0.954

e ηi ηi = –Loge (1 – 0.95ui ) — (1 + ηi 2) 0.248 1.580 0.173 0.729 1.239 0.541 1.022 0.318 0.051 2.234 1.035 1.998 0.276 0.254 1.667 1.790 0.614 0.118 1.632 0.212 0.327 0.070 0.064 1.213 2.366

1.207 1.389 1.154 1.354 1.362 1.329 1.359 1.248 1.049 1.558 1.359 1.477 1.225 1.211 1.401 1.425 1.342 1.110 1.396 1.183 1.253 1.067 1.061 1.361 1.615 Sum = 131.670 I = 1.251

81

83

CHAPTER

3

Variational Monte Carlo Method Applied to the Ground State of a Simple Harmonic Oscillator 3.1

THE VARIATIONAL METHOD OF QUANTUM MECHANICS APPLIED TO THE GROUND STATE OF ANY QUANTUM MECHANICAL SYSTEM

Let ‰0 , ‰1 , ‰2 , ‰3 ; : : : ; ‰N be exact, unknown, orthonormal eigenfunctions of known Hamiltonian H . Let E0 < E1 < E2 < E3 < : : : < EN be exact, unknown eigenvalues of energy, respectively. That is, we have eigenvalue equation H ‰n D En ‰n

(3.1)

which we cannot solve. Suppose we wish to estimate E0 and we wish to know approximate form of ‰0 . Let us choose by guess a suitable function ˆ0 that is expected to resemble unknown ‰0 and write N X Cn.0/ ‰n (3.2) ˆ0 D nD0

and let us evaluate the quantity Eˆ0 given by

84

3. VARIATIONAL MONTE CARLO METHOD

R  ˆ Hˆ0 d  .ˆ0 ; Hˆ0 / D R 0 .ˆ0 ; ˆ0 / ˆ0 ˆ0 d   P P .0/ .0/ C ‰ ; H C ‰ m n m n mD0 nD0  P P .0/ .0/ mD0 Cm ‰m ; nD0 Cn ‰n P  P .0/ .0/ C ‰ ; C H ‰ m n m n mD0 nD0 P  P .0/ .0/ C ‰ ; C ‰ m n mD0 m nD0 n  P P .0/ .0/ mD0 Cm ‰m ; nD0 Cn En ‰n P  P .0/ .0/ C ‰ ; C ‰ m n mD0 m nD0 n

Eˆ0 D D

D

D

(3.3)

using Equation (3.1) D

P

mD0

P

P

nD0

P

Cm.0/ Cn.0/ En .‰m ; ‰n /

Cm.0/ Cn.0/ .‰m ; ‰n / ˇ ˇ P ˇ .0/ ˇ2 P P .0/ .0/ C ˇ ˇ En n nD0 mD0 nD0 Cm Cn En ımn D P D ˇ ˇ : P .0/ .0/ P ˇ .0/ ˇ2 mD0 nD0 Cm Cn ımn C ˇ ˇ n nD0

Thus,

mD0

nD0

ˇ ˇ ˇ .0/ ˇ2 ˇCn ˇ .En E0 / E0 D : (3.4) Eˆ0 E0 D ˇ ˇ P ˇ .0/ ˇ2 nD0 ˇCn ˇ ˇ ˇ2 ˇ ˇ In Equation (3.4), En  E0 . ) En E0  0; moreover, ˇCn.0/ ˇ  0. Thus, Equation (3.4) shows that Eˆ0 E0  0 or, Eˆ0  E0 . We find that for arbitrary ˆ0 , calculated value of Eˆ0  E0 i.e., Eˆ0 is larger than or equal to ground state energy E0 . Eˆ0 gives upper bound of ground state energy E0 . If chosen ˆ0 is different from actual unknown ‰0 , calculated value of Eˆ0 calculated using Equation (3.3) will be larger than E0 . If chosen ˆ0 happens to be same as unknown ‰0 , value of Eˆ0 calculated using Equation (3.3) becomes equal to E0 . Thus, we can choose arbitrary ˆ0 and calculate Eˆ0 using Equation (3.3) and get upper bound of E0 . Usually we choose a suitable ˆ0 known as trail function containing one or more adjustable parameters ˛i.0/ ’s, e.g., ˇ ˇ ˇ .0/ ˇ2 ˇCn ˇ En ˇ ˇ P ˇ .0/ ˇ2 nD0 ˇCn ˇ

P

P

nD0

ˆ0 D e

nD0

.0/

˛1 x

;

3.2. THE VARIATIONAL METHOD OF QUANTUM MECHANICS

85

˛1.0/

where is an adjustable parameter. Then we use this expression of ˆ0 in Equation (3.3) and @E 0 calculated expression of Eˆ0 will contain the parameters ˛i.0/ ’s. We then stipulate ˆ .0/ D 0 for @˛i

˛i.0/ ,

˛i.0/ ’s

each separately. We thus get values of for which Eˆ0 is minimum. We can put these .0/ values of ˛i ’s in the expression of Eˆ0 . We then get the lowest value of Eˆ0 for the chosen ˆ0 . This upper bound to E0 will be close to actual unknown value of E0 if trail function ˆ0 has a form closely resembling that of actual unknown ‰0 . The entire process is repeated for several different functions as ˆ0 to see which function leads to much lower value of Eˆ0 . If we are happy that a calculated value of Eˆ0 is low enough and that it is likely to be close to actual unknown value of E0 , we can take E0  Eˆ0 and ‰0  ˆ0 within some approximation.

3.2

THE VARIATIONAL METHOD OF QUANTUM MECHANICS APPLIED TO THE GROUND STATE OF A SIMPLE HARMONIC OSCILLATOR

Suppose we know Hamiltonian operator H D

1 „2 d 2 C m!c2 x 2 2 2m dx 2

of a simple harmonic oscillator and we do not know eigenfunctions ‰n and eigenvalues En of H , because suppose we cannot solve eigenvalue equation H ‰n D En ‰n of the operator H . Suppose we wish to estimate ground state energy E0 and wish to know approximate form of corresponding eigenfunction ‰0 . 2 Let us choose ˆ0 D e ˛x by guessing as an approximate form of ‰0 and let us evaluate the quantity .ˆ0 ; Hˆ0 / Eˆ0 D : (3.5) .ˆ0 ; ˆ0 / Now Z C/   Z C/ 2 2 2 2 2 .ˆ0 ; ˆ0 / D e ˛x ; e ˛x D e ˛x e ˛x dx D e 2˛x dx / / Z Z Z C/ 1 C/ 1 2˛x 2 2 1 C/ 1 2˛x 2 2xdx D dx D x e x e x 1e D 2 / 2 / 0 Z C/ Z C/
1=2 1 2˛x 2 2
1=2 2˛x 2 2 dx D .2˛/ dx D x2 e D x2 e 0 0 r Z C/ p  1=2 D .2˛/ using x n 1 e ax dx D a n €n; D 2˛ 0

where € stands for a well-known Gamma function.

2˛x 2

1=2

dx 2

€.1=2/

86

3. VARIATIONAL MONTE CARLO METHOD

Thus, .ˆ0 ; ˆ0 / D

We also note that

Z

C/

e

2˛x 2

0

r

 : 2˛

1 dx D 2

(3.6)

r

 : 2˛

(3.7)

We now turn to .ˆ0 ; Hˆ0 /. H D T C V where T is operator of kinetic energy and V is operator of potential energy.  .ˆ0 ; Hˆ0 / D .ˆ0 ; .T C V /ˆ0 / D ˆ0 ;

 „2 d 2 ˆ0 1 2 2 C m!c x ˆ0 : 2m dx 2 2

(3.8)

Now     1 2 2 2 2 ˛x 2 ˛x 2 1 .ˆ0 ; V ˆ0 / D ˆ0 ; m!c x ˆ0 D e ; m!c x e 2 2 Z C/ Z C/   1 1 2 ˛x 2 ˛x 2 2 2 2 2 2˛x 2 D m!c ;x e dx D m!c x2e D m!c e x e 2 2 0 / Z C/ Z C/

1=2 2˛x 2 1 1 2 2˛x 2 2 D m!c 2xdx D m!c d x2 xe x2 e 2 2 0 0 Z C/

1 1 2 3=2 1 D m!c2 x2 e 2˛x d x 2 D m!c2 .2˛/ 3=2 €.3=2/ 2 2 0 r p 1 1 m!c2  D m!c2 .2˛/ 3=2 .1=2/€.1=2/ D m!c2 .2˛/ 3=2 .1=2/  D ; 2 2 8˛ 2˛

2˛x 2

dx

using Z

C/

xn

1

e

ax

0

dx D a

n

€n

and

Thus, m!c2 .ˆ0 ; V ˆ0 / D 8˛

We also note that

Z 0

C/

2

x e

2˛x 2

r

1 dx D 8˛

€.n C 1/ D n€n:

 : 2˛ r

 : 2˛

(3.9)

(3.10)

3.2. THE VARIATIONAL METHOD OF QUANTUM MECHANICS

87

Now  .ˆ0 ; T ˆ0 / D ˆ0 ;  D e

D D D D D D

˛x 2

„2 d 2 ˆ 0 2m dx 2 ;



„2 d 2 e 2m dx 2

˛x 2



 „2 e 2m

D

˛x 2

d2 ; 2e dx

˛x 2



Z Z „2 C/ ˛x 2 d 2 ˛x 2 „2 C/ ˛x 2 d 2 ˛x 2 e e e e dx D dx 2m / dx 2 m 0 dx 2 Z  „2 C/ ˛x 2 d  2 e 2˛xe ˛x dx m 0 dx   2 Z C/ „ 2 2 2 e ˛x 2˛e ˛x 2˛x. 2˛x/e ˛x dx m 0 Z  „2 C/ ˛x 2  2 2 e 2˛e ˛x C 4˛ 2 x 2 e ˛x dx m 0   Z C/ Z C/ „2 2 2 2˛x 2 2˛x 2 dx 2˛ dx C 4˛ x e e m 0 0 r r  r r  „2 ˛i  „2 ˛  1   „2 h 2 1 C 4˛ ˛C D ; 2˛ D m 2 2˛ 8˛ 2˛ m 2 2˛ 2m 2˛

using Equations (3.7) and (3.10). Thus, „2 ˛ .ˆ0 ; T ˆ0 / D 2m

r

 : 2˛

(3.11)

From Equations (3.11) and (3.9), we gather that  2 r „ ˛ m!c2  .ˆ0 ; Hˆ0 / D .ˆ0 ; .T C V /ˆ0 / D C : 2m 8˛ 2˛ Using Equations (3.12) and (3.6) in Equation (3.5), we get  2 r „ ˛ m!c2  C .ˆ0 ; Hˆ0 / „2 ˛ m!c2 2m 8˛ 2˛ D C : Eˆ0 D D r .ˆ0 ; ˆ0 / 2m 8˛  2˛

Eˆ0

We now stipulate that for

@Eˆ0 @˛

D 0. Hence, ˛D

„2 2m

m!c 2„

m!c2 8˛ 2

(3.12)

(3.13)

D 0. Thus, we get lowest value of

(3.14)

88

3. VARIATIONAL MONTE CARLO METHOD

for the chosen ˆ0 .D e of

˛x 2

). Using Equation (3.14) in Equation (3.13), we get the lowest value

Eˆ0 D

m!c2 „2 m!c m! 2 „2 ˛ C D C m!cc : 2m 8˛ 2m 2„ 8 2„

We take ground state energy as E0  Eˆ0 D

1 „!c : 2

(3.15)

This incidentally agrees exactly with known value of ground state energy E0 of simple harmonic 2 oscillator. The ground state eigenfunction ˆ0 D e ˛x becomes ‰0  ˆ0 D e

1 m!c 2 „

x2

(3.16)

which incidentally agrees exactly with known ground state eigenfunction ‰0 of energy of simple harmonic oscillator. Thus, to some extent, we have become familiar with use of the variational method.

3.3

GROUND STATE ENERGY OF A SIMPLE HARMONIC OSCILLATOR USING THE MONTE CARLO METHOD

We consider an electron residing in a simple harmonic oscillator potential V .x/ D

1 1 m!c2 x 2 D kx 2 : 2 2

(3.17)

We first gather exact results from Quantum Mechanics books. Eigenvalue equation of energy is H ‰ D E‰ , or   1 „2 d 2 2 2 C m! x ‰ D E‰: (3.18) c 2m dx 2 2 Eigenvalues of energy are En D .n C 1=2/„!c ;

(3.19)

where n D 0; 1; 2; 3; : : :. Ground state energy in unit of „!c is E0 =.„!c / D 0:5 and ground state eigenfunction is 1 m!c 2 1 2 ‰0 D e 2 „ x D e 2 Lx ; (3.20) where

m!c : (3.21) „ We now gather results from variational method of Quantum Mechanics detailed in Section 3.2: for trial wavefunction 2 2 ˆ0 D e ˛x D e ˇLx ; LD

3.3. GROUND STATE ENERGY OF A SIMPLE HARMONIC OSCILLATOR

89

ground state energy estimated or evaluated using Equation (3.5): .ˆ0 ; Hˆ0 / .ˆ0 ; ˆ0 /

Eˆ0 D

is E0 =.„!c / D 0:5 and value of ˇ turns out to be 0.5 so that ground state eigenfunction is ˆ0 D e

˛x 2

ˇLx 2

De

1 2 2 Lx

De

(3.22)

:

We now move onto variational Monte Carlo method of evaluating ground state eigenvalue and eigenfunction of energy. We have Hamiltonian operator „2 d 2 1 C m!c2 x 2 2m dx 2 2

H D

or, H D ˛x 2

which together with trial wavefunction ˆ0 D e Hˆ0 D He

˛x 2

D

„2 2m

„2 d 2 1 C kx 2 2m dx 2 2

lead to


2˛ C 4˛ 2 x 2 e

˛x 2

1 C kx 2 e 2

˛x 2

or, Hˆ0 D ˆ0

„2 2m


1 2˛ C 4˛ 2 x 2 C kx 2 2

(3.23)

and ground state energy in unit of „!c as E0 .ˆ0 ; Hˆ0 / D D „!c „!c .ˆ0 ; ˆ0 / D

Z

F .x/dx D

D< f >D
: „!c ˆ0

Thus, average or the so-called expectation value of energy in unit of „!c . Here,

(3.24)

1 Hˆ0 „!c ˆ0

is a good estimate of ground state

1 Hˆ0 „! ˆ0 rc 2˛ 2 ˆ p.x/ D  0

(3.25)

f D

and F .x/ D



1 Hˆ0 „!c ˆ0

(3.26)  r

! 2˛ 2 ˆ :  0

(3.27)

90

3. VARIATIONAL MONTE CARLO METHOD

Looking at Equation (3.24), ground state energy in unit of „!c is
1 E0 1 Hˆ0 1 „2 2˛ C 4˛ 2 x 2 C kx 2 > D< >D < „!c „!c ˆ0 „!c 2m 2
1 k „ x2 > 2˛ C 4˛ 2 x 2 C D< 2m!c 2 „!c

using Equation (3.23), or E0 D< „!c

using

k „!c


L 2˛ C 4˛ 2 x 2 C x 2 > 2

1 2L

D L obtained as follows: L2 D

m2 !c2 ; „2

L2 D

m2 k ; „2 m

see Equation (3.21), or

see Equation (3.17), or

(3.28)

p mk LD : „

Again,

p r k m mk k D D L: D „!c „ k „ c In Equation (3.28), ˛ D ˇL and L D m! . We note that ground state energy is given by „ E0 D< „!c

1 2L


L 2˛ C 4˛ 2 xi2 C xi2 >; 2

(3.29)

where xi ’s are random values of x obeying the Gaussian probability density function p.x/ given by Equation (3.26). Before we proceed to evaluate ground state energy using the prescription of Equation (3.29), we present expressions for F .x/ and p.x/ and plot them together to ascertain that they follow each other. As demonstrated in Section 1.7, p.x/ at least should follow F .x/, if not be proportional to F .x/ to get good result. According to Equation (3.26), r 2˛ 2˛x 2 (3.30) p.x/ D e  which is Gaussian function of x given by p.x/ D

1 p e  2

1 2

2

. x a / :

(3.31)

3.3. GROUND STATE ENERGY OF A SIMPLE HARMONIC OSCILLATOR

91

F(x) or Gaussian(x)

1.0 0.8 0.6 0.4 0.2 0.0 –15

–10

–5

0

5

10

15

L 2 x 2


q 2˛

2˛x 2

x

Figure 3.1: Showing F .x/ D

1 . 2L

2 2

2˛ C 4˛ x / C h as undashed curve and Gaussian p.x/ D p1 Exp 

2

1 2



x 


i a 2

e



of Equation (3.33)

of Equation (3.31) as dashed

curve obtained using Program 3.1. For chosen value of spring constant k D 10 18 that correc D 9:046  109 , ˇ D 0:8, ˛ D ˇL D 7:237  109 , a D 0, sponds to !c D 1.048 MHz, L D m! „  D 12 ˛ 1=2 D 5:878  10 6 . (Here x is in unit of 10 6 and F .x/ and p.x/ are in unit of 10 5 ). Comparison between Equations (3.30) and (3.31) shows that a D 0 and D

1 ˛ 2

1=2

(3.32)

:

And, according to Equation (3.27), F .x/ D



1 Hˆ0 „!c ˆ0

 r

2˛ 2 ˆ  0

!

or F .x/ D



1 2L

2˛ C 4˛ 2 x


2

L C x2 2

 r

2˛ e 

2˛x 2

!

(3.33)

using Equation (3.28). Using Program 3.1 in Mathematica 6.0, we plotted Figure 3.1 to ascertain that F .x/ and p.x/ follow each other. We have chosen “spring constant” k D 10 18 that corresponds to !c D 1:048 MHz. We find that F .x/ and p.x/ follow each other very well and, hence, Equation (3.29) E0 will give a good estimate of „! . c

92

3. VARIATIONAL MONTE CARLO METHOD

Program 3.1

beta=0.8; k=10^-18; m=9.1*10^-31; hcut=(6.626*10^-34)/(2*3.1416); omega=Sqrt[k/m] L=m*omega/hcut alpha=beta*L; a=0; sig=0.5*((beta*L)^(-0.5)) F=Plot[(10^-5)*((-1/(2*L))*(-2*alpha+4*alpha^2*(x*(10^-6))^2) +0.5*L*(x*(10^-6))^2)*(Sqrt[2*alpha/3.1416])* Exp[-2*alpha*(x*(10^-6))^2], {x,-3*sig/(10^-6), +3*sig/(10^-6)},PlotStyle->{Black},Frame->True, FrameLabel->{"x","F(x) or Gaussian (x)"},PlotRange->{0,1}]; p=Plot[(10^-5)*(Sqrt[2*alpha/3.1416])*Exp[-2*alpha*(x*(10^-6))^2], {x,-3*sig/(10^-6),+3*sig/(10^-6)},PlotStyle->{Dashed,Black}, PlotRange->{0,1}]; Show[F,p]

We now generate random variables G obeying 1 p.G/ D p Exp  2

with a D 0 and  D 5:878  10 obeying

6

"

1 2



G

a

2 #



using the tabulated values of random variable g of Table 1.4

p.g/ D

1 p Exp  2



1  g a 2 2 



for a D 0 and  D 1. The connection is G D a C g D 0 C 5:878  10 6 g .

3.3. GROUND STATE ENERGY OF A SIMPLE HARMONIC OSCILLATOR

93

1.4 1.2

E 0 /(ћwc)

1.0 0.8 0.6 0.4 0.2 0.0 0.0

0.5

1.0

1.5

2.0

β

Figure 3.2: Showing values of ground state energy E0 =.„!c / of an electron in simple harmonic oscillator potential V .x/ D 12 kx 2 with k D 10 18 , for various values of variational parameter m!c 2 2 ˇ contained in trial wavefunction ˆ0 D e ˇLx D e ˇ „ x . Minimum value of E0 =.„!c / D 1 2 2 0:5 occurs for ˇ D 0:5. Thus, the ground state eigenfunction is ˆ0 D e ˇLx D e 2 Lx D 1 m!c 2 e 2 „ x while ground state eigenvalue of energy is E0 D 0:5„!c . Using Table 3.1 in Microsoft Excel, we have evaluated the expectation value set forth in Equation (3.29)
L E0 1 D< 2˛ C 4˛ 2 xi2 C xi2 > „!c 2L 2 as a value of the ground state energy. We have run Program 3.1 and executed the Microsoft Excel spreadsheet for Table 3.1 for various values of ˇ namely 0.1–2.0 and obtained values of E0 =.„!c / in each case. Table 3.2 shows the results which are plotted in Figure 3.2. We find from Figure 3.2 that minimum value 2 of E0 =.„!c / D 0:5 occurs for ˇ D 0:5. Thus, the ground state eigenfunction is ˆ0 D e ˇLx D m! 1 1 c 2 2 e 2 Lx D e 2 „ x while ground state eigenvalue of energy is E0 D 0:5„!c . We have repeated the run with Program 3.1 with k D 10 16 . Figure 3.3 shows that F .x/ and p.x/ follow each other very well and hence good results are expected. We use ˇ D 0:8, !c D 10:483  106 , L D 9:046  1010 , ˛ D ˇL D 7:237  1010 , a D 0,  D 1:859  10 6 . We now generate random variables G obeying 1 p.G/ D p Exp  2

"

1 2



G

a 

2 #

94

3. VARIATIONAL MONTE CARLO METHOD


1 Table 3.1: Showing tabulated values of ui , gi , Gi , 2L 2˛ C 4˛ 2 Gi2 C L2 Gi2 from which we
E0 1 get „! D< 2L 2˛ C 4˛ 2 Gi2 C L2 Gi2 > : Using a D 0;  D 5:878  10 6 , ˇ D 0:8; L D c 9:046  109 ; ˛ D ˇL.

i

ui

gi

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

0.169 0.050 0.844 0.182 0.599 0.985 0.726 0.148 0.788 0.698 0.010 0.397 0.852 0.761 0.309 0.113 0.212 0.809 0.084 0.276 0.243 0.989 0.508 0.817 0.174

–0.955 –1.635 1.015 –0.905 0.255 2.195 0.605 –1.040 0.805 0.520 –2.295 –0.255 1.050 0.715 –0.495 –1.205 –0.795 0.880 –1.370 –0.590 –0.695 2.325 0.020 0.910 –0.935

1 (–2α + 4α2G 2) + – LG 2 Gi = a + σ g i – — i i 2L

–5.613E–06 –9.611E–06 5.966E–06 –5.320E–06 1.499E–06 1.290E–05 3.556E–06 –6.113E–06 4.732E–06 3.057E–06 –1.349E–05 –1.499E–06 6.172E–06 4.203E–06 –2.910E–06 –7.083E–06 –4.673E–06 5.173E–06 –8.053E–06 –3.468E–06 –4.085E–06 1.367E–05 1.176E–07 5.349E–06 –5.496E–06

2

5.777E–01 1.483E–01 5.488E–01 6.003E–01 7.841E–01 –3.746E–01 7.108E–01 5.363E–01 6.420E–01 7.341E–01 –4.840E–01 7.841E–01 5.312E–01 6.754E–01 7.403E–01 4.460E–01 6.459E–01 6.112E–01 3.424E–01 7.151E–01 6.822E–01 –5.178E–01 7.999E–01 5.981E–01 5.869E–01

3.3. GROUND STATE ENERGY OF A SIMPLE HARMONIC OSCILLATOR

i

ui

gi

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

0.225 0.687 0.060 0.070 0.835 0.466 0.271 0.813 0.376 0.983 0.181 0.583 0.612 0.468 0.614 0.768 0.811 0.461 0.985 0.735 0.559 0.884 0.229 0.920 0.524

–0.750 0.490 –1.540 –1.465 0.980 –0.085 –0.605 0.895 –0.315 2.145 –0.905 0.210 0.285 –0.080 0.290 0.735 0.885 –0.095 2.195 0.630 0.150 1.200 –0.740 1.415 0.060

1 (–2α + 4α2G 2) + – LG 2 Gi = a + σ g i – — i i 2L

–4.409E–06 2.880E–06 –9.052E–06 –8.611E–06 5.760E–06 –4.996E–07 –3.556E–06 5.261E–06 –1.852E–06 1.261E–05 –5.320E–06 1.234E–06 1.675E–06 –4.702E–07 1.705E–06 4.320E–06 5.202E–06 –5.584E–07 1.290E–05 3.703E–06 8.817E–07 7.054E–06 –4.350E–06 8.317E–06 3.527E–07

2

6.629E–01 7.415E–01 2.218E–01 2.768E–01 5.659E–01 7.982E–01 7.108E–01 6.047E–01 7.758E–01 –3.217E–01 6.003E–01 7.892E–01 7.802E–01 7.984E–01 7.795E–01 6.683E–01 6.091E–01 7.978E–01 –3.746E–01 7.032E–01 7.945E–01 4.489E–01 6.665E–01 3.119E–01 7.991E–01

95

96

3. VARIATIONAL MONTE CARLO METHOD

i

ui

gi

51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75

0.393 0.306 0.819 0.620 0.875 0.773 0.492 0.332 0.185 0.565 0.434 0.665 0.516 0.659 0.709 0.515 0.683 0.092 0.417 0.598 0.682 0.948 0.125 0.149 0.246

–0.270 –0.505 0.915 0.310 1.155 0.750 –0.020 –0.430 –0.890 0.165 –0.165 0.430 0.040 0.410 0.555 0.040 0.480 –1.320 –0.205 0.250 0.475 1.635 –1.145 –1.035 –0.685

1 (–2α + 4α2G 2) + – LG 2 Gi = a + σ g i – — i i 2L

–1.587E–06 –2.968E–06 5.378E–06 1.822E–06 6.789E–06 4.409E–06 –1.176E–07 –2.528E–06 –5.231E–06 9.699E–07 –9.699E–07 2.528E–06 2.351E–07 2.410E–06 3.262E–06 2.351E–07 2.821E–06 –7.759E–06 –1.205E–06 1.470E–06 2.792E–06 9.611E–06 –6.730E–06 –6.084E–06 –4.026E–06

2

7.822E–01 7.378E–01 5.959E–01 7.766E–01 4.748E–01 6.629E–01 7.999E–01 7.549E–01 6.069E–01 7.934E–01 7.934E–01 7.549E–01 7.996E–01 7.590E–01 7.249E–01 7.996E–01 7.438E–01 3.752E–01 7.898E–01 7.848E–01 7.450E–01 1.483E–01 4.804E–01 5.388E–01 6.856E–01

3.3. GROUND STATE ENERGY OF A SIMPLE HARMONIC OSCILLATOR

i

ui

gi

76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

0.231 0.836 0.167 0.545 0.748 0.440 0.674 0.287 0.052 0.940 0.679 0.910 0.254 0.236 0.854 0.877 0.483 0.117 0.847 0.201 0.294 0.071 0.065 0.740 0.954

–0.730 0.985 –0.960 0.115 0.670 –0.150 0.455 –0.560 –1.615 1.565 0.470 1.350 –0.660 –0.715 1.060 1.165 –0.040 –1.185 1.030 –0.835 –0.540 –1.460 –1.505 0.645 1.695

1 (–2α + 4α2G 2) + – LG 2 Gi = a + σ g i – — i i 2L

–4.291E–06 5.790E–06 –5.643E–06 6.760E–07 3.938E–06 –8.817E–07 2.674E–06 –3.292E–06 –9.493E–06 9.199E–06 2.763E–06 7.935E–06 –3.879E–06 –4.203E–06 6.231E–06 6.848E–06 –2.351E–07 –6.965E–06 6.054E–06 –4.908E–06 –3.174E–06 –8.582E–06 –8.846E–06 3.791E–06 9.963E–06

6.701E–01 5.635E–01 5.753E–01 7.968E–01 6.906E–01 7.945E–01 7.495E–01 7.235E–01 1.641E–01 2.029E–01 7.461E–01 3.557E–01 6.938E–01 6.754E–01 5.261E–01 4.691E–01 7.996E–01 4.577E–01 5.414E–01 6.300E–01 7.289E–01 2.803E–01 2.478E–01 6.986E–01 9.959E–02 Sum = 57.064 E0 /(ħwc) = 0.571

2

97

98

3. VARIATIONAL MONTE CARLO METHOD

Table 3.2: Showing values of ground state energy E0 =.„!c / of an electron in simple harmonic oscillator potential V .x/ D 21 kx 2 with k D 10 18 , for various values of variational parameter ˇ m!c 2 2 contained in trial wave function ˆ0 D e ˇLx D e ˇ „ x β

E0 /(ħwc)

0.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

1.229 0.694 0.551 0.506 0.500 0.514 0.571 0.647 0.734 0.825 0.921 1.019 1.118

3.3. GROUND STATE ENERGY OF A SIMPLE HARMONIC OSCILLATOR

99

3.0

F(x) or Gaussian(x)

2.5 2.0 1.5 1.0 0.5 0.0 –4

–2

0

2

4

x

Figure 3.3: Showing F .x/ D

1 2L

2 2

2˛ C 4˛ x

as undashed curve and Gaussian p.x/ D






p1 Exp 2

C h

L 2 x 2


q 2˛

1 2



x


i a 2



e

2˛x 2



of Equation (3.33)

of Equation (3.31) as dashed

curve obtained using Program 3.1. For chosen value of spring constant k D 10 16 that corc D 9:046  1010 , ˇ D 0:8, ˛ D ˇL D 7:237  1010 , responds to !c D 10:483 MHz, L D m! „ a D 0,  D 12 ˛ 1=2 D 1:859  10 6 . (Here x is in unit of 10 6 and F .x/ and p.x/ are in unit of 10 5 .) with a D 0 and  D 1:859  10 obeying

6

using the tabulated values of random variable g of Table 1.4

  1  g a 2 1 p.g/ D p Exp 2   2 for a D 0 and  D 1. The connection is G D a C g D 0 C 1:859  10 6 g . Using Table 3.3 in Microsoft Excel, we have evaluated the expectation value set forth in Equation (3.29)
L E0 1 2˛ C 4˛ 2 xi2 C xi2 > D< „!c 2L 2 as a value of the ground state energy. We have run Program 3.1 and executed the Microsoft Excel spreadsheet for Table 3.3 for various values of ˇ namely 0.1–2.0 and obtained values of E0 =.„!c / in each case. Table 3.4 shows the results which are plotted in Figure 3.4. We find from Figure 3.4 that minimum value of E0 =.„!c / D 0:5 occurs for ˇ D 0:5. Thus, the ground state eigenfunction is ˆ0 D e

ˇLx 2

De

1 2 2 Lx

De

while ground state eigenvalue of energy is E0 D 0:5„!c .

1 m!c 2 „

x2

100

3. VARIATIONAL MONTE CARLO METHOD

1 Table 3.3: Showing tabulated values of ui , gi , Gi , 2L . 2˛ C 4˛ 2 Gi2 / C L2 Gi2 from which
L 2 E0 1 2 2 we get „!c D< 2L 2˛ C 4˛ Gi C 2 Gi >. Using a D 0,  D 1:859  10 6 , ˇ D 0:8, k D 10 16 , L D 9:046  1010 , ˛ D ˇL D 7:237  1010 , !c D 10:483 MHz.

i

ui

gi

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

0.169 0.050 0.844 0.182 0.599 0.985 0.726 0.148 0.788 0.698 0.010 0.397 0.852 0.761 0.309 0.113 0.212 0.809 0.084 0.276 0.243 0.989 0.508 0.817 0.174

–0.955 –1.635 1.015 –0.905 0.255 2.195 0.605 –1.040 0.805 0.520 –2.295 –0.255 1.050 0.715 –0.495 –1.205 –0.795 0.880 –1.370 –0.590 –0.695 2.325 0.020 0.910 –0.935

1 (–2α + 4α2G 2) + – LG 2 Gi = a + σ g i – — i i 2L

–1.775E–06 –3.039E–06 1.887E–06 –1.682E–06 4.740E–07 4.081E–06 1.125E–06 –1.933E–06 1.496E–06 9.667E–07 –4.266E–06 –4.740E–07 1.952E–06 1.329E–06 –9.202E–07 –2.240E–06 –1.478E–06 1.636E–06 –2.547E–06 –1.097E–06 –1.292E–06 4.322E–06 3.718E–08 1.692E–06 –1.738E–06

2

5.776E–01 1.482E–01 5.488E–01 6.003E–01 7.841E–01 –3.748E–01 7.107E–01 5.363E–01 6.420E–01 7.341E–01 –4.843E–01 7.841E–01 5.312E–01 6.753E–01 7.403E–01 4.459E–01 6.459E–01 6.112E–01 3.423E–01 7.151E–01 6.822E–01 –5.181E–01 7.999E–01 5.981E–01 5.868E–01

3.3. GROUND STATE ENERGY OF A SIMPLE HARMONIC OSCILLATOR

i

ui

gi

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

0.225 0.687 0.060 0.070 0.835 0.466 0.271 0.813 0.376 0.983 0.181 0.583 0.612 0.468 0.614 0.768 0.811 0.461 0.985 0.735 0.559 0.884 0.229 0.920 0.524

–0.750 0.490 –1.540 –1.465 0.980 –0.085 –0.605 0.895 –0.315 2.145 –0.905 0.210 0.285 –0.080 0.290 0.735 0.885 –0.095 2.195 0.630 0.150 1.200 –0.740 1.415 0.060

1 (–2α + 4α2G 2) + – LG 2 Gi = a + σ g i – — i i 2L

–1.394E–06 9.109E–07 –2.863E–06 –2.723E–06 1.822E–06 –1.580E–07 –1.125E–06 1.664E–06 –5.856E–07 3.988E–06 –1.682E–06 3.904E–07 5.298E–07 –1.487E–07 5.391E–07 1.366E–06 1.645E–06 –1.766E–07 4.081E–06 1.171E–06 2.789E–07 2.231E–06 –1.376E–06 2.630E–06 1.115E–07

2

6.628E–01 7.415E–01 2.217E–01 2.767E–01 5.658E–01 7.982E–01 7.107E–01 6.047E–01 7.758E–01 –3.219E–01 6.003E–01 7.892E–01 7.802E–01 7.984E–01 7.795E–01 6.683E–01 6.090E–01 7.978E–01 –3.748E–01 7.032E–01 7.945E–01 4.489E–01 6.665E–01 3.118E–01 7.991E–01

101

102

3. VARIATIONAL MONTE CARLO METHOD

i

ui

gi

51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75

0.393 0.306 0.819 0.620 0.875 0.773 0.492 0.332 0.185 0.565 0.434 0.665 0.516 0.659 0.709 0.515 0.683 0.092 0.417 0.598 0.682 0.948 0.125 0.149 0.246

–0.270 –0.505 0.915 0.310 1.155 0.750 –0.020 –0.430 –0.890 0.165 –0.165 0.430 0.040 0.410 0.555 0.040 0.480 –1.320 –0.205 0.250 0.475 1.635 –1.145 –1.035 –0.685

1 (–2α + 4α2G 2) + – LG 2 Gi = a + σ g i – — i i 2L

–5.019E–07 –9.388E–07 1.701E–06 5.763E–07 2.147E–06 1.394E–06 –3.718E–08 –7.994E–07 –1.655E–06 3.067E–07 –3.067E–07 7.994E–07 7.436E–08 7.622E–07 1.032E–06 7.436E–08 8.923E–07 –2.454E–06 –3.811E–07 4.648E–07 8.830E–07 3.039E–06 –2.129E–06 –1.924E–06 –1.273E–06

2

7.822E–01 7.378E–01 5.958E–01 7.766E–01 4.747E–01 6.628E–01 7.999E–01 7.549E–01 6.069E–01 7.934E–01 7.934E–01 7.549E–01 7.996E–01 7.590E–01 7.249E–01 7.996E–01 7.438E–01 3.751E–01 7.898E–01 7.848E–01 7.450E–01 1.482E–01 4.803E–01 5.388E–01 6.856E–01

3.3. GROUND STATE ENERGY OF A SIMPLE HARMONIC OSCILLATOR

i

ui

gi

76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

0.231 0.836 0.167 0.545 0.748 0.440 0.674 0.287 0.052 0.940 0.679 0.910 0.254 0.236 0.854 0.877 0.483 0.117 0.847 0.201 0.294 0.071 0.065 0.740 0.954

–0.730 0.985 –0.960 0.115 0.670 –0.150 0.455 –0.560 –1.615 1.565 0.470 1.350 –0.660 –0.715 1.060 1.165 –0.040 –1.185 1.030 –0.835 –0.540 –1.460 –1.505 0.645 1.695

1 (–2α + 4α2G 2) + – LG 2 Gi = a + σ g i – — i i 2L

–1.357E–06 1.831E–06 –1.785E–06 2.138E–07 1.246E–06 –2.789E–07 8.458E–07 –1.041E–06 –3.002E–06 2.909E–06 8.737E–07 2.510E–06 –1.227E–06 –1.329E–06 1.971E–06 2.166E–06 –7.436E–08 –2.203E–06 1.915E–06 –1.552E–06 –1.004E–06 –2.714E–06 –2.798E–06 1.199E–06 3.151E–06

6.701E–01 5.634E–01 5.753E–01 7.968E–01 6.905E–01 7.945E–01 7.495E–01 7.235E–01 1.640E–01 2.028E–01 7.461E–01 3.556E–01 6.938E–01 6.753E–01 5.260E–01 4.691E–01 7.996E–01 4.576E–01 5.413E–01 6.300E–01 7.289E–01 2.802E–01 2.477E–01 6.986E–01 9.943E–02 Sum = 57.058 E0 /(ħwc) = 0.571

2

103

104

3. VARIATIONAL MONTE CARLO METHOD

Table 3.4: Showing values of ground state energy E0 =.„!c / of an electron in simple harmonic oscillator potential V .x/ D 21 kx 2 with k D 10 16 , for various values of variational parameter ˇ m!c 2 2 contained in trial wavefunction ˆ0 D e ˇLx D e ˇ „ x β

E0 /(ħwc)

0.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

1.229 0.694 0.551 0.506 0.500 0.514 0.571 0.647 0.733 0.825 0.921 1.019 1.117

3.3. GROUND STATE ENERGY OF A SIMPLE HARMONIC OSCILLATOR

105

1.4 1.2

E 0 /(ћwc)

1.0 0.8 0.6 0.4 0.2 0.0 0.0

0.5

1.0

1.5

2.0

β

Figure 3.4: Showing values of ground state energy E0 =.„!c / of an electron in simple harmonic oscillator potential V .x/ D 12 kx 2 with k D 10 16 , for various values of variational parameter m!c 2 2 ˇ contained in trial wavefunction ˆ0 D e ˇLx D e ˇ „ x . Minimum value of E0 =.„!c / D 1 2 2 0:5 occurs for ˇ D 0:5. Thus, the ground state eigenfunction is ˆ0 D e ˇLx D e 2 Lx D 1 m!c 2 e 2 „ x while ground state eigenvalue of energy is E0 D 0:5„!c .

107

CHAPTER

4

Variational Monte Carlo Method Applied to the Ground State of a Hydrogen Atom 4.1

THE VARIATIONAL METHOD OF QUANTUM MECHANICS APPLIED TO THE GROUND STATE OF ANY QUANTUM MECHANICAL SYSTEM (AGAIN)

Let ‰0 ; ‰1 ; ‰2 ; ‰3 ; : : : ; ‰N be exact, unknown, orthonormal eigenfunctions of known Hamiltonian H . Let E0 < E1 < E2 < E3 < : : : < EN be exact, unknown eigenvalues of energy, respectively. That is, we have eigenvalue equation H ‰n D En ‰n

(4.1)

which we cannot solve. Suppose we wish to estimate E0 and we wish to know approximate form of ‰0 . Let us choose by guess a suitable function ˆ0 that is expected to resemble unknown ‰0 and write N X ˆ0 D Cn.0/ ‰n (4.2) nD0

and let us evaluate the quantity Eˆ0 given by R ˆ0  Hˆ0 d  .ˆ0 ; Hˆ0 / Eˆ0 D (4.3) D R .ˆ0 ; ˆ0 / ˆ0  ˆ0 d   P P  P P .0/ .0/ .0/ .0/ C ‰ ; C H ‰ C ‰ ; H C ‰ m n m n m n m n mD0 nD0 mD0 nD0  D P  D P P P .0/ .0/ .0/ .0/ mD0 Cm ‰m ; nD0 Cn ‰n mD0 Cm ‰m ; nD0 Cn ‰n  P P .0/ .0/ mD0 Cm ‰m ; nD0 Cn En ‰n  D P P .0/ .0/ C ‰ ; C ‰ m n mD0 m nD0 n

108

4. VARIATIONAL MONTE CARLO METHOD

using Equation (4.1) D

P

mD0

P

P

mD0

nD0

P

Cm.0/ Cn.0/ En .‰m ; ‰n /

nD0

Cm.0/ Cn.0/ .‰m ; ‰n /

D

D

P

mD0

P

P

nD0

P

Cm.0/ Cn.0/ En ımn

.0/ .0/ nD0 Cm Cn ımn ˇ ˇ P ˇ .0/ ˇ2 C ˇ n ˇ En nD0 mD0

ˇ ˇ : ˇ .0/ ˇ2 C nD0 ˇ n ˇ

P

Thus,

Eˆ0

ˇ ˇ ˇ .0/ ˇ2 nD0 ˇCn ˇ En E0 D E0 ˇ ˇ P ˇ .0/ ˇ2 nD0 ˇCn ˇ ˇ ˇ P ˇ .0/ ˇ2 C E0 / ˇ n ˇ .En nD0 D : ˇ ˇ P ˇ .0/ ˇ2 C nD0 ˇ n ˇ P

(4.4)

ˇ2 ˇ ˇ ˇ In Equation (4.4), En  E0 . ) En E0  0; moreover, ˇCn.0/ ˇ  0. Thus, Equation (4.4) shows that Eˆ0 E0  0 or, Eˆ0  E0 . We find that for arbitrary ˆ0 , calculated value of Eˆ0  E0 , i.e., Eˆ0 is larger than or equal to ground state energy E0 . Eˆ0 gives upper bound of ground state energy E0 . If chosen ˆ0 is different from actual unknown ‰0 , calculated value of Eˆ0 calculated using Equation (4.3) will be larger than E0 . If chosen ˆ0 happens to be same as unknown ‰0 , value of Eˆ0 calculated using Equation (4.3) becomes equal to E0 . Thus, we can choose arbitrary ˆ0 and calculate Eˆ0 using Equation (4.3) and get upper bound of E0 . Usually we choose a suitable ˆ0 known as trail function containing one or more adjustable .0/ parameters ˛i.0/ ’s, e.g., ˆ0 D e ˛1 x where ˛1.0/ is an adjustable parameter. Then we use this expression of ˆ0 in Equation (4.3) and calculated expression of Eˆ0 will contain the parameters @E 0 .0/ ˛i.0/ ’s. We then stipulate ˆ separately. We thus get values of ˛i.0/ ’s for which .0/ D 0 for each ˛i @˛i

Eˆ0 is minimum. We can put these values of ˛i.0/ ’s in the expression of Eˆ0 . We then get the lowest value of Eˆ0 for the chosen ˆ0 . This upper bound to E0 will be close to actual unknown value of E0 if trail function ˆ0 has a form closely resembling that of actual unknown ‰0 . The entire process is repeated for several different functions as ˆ0 to see which function leads to much lower value of Eˆ0 . If we are happy that a calculated value of Eˆ0 is low enough and that it is likely to be close to actual unknown value of E0 , we can take E0  Eˆ0 and ‰0  ˆ0 within some approximation.

4.2. THE VARIATIONAL METHOD OF QUANTUM MECHANICS: GROUND STATE

4.2

109

THE VARIATIONAL METHOD OF QUANTUM MECHANICS APPLIED TO THE GROUND STATE OF A HYDROGEN ATOM 2

2

„ e 1 We have a Hamiltonian operator H D 2m r 2 4" of a hydrogen atom. More explicitly, 0 r we have in a spherical polar coordinate system the Hamiltonian operator as       „2 1 @ 1 @ @ 1 @2 e2 1 2 @ H D r C Sin  C ; 2m r 2 @r @r r 2 Sin @ @ 4"0 r r 2 Sin2  @ 2

where m D 9:095  10 31 kg is a reduced mass of a proton-electron two-body system. Let us choose ˆ0 D e ˛r D e r=a0 by guess as an approximate form of ground state eigenfunction ˆ0 and let us evaluate the quantity E0 D

.ˆ0 ; Hˆ0 / : .ˆ0 ; ˆ0 /

(4.5)

Here, a0 D 0:53 Å is the so-called Bohr radius. Now, .ˆ0 ; ˆ0 / D .e ˛r ; e ˛r / Z 2 Z  Z D D0

D 4

Z

D0

/

rD0

e

/

e

2˛r 2

rD0

.2˛/r 3 1

r

r dr Sindd D 4

dr D 4.2˛/

.ˆ0 ; ˆ0 / D

3

2Š D

Z

/

e

2˛r 2

r dr

rD0

 : ˛3

 : ˛3

(4.6)

Now, Hˆ0 D

      „2 1 @ 1 @ @ˆ0 1 @ 2 ˆ0 2 @ˆ0 r C Sin  C 2m r 2 @r @r r 2 Sin @ @ r 2 Sin2  @ 2

or Hˆ0 D

    „2 1 @ 2 @ˆ0 r C0C0 2m r 2 @r @r

Hˆ0 D

   „2 1 @ 2 @ˆ0 r 2m r 2 @r @r

Hˆ0 D

  ˛r  „2 1 @ 2 @e r 2m r 2 @r @r

e2 1 ˆ0 4"0 r

or e2 1 ˆ0 4"0 r

or e2 1 e 4"0 r

˛r

e2 1 ˆ0 4"0 r

110

4. VARIATIONAL MONTE CARLO METHOD

or  „2 ˛2 2m

Hˆ0 D

 2 ˛ e r

˛r

e2 1 e 4"0 r

˛r

(4.7)

:

As such, E0 D

.ˆ0 ; Hˆ0 / .ˆ0 ; ˆ0 /

E0 D

.ˆ0 ; Hˆ0 / ; =˛ 3

gives

or 4



R/

rD0

E0 D

 „2 ˛2 2m

 2 ˛ r =˛ 3

 e2 1 e 4"0 r

2˛r 2

r dr

or E0 D

Z

/

4˛

3



rD0

 „2 ˛2 2m

 2 ˛ e r

2˛r

e2 1 e 4"0 r

2˛r



r 2 dr:

(4.8)

This integral is evaluated in the next section using the Monte Carlo method.

4.3

GROUND STATE ENERGY OF A HYDROGEN ATOM USING THE VARIATIONAL MONTE CARLO METHOD

We first gather the exact results from the Quantum Mechanics books. The ground state eigenfunction is ˆ0 D e ˛r D e r=a0 with a0 called Bohr radius given by 0.53 Å and eigenvalue of energy equal to 13:6 eV. As to the Monte Carlo method, we take up Equation (4.8):     Z / „2 2 e 2 1 2˛r 2 2 2˛r 3 ˛ ˛ e e r dr; E0 D 4˛ 2m r 4"0 r rD0 and rewrite it as E0 D

where F .r/ D 4˛

3



 „2 ˛2 2m

Z

/

(4.9)

F .r/dr;

rD0

 2 ˛ e r

2˛r

e2 1 e 4"0 r

2˛r



r 2:

(4.10)

4.3. GROUND STATE ENERGY OF HYDROGEN ATOM

111

2.0 × 1010

F(r) or p(r)

1.5 × 1010

1.0 × 1010

5.0 × 109

0 0

1

2

r/a 0

3

4

5

Figure 4.1:h Smooth curve shows 0:55  1018 F .r/ and dashed curve shows Gaussian p.r/ D i
1 r a 2 p1 Exp for a D 1:0a0 and  D 0:43a0 , a0 D 0:5 Å. We find that F .r/ and Gaus2   2 sian p.r/ are nearly proportional to each other and hence Gaussian p.r/ is an ideal choice for the Monte Carlo calculation. We now re-write Equation (4.9) as E0 D

Z

/

rD0

F .r/ p.r/dr; p.r/

(4.11)

where p.r/ is a probability density function to be taken such that it resembles F .r/. Thus, E0 is .r/ , i.e., given by average or expectation value of Fp.r/ E0 D
: p.r/

(4.12)

Thus, what we need to do is to see the functional form of F .r/ and get a probability density .r/ function p.r/ that looks like F .r/ and calculate average value of Fp.r/ for various values of a0 in .r/ the range 0.3–1 Å. We need to find out the lowest value of < Fp.r/ > which is expected to yield about 13:6 eV. Using Program 4.1 in Mathematica 6.0, for a0 D 0:5 Å, we have plotted 0:55  1018 F .r/ and Gaussian   1 1  r a 2 p.r/ D p Exp 2   2

for a D 1:0a0 and  D 0:43a0 . We find that F .r/ and Gaussian p.r/ are nearly proportional to each other and hence Gaussian p.r/ is an ideal choice for the Monte Carlo calculation.

112

4. VARIATIONAL MONTE CARLO METHOD

Program 4.1

a0=0.5*10^-10; alpha=1/a0; m=1837*(9.1*(10^-31))/1838; hcut=(6.626*(10^-34))/(2*3.1416); ce=1.6*(10^-19); ep0=8.85*(10^-12); F=Plot[(-0.55*10^18)*(4*(alpha^3)*(-(hcut^2)/ (2*m))*(alpha^2-2*alpha/(r*a0))* Exp[-2*alpha*(r*a0)]-(4*(alpha^3))*((ce^2)/(4*3.1416*ep0*(r*a0)))* Exp[-2*alpha*(r*a0)])*(r*a0)^2,{r,0,5},PlotStyle->{Black},Frame->True, FrameLabel->{"r/a0","F(r) or p(r)"},PlotRange->{0,2*10^10}] sig=0.43*a0; a=1.0*a0; p=Plot[(1/(sig*Sqrt[2*Pi]))*Exp[-0.5*((r*a0-a)/sig)^2],{r,0,5}, PlotStyle->{Dashed,Black},Frame->True, FrameLabel->{"r/a0","F(r) or p(r)"}, PlotRange->{0,2*10^10}] Show[F,p]

For values of a0 in the range 0.3–1 Å, we have run the Program 4.1 and find that F .r/ and p.r/ match well for different pairs of values of a and  tabulated in Table 4.1. Using Table 4.2 in Microsoft Excel, we have evaluated the expectation value set forth in .r/ Equation (4.12): E0 D< Fp.r/ > as a value of the ground state energy. Here, F .r/ D 4˛ 3



 „2 ˛2 2m

 2 ˛ e r

and 1 p.r/ D p Exp  2



2˛r

e2 1 e 4"0 r

2˛r



r2

 1  r a 2 : 2 

We have run Program 4.1 and executed the Microsoft Excel spreadsheet for Table 4.2 for various values of a0 namely 0.3–1.0 Å and obtained values of E0 in each case. Table 4.3 shows the results which are plotted in Figure 4.2. We find from Figure 4.2 that E0 reaches a minimum for a0 around 0.5 Å as expected. The minimum value is about 11:34 eV for a0 D

4.3. GROUND STATE ENERGY OF HYDROGEN ATOM

113

Table 4.1: Pairs of values of a and  for which F .r/ and p.r/ match with each other within a constant multiple, ascertained using Program 4.1 a0 in Å 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

a 1.6a0 1.3a0 1.0a0 0.9a0 0.8a0 0.7a0 0.7a0 0.6a0

0.50a0 0.50a0 0.43a0 0.40a0 0.35a0 0.35a0 0.30a0 0.25a0

0.6

0.7

σ

–6 –7

E 0 (eV)

–8 –9

–10 –11 –12 0.3

0.4

0.5

0.8

0.9

1.0

a 0 (Å)

Figure 4.2: Showing values of ground state energy E0 of a hydrogen atom for various values of Bohr radius a0 contained in trial wavefunction ˆ0 D e r=a0 . Minimum value of E0 D 11:34 eV occurs around a0 D 0:5 Å. 0:5 Å. The deviation from the calculations.

13:6 eV is not surprising given only 100 random numbers used in

114

4. VARIATIONAL MONTE CARLO METHOD

 2 h i „ Table 4.2: Showing tabulated values of ui , gi , Gi , F .Gi / D 4˛ 3 ˛ 2 G2i ˛ e 2˛Gi 2m    2  Gi a e2 1 1 1 2 2˛Gi Gi ; and p.Gi / D p Exp ; F .Gi /=p.Gi / from which we e 4"0 Gi 2   2

get the average value of E0 D< a D 1:0a0 ,  D 0:43a0 .

F p

> as a value of the ground state energy. Using a0 D 0:5 Å,

i

ui

gi

Gi

Fi

pi

Fi /pi

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

0.169 0.050 0.844 0.182 0.599 0.985 0.726 0.148 0.788 0.698 0.010 0.397 0.852 0.761 0.309 0.113 0.212 0.809 0.084 0.276 0.243 0.989 0.508 0.817 0.174

–0.955 –1.635 1.015 –0.905 0.255 2.195 0.605 –1.040 0.805 0.520 –2.295 –0.255 1.050 0.715 –0.495 –1.205 –0.795 0.880 –1.370 –0.590 –0.695 2.325 0.020 0.910 –0.935

2.95E–11 1.48E–11 7.18E–11 3.05E–11 5.55E–11 9.72E–11 6.30E–11 2.76E–11 6.73E–11 6.12E–11 6.58E–13 4.45E–11 7.26E–11 6.54E–11 3.94E–11 2.41E–11 3.29E–11 6.89E–11 2.05E–11 3.73E–11 3.51E–11 1.00E–10 5.04E–11 6.96E–11 2.99E–11

–1.67E–08 –5.76E–09 –2.10E–08 –1.74E–08 –2.34E–08 –1.42E–08 –2.27E–08 –1.56E–08 –2.19E–08 –2.29E–08 2.61E–10 –2.27E–08 –2.08E–08 –2.23E–08 –2.14E–08 –1.31E–08 –1.87E–08 –2.16E–08 –1.04E–08 –2.06E–08 –1.97E–08 –1.35E–08 –2.34E–08 –2.15E–08 –1.70E–08

1.18E+10 4.88E+09 1.11E+10 1.23E+10 1.80E+10 1.67E+09 1.55E+10 1.08E+10 1.34E+10 1.62E+10 1.33E+09 1.80E+10 1.07E+10 1.44E+10 1.64E+10 8.98E+09 1.35E+10 1.26E+10 7.26E+09 1.56E+10 1.46E+10 1.24E+09 1.86E+10 1.23E+10 1.20E+10

–1.42E–18 –1.18E–18 –1.89E–18 –1.41E–18 –1.30E–18 –8.53E–18 –1.47E–18 –1.44E–18 –1.63E–18 –1.41E–18 1.96E–19 –1.26E–18 –1.94E–18 –1.55E–18 –1.30E–18 –1.46E–18 –1.38E–18 –1.71E–18 –1.43E–18 –1.32E–18 –1.35E–18 –1.09E–17 –1.26E–18 –1.75E–18 –1.42E–18

4.3. GROUND STATE ENERGY OF HYDROGEN ATOM

i

ui

gi

Gi

Fi

pi

Fi /pi

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

0.225 0.687 0.060 0.070 0.835 0.466 0.271 0.813 0.376 0.983 0.181 0.583 0.612 0.468 0.614 0.768 0.811 0.461 0.985 0.735 0.559 0.884 0.229 0.920 0.524

–0.750 0.490 –1.540 –1.465 0.980 –0.085 –0.605 0.895 –0.315 2.145 –0.905 0.210 0.285 –0.080 0.290 0.735 0.885 –0.095 2.195 0.630 0.150 1.200 –0.740 1.415 0.060

3.39E–11 6.05E–11 1.69E–11 1.85E–11 7.11E–11 4.82E–11 3.70E–11 6.92E–11 4.32E–11 9.61E–11 3.05E–11 5.45E–11 5.61E–11 4.83E–11 5.62E–11 6.58E–11 6.90E–11 4.80E–11 9.72E–11 6.35E–11 5.32E–11 7.58E–11 3.41E–11 8.04E–11 5.13E–11

–1.91E–08 –2.30E–08 –7.41E–09 –8.72E–09 –2.11E–08 –2.32E–08 –2.05E–08 –2.15E–08 –2.24E–08 –1.45E–08 –1.74E–08 –2.34E–08 –2.34E–08 –2.32E–08 –2.34E–08 –2.22E–08 –2.16E–08 –2.32E–08 –1.42E–08 –2.26E–08 –2.35E–08 –2.00E–08 –1.93E–08 –1.88E–08 –2.34E–08

1.40E+10 1.65E+10 5.67E+09 6.34E+09 1.15E+10 1.85E+10 1.55E+10 1.24E+10 1.77E+10 1.86E+09 1.23E+10 1.82E+10 1.78E+10 1.85E+10 1.78E+10 1.42E+10 1.25E+10 1.85E+10 1.67E+09 1.52E+10 1.83E+10 9.03E+09 1.41E+10 6.82E+09 1.85E+10

–1.37E–18 –1.40E–18 –1.31E–18 –1.38E–18 –1.84E–18 –1.26E–18 –1.33E–18 –1.73E–18 –1.27E–18 –7.81E–18 –1.41E–18 –1.29E–18 –1.31E–18 –1.26E–18 –1.31E–18 –1.57E–18 –1.72E–18 –1.26E–18 –8.53E–18 –1.48E–18 –1.28E–18 –2.21E–18 –1.36E–18 –2.76E–18 –1.27E–18

115

116

4. VARIATIONAL MONTE CARLO METHOD

i

ui

gi

Gi

Fi

pi

Fi /pi

51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75

0.393 0.306 0.819 0.620 0.875 0.773 0.492 0.332 0.185 0.565 0.434 0.665 0.516 0.659 0.709 0.515 0.683 0.092 0.417 0.598 0.682 0.948 0.125 0.149 0.246

–0.270 –0.505 0.915 0.310 1.155 0.750 –0.020 –0.430 –0.890 0.165 –0.165 0.430 0.040 0.410 0.555 0.040 0.480 –1.320 –0.205 0.250 0.475 1.635 –1.145 –1.035 –0.685

4.42E–11 3.91E–11 6.97E–11 5.67E–11 7.48E–11 6.61E–11 4.96E–11 4.08E–11 3.09E–11 5.35E–11 4.65E–11 5.92E–11 5.09E–11 5.88E–11 6.19E–11 5.09E–11 6.03E–11 2.16E–11 4.56E–11 5.54E–11 6.02E–11 8.52E–11 2.54E–11 2.77E–11 3.53E–11

–2.26E–08 –2.13E–08 –2.14E–08 –2.33E–08 –2.02E–08 –2.21E–08 –2.33E–08 –2.18E–08 –1.76E–08 –2.35E–08 –2.30E–08 –2.31E–08 –2.34E–08 –2.32E–08 –2.28E–08 –2.34E–08 –2.30E–08 –1.12E–08 –2.29E–08 –2.34E–08 –2.30E–08 –1.75E–08 –1.40E–08 –1.57E–08 –1.98E–08

1.79E+10 1.63E+10 1.22E+10 1.77E+10 9.52E+09 1.40E+10 1.86E+10 1.69E+10 1.25E+10 1.83E+10 1.83E+10 1.69E+10 1.85E+10 1.71E+10 1.59E+10 1.85E+10 1.65E+10 7.76E+09 1.82E+10 1.80E+10 1.66E+10 4.88E+09 9.63E+09 1.09E+10 1.47E+10

–1.26E–18 –1.30E–18 –1.76E–18 –1.32E–18 –2.13E–18 –1.58E–18 –1.26E–18 –1.29E–18 –1.41E–18 –1.28E–18 –1.26E–18 –1.37E–18 –1.26E–18 –1.36E–18 –1.43E–18 –1.26E–18 –1.39E–18 –1.45E–18 –1.26E–18 –1.30E–18 –1.39E–18 –3.59E–18 –1.46E–18 –1.44E–18 –1.35E–18

4.3. GROUND STATE ENERGY OF HYDROGEN ATOM

i

ui

gi

Gi

Fi

76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

0.231 0.836 0.167 0.545 0.748 0.440 0.674 0.287 0.052 0.940 0.679 0.910 0.254 0.236 0.854 0.877 0.483 0.117 0.847 0.201 0.294 0.071 0.065 0.740 0.954

–0.730 0.985 –0.960 0.115 0.670 –0.150 0.455 –0.560 –1.615 1.565 0.470 1.350 –0.660 –0.715 1.060 1.165 –0.040 –1.185 1.030 –0.835 –0.540 –1.460 –1.505 0.645 1.695

3.43E–11 7.12E–11 2.94E–11 5.25E–11 6.44E–11 4.68E–11 5.98E–11 3.80E–11 1.53E–11 8.36E–11 6.01E–11 7.90E–11 3.58E–11 3.46E–11 7.28E–11 7.50E–11 4.91E–11 2.45E–11 7.21E–11 3.20E–11 3.84E–11 1.86E–11 1.76E–11 6.39E–11 8.64E–11

–1.94E–08 –2.11E–08 –1.67E–08 –2.35E–08 –2.24E–08 –2.31E–08 –2.31E–08 –2.09E–08 –6.10E–09 –1.79E–08 –2.30E–08 –1.92E–08 –2.00E–08 –1.95E–08 –2.07E–08 –2.02E–08 –2.33E–08 –1.34E–08 –2.09E–08 –1.82E–08 –2.10E–08 –8.81E–09 –8.02E–09 –2.25E–08 –1.72E–08

pi

Fi /pi

1.42E+10 –1.36E–18 1.14E+10 –1.85E–18 1.17E+10 –1.42E–18 1.84E+10 –1.27E–18 1.48E+10 –1.51E–18 1.83E+10 –1.26E–18 1.67E+10 –1.38E–18 1.59E+10 –1.32E–18 5.04E+09 –1.21E–18 5.45E+09 –3.29E–18 1.66E+10 –1.39E–18 7.46E+09 –2.57E–18 1.49E+10 –1.34E–18 1.44E+10 –1.36E–18 1.06E+10 –1.96E–18 9.41E+09 –2.14E–18 1.85E+10 –1.26E–18 9.19E+09 –1.46E–18 1.09E+10 –1.91E–18 1.31E+10 –1.39E–18 1.60E+10 –1.31E–18 6.39E+09 –1.38E–18 5.98E+09 –1.34E–18 1.51E+10 –1.49E–18 4.41E+09 –3.89E–18 Sum = –1.81E–16 E0 = –11.33 eV

117

118

4. VARIATIONAL MONTE CARLO METHOD

Table 4.3: Showing values of ground state energy E0 of a hydrogen atom for various values of Bohr radius a0 contained in trial wavefunction ˆ0 D e r=a0 a0 (Å) 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

E0 (eV) –9.83 –11.21 –11.34 –11.03 –9.98 –9.16 –8.34 –6.98

119

CHAPTER

5

Concluding Remarks 1. This book is intended for undergraduate students of Mathematics, Statistics, and Physics who know nothing about the Monte Carlo method but wish to know how it works. 2. All treatments have been done as much manually as is practicable. 3. The treatments are deliberately manual to let the readers get the real feel of how the Monte Carlo method works. 4. Definite integrals of a total of five functions F .x/, namely Sin.x/; Cos.x/; e x , loge .x/, 1 , have been evaluated using constant, linear, Gaussian, and exponential probability 1Cx 2 density functions p.x/. 5. It is shown that results agree with known exact values better if p.x/ is proportional to F .x/. 6. Deviation from the proportionality results in worse agreement. 7. Necessary background materials are covered in Chapter 1 while the integrals are evaluated in Chapter 2. 8. Two separate chapters have been dedicated to the variational Monte Carlo method applied to the ground state of a simple harmonic oscillator and of a hydrogen atom with remarkable success given only 100 random numbers used. 9. The book is a good read, and is intended to make the readers adept at using the method.

121

Bibliography [1] IIya M. Sobol, A Primer for the Monte Carlo Method, CRC Press, 1994. [2] IIya M. Sobol, The Monte Carlo Method, The University of Chicago Press, 1974. [3] Yu. A. Shreider, Ed., The Monte Carlo Method: The Method of Statistical Trials, Pergamon Press, 1966.

123

Author’s Biography SUJAUL CHOWDHURY Sujaul Chowdhury is a Professor in the Department of Physics at the Shahjalal University of Science and Technology (SUST), Sylhet, Bangladesh (www.sust.edu). He obtained a B.Sc. (Honours) in Physics in 1994 and an M.Sc. in Physics in 1996 from SUST. He obtained a Ph.D. in Physics from The University of Glasgow, UK in 2001. He was a Humboldt Research Fellow for one year at The Max Planck Institute, Stuttgart, Germany.