ISBN-10:
9814425036
ISBN-13:
9789814425032
Pub. Date:
Publisher:
Guide To Pamir, The: Theory And Use Of Parameterized Adaptive Multidimensional Integration Routines

Guide To Pamir, The: Theory And Use Of Parameterized Adaptive Multidimensional Integration Routines

by Stephen L Adler

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Overview

PAMIR (Parameterized Adaptive Multidimensional Integration Routines) is a suite of Fortran programs for multidimensional numerical integration over hypercubes, simplexes, and hyper-rectangles in general dimension p, intended for use by physicists, applied mathematicians, computer scientists, and engineers. The programs, which are available on the internet at www.pamir-integrate.com and are free for non-profit research use, are capable of following localized peaks and valleys of the integrand. Each program comes with a Message-Passing Interface (MPI) parallel version for cluster use as well as serial versions.The first chapter presents introductory material, similar to that on the PAMIR website, and the next is a “manual” giving much more detail on the use of the programs than is on the website. They are followed by many examples of performance benchmarks and comparisons with other programs, and a discussion of the computational integration aspects of PAMIR, in comparison with other methods in the literature. The final chapter provides details of the construction of the algorithms, while the Appendices give technical details and certain mathematical derivations.


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Product Details

ISBN-13: 9789814425032
Publisher: World Scientific Publishing Company, Incorporated
Publication date: 12/21/2012
Pages: 216
Product dimensions: 6.20(w) x 9.10(h) x 0.80(d)

Table of Contents

1 Introduction 1

1.1 Copyright, credits, feedback and updates, licensing, and disclaimer 2

1.2 Program language and systems considerations 4

1.3 Acknowledgements 5

2 Using PAMIR 7

2.1 Base regions for the PAMIR programs, and normalization conventions 7

2.1.1 Base region definitions 7

2.1.2 Normalization conventions 11

2.1.3 Base region applications 11

2.1.4 Remarks on strategy 13

2.2 Sketch of the algorithms 14

2.2.1 Folders in pamir_archive 14

2.2.2 Entering the function and the spatial dimension 15

2.2.3 Basic algorithm module 16

2.2.4 Three versions of the algorithm 17

2.2.5 Default double precision and real(16) option 17

2.2.6 Linking main programs to subroutines 18

2.2.7 Error estimates and thinning 18

2.2.8 User options 20

2.3 Contents of programs in the folders 21

2.3.1 Folder simplex123 21

2.3.2 Folder simplex4 22

2.3.3 Folder simplex579 22

2.3.4 Folder simplex579_16 22

2.3.5 Folder cubel3 23

2.3.6 Folder cube357 23

2.3.7 Folder cube357_16 23

2.3.8 Folder cube579 24

2.3.9 Folder cube579_l6 24

2.3.10 Folder constant_jacobian_map 24

2.3.11 Folder loop_exaraples 25

2.3.12 Folder readme 26

2.4 Program inputs and outputs 26

2.4.1 Inputs 26

2.4.2 Outputs 30

2.5 Function calls, timing estimates, and memory estimates 33

2.5.1 Function call counting 33

2.5.2 Timing estimates 35

2.5.3 Memory estimates 38

2.5.4 Projection of future performance 39

2.6 False positives and their avoidance 40

2.7 Programming remarks 43

3 Benchmark examples and comparisons 51

3.1 Introduction 51

3.2 Lattice Green's Function near the branch cut: comparison of PAMIR with CUBPACK 52

3.3 Double Gaussian in seven dimensions: comparison of PAMIR with VEGAS and MISER 58

3.4 Two loop self-energy master function 60

3.5 Comparison of PAMIR with CUBPACK for polynomial integrals over standard simplex in five dimensions 68

3.6 Comparison of PAMIR with CUBPACK for the Feynman-Schwinger integral over standard simplex in five dimensions 72

3.7 Comparisons of PAMIR with CUBPACK for hypercube test integrals in five dimensions 75

3.7.1 Double Gaussian 77

3.7.2 Tsuda functions 79

3.7.3 Eighth order polynomial 83

3.7.4 Single Gaussian with generic center in five dimensions 83

3.7.5 Lorentzians in five dimensions: corner peak, edge peak, generic internal peak 86

3.7.6 Oscillatory cosine integrals in five dimensions 89

3.7.7 C0 function in five dimensions 90

3.8 Attenuation function of radiation from a disk source 95

3.9 High accuracy calculations of double Gaussian integral in seven and nine dimensions on a 64 process cluster 95

4 Computational integration theory and PAMIR 101

4.1 Overview: The philosophy of PAMIR 101

4.2 Choice of integration method 102

4.3 Choice of integration rule 103

4.4 Choice of error estimator 112

4.5 Choice of subdivision method and decision method 114

5 Details of construction of the PAMIR algorithms and programs 117

5.1 Simplex properties 117

5.2 Simplex subdivision 120

5.2.1 Simplex subdivision algorithms 120

5.2.2 Simplex subdivision properties 125

5.3 Hypercube and hyper-rectangle subdivision 130

5.3.1 Hypercube subdivision algorithm 130

5.3.2 Hypercube and subdivision properties 132

5.3.3 Hyper-rectangle subdivision algorithm 134

5.4 Vandermonde solvers 135

5.5 Parameterized higher order integration formulas for a general simplex 136

5.5.1 Simplex integrals in terms of moments 136

5.5.2 First through third order simplex formulas 142

5.5.3 Fourth order simplex formula 145

5.5.4 Fifth order simplex formula 145

5.5.5 Seventh order simplex formula 146

5.5.6 Ninth order simplex formula 148

5.5.7 Leading term in higher order for simplexes 149

5.6 Parameterized higher order integration formulas for axis-parallel hypercubes from moments 149

5.6.1 First and third order formulas 152

5.6.2 Fifth order formula 153

5.6.3 Seventh order formula 154

5.6.4 Ninth order formula 155

5.6.5 Leading term in higher order for hypercubes 155

5.6.6 Redundant function calls in low dimensions p 156

5.7 Some details of the cube357 programs 157

5.7.1 Integration over hyper-rectangles 157

5.7.2 Measure var(i) of variation along axis i 158

5.8 Some samples of code 158

5.8.1 Basic module 159

5.8.2 Simplex and hypercube rescaling for the "r" and "m" programs 162

5.8.3 Distributing residual subregions to processes in the "m" programs 164

5.9 Programming extensions and open questions 165

Appendix A Test integrals 167

Appendix B Derivation of the simplex generating function 173

Appendix C Derivation of the hypercube generating function 175

Appendix D Mappings between base regions 177

Appendix E Rule for determining where a point lies with respect to a simplex 181

Appendix F Expansion for ∑4 185

Appendix G Fourth order simplex formula 189

Appendix H Ninth order simplex formula 193

Appendix I Ninth order hypercube formula 195

Bibliography 197

Index 201

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