Engineering and Scientific Subroutine Library for AIX: Guide and Reference

Part 1. Guide Information
Looking for a Subroutine?
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 xvii
About This Book . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 xix
How to Use This Book . . . . . . . . . . . . . . . . . . . . . . . . .
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 . xix
How to Find a Subroutine Description . . . . . . . . . . . . . . . .
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 . xx
Where to Find Related Publications . . . . . . . . . . . . . . . . .
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 . xxi
How to Look Up a Bibliography Reference . . . . . . . . . . . . .
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 . xxi
Special Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 . xxi
Short and Long Precision
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 . xxi
Subroutines and Subprograms
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 . xxi
How to Interpret the Subroutine Names with a Prefix Underscore
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 . xxii
Abbreviated Names
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 . xxii
Fonts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 . xxii
Special Notations and Conventions . . . . . . . . . . . . . . . . .
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 xxiii
Scalar Data
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 xxiii
Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 xxiii
Matrices
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 xxiii
Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 xxiv
Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 . xxv
Special Characters, Symbols, Expressions, and Abbreviations
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 xxvi
How to Interpret the Subroutine Descriptions . . . . . . . . . . . .
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 xxviii
Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 xxviii
Syntax
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 xxviii
On Entry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 xxviii
On Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 xxix
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 xxix
Function
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 xxix
Special Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 xxix
Error Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 xxix
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 xxix
What's New for ESSL for AIX . . . . . . .
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 xxxi
What's New for ESSL Version 3 Release 2
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 xxxi
Changes for ESSL Version 3 Release 1.2
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 xxxi
Changes for ESSL Version 3 Release 1.1
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 xxxii
Changes for ESSL Version 3 . . . . . . . .
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 xxxii
Future Migration
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 xxxiii
In Brief—What's Provided in ESSL for AIX
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 xxxv
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 1
Chapter 1. Introduction and Requirements
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 3
Overview of ESSL . . . . . . . . . . . . . . .
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Performance and Functional Capability
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Usability
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The Variety of Mathematical Functions . .
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ESSL—Processing Capabilities . . . . . .
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Accuracy of the Computations . . . . . . .
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2000
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High Performance of ESSL
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 6
The Fortran Language Interface to the Subroutines . . . . . . .
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 7
Software and Hardware Products That Can Be Used with ESSL .
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 7
For ESSL—Hardware . . . . . . . . . . . . . . . . . . . . . . . . .
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 7
ESSL—Operating Systems
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 7
ESSL—Software Products . . . . . . . . . . . . . . . . . . . . . .
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 8
Installation and Customization Products . . . . . . . . . . . . . .
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 8
Software Products for Displaying ESSL Online Information . . .
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 8
ESSL Internet Resources . . . . . . . . . . . . . . . . . . . . . . . .
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 8
Obtaining Documentation
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 9
Accessing ESSL's Product Home Pages . . . . . . . . . . . . . .
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 9
Getting on the ESSL Mailing List . . . . . . . . . . . . . . . . . . . .
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 9
List of ESSL Subroutines . . . . . . . . . . . . . . . . . . . . . . . .
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 10
Linear Algebra Subprograms
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 10
Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 13
Linear Algebraic Equations . . . . . . . . . . . . . . . . . . . . . .
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 14
Eigensystem Analysis . . . . . . . . . . . . . . . . . . . . . . . . .
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 18
Fourier Transforms, Convolutions and Correlations, and Related
Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 18
Sorting and Searching . . . . . . . . . . . . . . . . . . . . . . . .
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 20
Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 21
Numerical Quadrature
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 21
Random Number Generation . . . . . . . . . . . . . . . . . . . .
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 21
Utilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 22
Chapter 2. Planning Your Program . . . . . . . . . . . . . . . . . . . . . . . .
Selecting an ESSL Subroutine . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Which ESSL Library Do You Want to Use? . . . . . . . . . . . . . . . . . . .
What Type of Data Are You Processing in Your Program? . . . . . . . . . .
How Is Your Data Structured? And What Storage Technique Are You
Using? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
What about Performance and Accuracy?
 . . . . . . . . . . . . . . . . . . . .
Avoiding Conflicts with Internal ESSL Routine Names That are Exported . . .
Setting Up Your Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
How Do You Set Up Your Scalar Data? . . . . . . . . . . . . . . . . . . . . .
How Do You Set Up Your Arrays? . . . . . . . . . . . . . . . . . . . . . . . .
How Should Your Array Data Be Aligned? . . . . . . . . . . . . . . . . . . . .
What Storage Mode Should You Use for Your Data? . . . . . . . . . . . . .
How Do You Convert from One Storage Mode to Another? . . . . . . . . . .
Setting Up Your ESSL Calling Sequences . . . . . . . . . . . . . . . . . . . . .
What Is an Input-Output Argument? . . . . . . . . . . . . . . . . . . . . . . .
What Are the General Rules to Follow when Specifying Data for the
Arguments? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
What Happens When a Value of 0 Is Specified for N? . . . . . . . . . . . . .
How Do You Specify the Beginning of the Data Structure in the ESSL Calling
Sequence? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Using Auxiliary Storage in ESSL . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dynamic Allocation of Auxiliary Storage . . . . . . . . . . . . . . . . . . . . .
Setting Up Auxiliary Storage When Dynamic Allocation Is Not Used . . . . .
Who Do You Want to Calculate the Size? You or ESSL?
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How Do You Calculate the Size Using the Formulas? . . . . . . . . . . . . .
How Do You Get ESSL to Calculate the Size Using ESSL Error Handling?
Providing a Correct Transform Length to ESSL . . . . . . . . . . . . . . . . . .
What ESSL Subroutines Require Transform Lengths? . . . . . . . . . . . . .
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 Engineering and Scientific Subroutine Library Version 3 Guide and Reference
Who Do You Want to Calculate the Length? You or ESSL? . . . . . . . . .
 37
How Do You Calculate the Length Using the Table or Formula? . . . . . . .
 37
How Do You Get ESSL to Calculate the Length Using ESSL Error Handling?
 37
Getting the Best Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 42
What Precisions Do ESSL Subroutines Operate On? . . . . . . . . . . . . .
 42
How does the Nature of the ESSL Computation Affect Accuracy? . . . . . .
 42
What Data Type Standards Are Used by ESSL, and What Exceptions Should
You Know About? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 43
How is Underflow Handled? . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Where Can You Find More Information on Accuracy? . . . . . . . . . . . . .
 43
Getting the Best Performance
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 43
What General Coding Techniques Can You Use to Improve Performance?
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Where Can You Find More Information on Performance? . . . . . . . . . . .
 44
Dealing with Errors when Using ESSL
 . . . . . . . . . . . . . . . . . . . . . . .
 45
What Can You Do about Program Exceptions? . . . . . . . . . . . . . . . . .
 45
What Can You Do about ESSL Input-Argument Errors? . . . . . . . . . . . .
 45
What Can You Do about ESSL Computational Errors? . . . . . . . . . . . .
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What Can You Do about ESSL Resource Errors? . . . . . . . . . . . . . . .
 48
What Can You Do about ESSL Attention Messages? . . . . . . . . . . . . .
 49
How Do You Control Error Handling by Setting Values in the ESSL Error
Option Table? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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How does Error Handling Work in a Threaded Environment? . . . . . . . . .
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Where Can You Find More Information on Errors? . . . . . . . . . . . . . . .
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Chapter 3. Setting Up Your Data Structures . . . . . . . . . . . . . . .
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 53
Concepts
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 53
Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 53
Transpose of a Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Conjugate Transpose of a Vector . . . . . . . . . . . . . . . . . . . . . .
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In Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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How Stride Is Used for Vectors . . . . . . . . . . . . . . . . . . . . . . .
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 56
Sparse Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 60
Transpose of a Matrix
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Conjugate Transpose of a Matrix . . . . . . . . . . . . . . . . . . . . . .
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 60
In Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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How Leading Dimension Is Used for Matrices . . . . . . . . . . . . . . .
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Symmetric Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Positive Definite or Negative Definite Symmetric Matrix . . . . . . . . .
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 Symmetric Indefinite Matrix
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Complex Hermitian Matrix . . . . . . . . . . . . . . . . . . . . . . . . . .
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Positive Definite or Negative Definite Complex Hermitian Matrix . . . .
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Positive Definite or Negative Definite Symmetric Toeplitz Matrix . . . .
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Positive Definite or Negative Definite Complex Hermitian Toeplitz Matrix
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 70
Triangular Matrix
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General Band Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Symmetric Band Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Positive Definite Symmetric Band Matrix . . . . . . . . . . . . . . . . . .
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Complex Hermitian Band Matrix . . . . . . . . . . . . . . . . . . . . . . .
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Triangular Band Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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General Tridiagonal Matrix . . . . . . . . . . . . . . . . . . . . . . . . . .
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Symmetric Tridiagonal Matrix . . . . . . . . . . . . . . . . . . . . . . . .
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Positive Definite Symmetric Tridiagonal Matrix . . . . . . . . . . . . . .
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 90
Sparse Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 90
Contents
 v
Sequences
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Real and Complex Elements in Storage . . . . . . . .
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One-Dimensional Sequences . . . . . . . . . . . . . .
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Two-Dimensional Sequences . . . . . . . . . . . . . .
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Three-Dimensional Sequences . . . . . . . . . . . . .
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How Stride Is Used for Three-Dimensional Sequences
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Chapter 4. Coding Your Program . . . . . . . . . . . . . . . . . . . . . . . .
Fortran Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Calling ESSL Subroutines and Functions in Fortran . . . . . . . . . . . . .
Setting Up a User-Supplied Subroutine for ESSL in Fortran
 . . . . . . . .
Setting Up Scalar Data in Fortran . . . . . . . . . . . . . . . . . . . . . . . .
Setting Up Arrays in Fortran . . . . . . . . . . . . . . . . . . . . . . . . . . .
Creating Multiple Threads and Calling ESSL from Your Fortran Program .
Handling Errors in Your Fortran Program . . . . . . . . . . . . . . . . . . .
Example of Handling Errors in a Multithreaded Application Program . . . .
C Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Calling ESSL Subroutines and Functions in C
 . . . . . . . . . . . . . . . .
Passing Arguments in C . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Setting Up a User-Supplied Subroutine for ESSL in C . . . . . . . . . . . .
Setting Up Scalar Data in C . . . . . . . . . . . . . . . . . . . . . . . . . . .
Setting Up Complex and Logical Data Types in C . . . . . . . . . . . . . .
Setting Up Arrays in C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Creating Multiple Threads and Calling ESSL from Your C Program . . . .
Handling Errors in Your C Program . . . . . . . . . . . . . . . . . . . . . . .
C++ Programs
 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Calling ESSL Subroutines and Functions in C++ . . . . . . . . . . . . . . .
Passing Arguments in C++ . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Setting Up a User-Supplied Subroutine for ESSL in C++ . . . . . . . . . .
Setting Up Scalar Data in C++
 . . . . . . . . . . . . . . . . . . . . . . . . .
Setting Up Short-Precision Complex Data Types and Logical Data Types in
C++ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Setting Up Arrays in C++ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Creating Multiple Threads and Calling ESSL from Your C++ Program
 . .
Handling Errors in Your C++ Program . . . . . . . . . . . . . . . . . . . . .
PL/I Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 5. Processing Your Program . . . . . . . . . . . . . . . .
 . . . . .
Compiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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General Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . .
 . . . . .
Using Your Own Complex Data Definitions in C Programs . . . .
 . . . . .
Using Your Own Short Complex Data Definitions in C++ Programs
 . . . .
Compiling and Linking . . . . . . . . . . . . . . . . . . . . . . . . . . .
 . . . . .
64-bit environment
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Fortran Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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C Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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C++ Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 6. Migrating Your Programs . . . . . . . . . . . . .
 . . . . . . . . .
Migrating ESSL Version 3 Programs to Version 3 Release 2
 . . . . . . . . .
ESSL Subroutines . . . . . . . . . . . . . . . . . . . . . . . .
 . . . . . . . . .
Migrating ESSL Version 3 Programs to Version 3 Release 1.2
 . . . . . . . .
ESSL Subroutines . . . . . . . . . . . . . . . . . . . . . . . .
 . . . . . . . . .
Migrating ESSL Version 3 Programs to Version 3 Release 1.1
 . . . . . . . .
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 Engineering and Scientific Subroutine Library Version 3 Guide and Reference
ESSL Subroutines . . . . . . . . . . . . . . . . .
 . . . . . . . . . . . . . . . .
Migrating ESSL Version 2 Programs to Version 3
 . . . . . . . . . . . . . . . .
ESSL Subroutines . . . . . . . . . . . . . . . . .
 . . . . . . . . . . . . . . . .
ESSL Messages . . . . . . . . . . . . . . . . . .
 . . . . . . . . . . . . . . . .
Planning for Future Migration . . . . . . . . . . . .
 . . . . . . . . . . . . . . . .
Migrating between RS/6000 Processors . . . . . .
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Auxiliary Storage
 . . . . . . . . . . . . . . . . .
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Bitwise-Identical Results . . . . . . . . . . . . .
 . . . . . . . . . . . . . . . .
Migrating from Other Libraries to ESSL . . . . . .
 . . . . . . . . . . . . . . . .
Migrating from ESSL/370 . . . . . . . . . . . . .
 . . . . . . . . . . . . . . . .
Migrating from Another IBM Subroutine Library
 . . . . . . . . . . . . . . . .
Migrating from LAPACK . . . . . . . . . . . . .
 . . . . . . . . . . . . . . . .
Migrating from a Non-IBM Subroutine Library .
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Chapter 7. Handling Problems
 . . . . . . . . . . . . . .
 . . . . . . . . . . .
Where to Find More Information About Errors . . . . . . .
 . . . . . . . . . . .
Getting Help from IBM Support . . . . . . . . . . . . . . . .
 . . . . . . . . . . .
National Language Support . . . . . . . . . . . . . . . . . .
 . . . . . . . . . . .
Dealing with Errors . . . . . . . . . . . . . . . . . . . . . . .
 . . . . . . . . . . .
Program Exceptions
 . . . . . . . . . . . . . . . . . . . .
 . . . . . . . . . . .
ESSL Input-Argument Error Messages . . . . . . . . . .
 . . . . . . . . . . .
ESSL Computational Error Messages . . . . . . . . . .
 . . . . . . . . . . .
ESSL Resource Error Messages . . . . . . . . . . . . .
 . . . . . . . . . . .
ESSL Informational and Attention Messages . . . . . .
 . . . . . . . . . . .
Miscellaneous Error Messages . . . . . . . . . . . . . .
 . . . . . . . . . . .
Messages . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 . . . . . . . . . . .
Message Conventions . . . . . . . . . . . . . . . . . . .
 . . . . . . . . . . .
Input-Argument Error Messages(2001-2099) . . . . . .
 . . . . . . . . . . .
Computational Error Messages(2100-2199) . . . . . . .
 . . . . . . . . . . .
Input-Argument Error Messages(2200-2299) . . . . . .
 . . . . . . . . . . .
Resource Error Messages(2400-2499) . . . . . . . . . .
 . . . . . . . . . . .
Informational and Attention Error Messages(2600-2699)
 . . . . . . . . . .
Miscellaneous Error Messages(2700-2799) . . . . . . .
 . . . . . . . . . . .
171
171
171
172
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174
175
176
176
176
177
185
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188
188
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Part 2. Reference Information
 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 191
Chapter 8. Linear Algebra Subprograms . . . . . . . . . . . . . . . . . . .
Overview of the Linear Algebra Subprograms . . . . . . . . . . . . . . . . . .
Vector-Scalar Linear Algebra Subprograms . . . . . . . . . . . . . . . . . .
Sparse Vector-Scalar Linear Algebra Subprograms . . . . . . . . . . . . .
Matrix-Vector Linear Algebra Subprograms . . . . . . . . . . . . . . . . . .
Sparse Matrix-Vector Linear Algebra Subprograms
 . . . . . . . . . . . . .
Use Considerations
 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Performance and Accuracy Considerations . . . . . . . . . . . . . . . . . . . .
Vector-Scalar Subprograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ISAMAX, IDAMAX, ICAMAX, and IZAMAX—Position of the First or Last
Occurrence of the Vector Element Having the Largest Magnitude . . .
ISAMIN and IDAMIN—Position of the First or Last Occurrence of the
Vector Element Having Minimum Absolute Value . . . . . . . . . . . . .
ISMAX and IDMAX—Position of the First or Last Occurrence of the
Vector Element Having the Maximum Value . . . . . . . . . . . . . . . . .
ISMIN and IDMIN—Position of the First or Last Occurrence of the Vector
Element Having Minimum Value . . . . . . . . . . . . . . . . . . . . . . . .
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Contents
 vii
SASUM, DASUM, SCASUM, and DZASUM—Sum of the Magnitudes of
the Elements in a Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 211
SAXPY, DAXPY, CAXPY, and ZAXPY—Multiply a Vector X by a Scalar,
Add to a Vector Y, and Store in the Vector Y . . . . . . . . . . . . . . . .
 214
SCOPY, DCOPY, CCOPY, and ZCOPY—Copy a Vector . . . . . . . . . .
 217
SDOT, DDOT, CDOTU, ZDOTU, CDOTC, and ZDOTC—Dot Product of
Two Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 220
SNAXPY and DNAXPY—Compute SAXPY or DAXPY N Times . . . . . .
 224
SNDOT and DNDOT—Compute Special Dot Products N Times . . . . . .
 229
SNRM2, DNRM2, SCNRM2, and DZNRM2—Euclidean Length of a Vector
with Scaling of Input to Avoid Destructive Underflow and Overflow . . .
 234
SNORM2, DNORM2, CNORM2, and ZNORM2—Euclidean Length of a
Vector with No Scaling of Input . . . . . . . . . . . . . . . . . . . . . . . .
 237
SROTG, DROTG, CROTG, and ZROTG—Construct a Givens Plane
Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 240
SROT, DROT, CROT, ZROT, CSROT, and ZDROT—Apply a Plane
Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 247
SSCAL, DSCAL, CSCAL, ZSCAL, CSSCAL, and ZDSCAL—Multiply a
Vector X by a Scalar and Store in the Vector X . . . . . . . . . . . . . .
 251
SSWAP, DSWAP, CSWAP, and ZSWAP—Interchange the Elements of
Two Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 254
SVEA, DVEA, CVEA, and ZVEA—Add a Vector X to a Vector Y and
Store in a Vector Z
 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 257
SVES, DVES, CVES, and ZVES—Subtract a Vector Y from a Vector X
and Store in a Vector Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 261
SVEM, DVEM, CVEM, and ZVEM—Multiply a Vector X by a Vector Y and
Store in a Vector Z
 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 265
SYAX, DYAX, CYAX, ZYAX, CSYAX, and ZDYAX—Multiply a Vector X
by a Scalar and Store in a Vector Y
 . . . . . . . . . . . . . . . . . . . . .
 269
SZAXPY, DZAXPY, CZAXPY, and ZZAXPY—Multiply a Vector X by a
Scalar, Add to a Vector Y, and Store in a Vector Z . . . . . . . . . . . . .
 272
Sparse Vector-Scalar Subprograms . . . . . . . . . . . . . . . . . . . . . . . .
 276
SSCTR, DSCTR, CSCTR, ZSCTR—Scatter the Elements of a Sparse
Vector X in Compressed-Vector Storage Mode into Specified Elements
of a Sparse Vector Y in Full-Vector Storage Mode . . . . . . . . . . . .
 277
SGTHR, DGTHR, CGTHR, and ZGTHR—Gather Specified Elements of a
Sparse Vector Y in Full-Vector Storage Mode into a Sparse Vector X in
Compressed-Vector Storage Mode . . . . . . . . . . . . . . . . . . . . . .
 280
SGTHRZ, DGTHRZ, CGTHRZ, and ZGTHRZ—Gather Specified Elements
of a Sparse Vector Y in Full-Vector Mode into a Sparse Vector X in
Compressed-Vector Mode, and Zero the Same Specified Elements of Y
 283
SAXPYI, DAXPYI, CAXPYI, and ZAXPYI—Multiply a Sparse Vector X in
Compressed-Vector Storage Mode by a Scalar, Add to a Sparse Vector
Y in Full-Vector Storage Mode, and Store in the Vector Y . . . . . . . .
 286
SDOTI, DDOTI, CDOTUI, ZDOTUI, CDOTCI, and ZDOTCI—Dot Product
of a Sparse Vector X in Compressed-Vector Storage Mode and a
Sparse Vector Y in Full-Vector Storage Mode . . . . . . . . . . . . . . . .
 289
Matrix-Vector Subprograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 293
SGEMV, DGEMV, CGEMV, ZGEMV, SGEMX, DGEMX, SGEMTX, and
DGEMTX—Matrix-Vector Product for a General Matrix, Its Transpose, or
Its Conjugate Transpose . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 294
SGER, DGER, CGERU, ZGERU, CGERC, and ZGERC—Rank-One
Update of a General Matrix
 . . . . . . . . . . . . . . . . . . . . . . . . . .
 305
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 Engineering and Scientific Subroutine Library Version 3 Guide and Reference
SSPMV, DSPMV, CHPMV, ZHPMV, SSYMV, DSYMV, CHEMV, ZHEMV,
SSLMX, and DSLMX—Matrix-Vector Product for a Real Symmetric or
Complex Hermitian Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . .
SSPR, DSPR, CHPR, ZHPR, SSYR, DSYR, CHER, ZHER, SSLR1, and
DSLR1 —Rank-One Update of a Real Symmetric or Complex Hermitian
Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SSPR2, DSPR2, CHPR2, ZHPR2, SSYR2, DSYR2, CHER2, ZHER2,
SSLR2, and DSLR2—Rank-Two Update of a Real Symmetric or
Complex Hermitian Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . .
SGBMV, DGBMV, CGBMV, and ZGBMV—Matrix-Vector Product for a
General Band Matrix, Its Transpose, or Its Conjugate Transpose
 . . .
SSBMV, DSBMV, CHBMV, and ZHBMV—Matrix-Vector Product for a
Real Symmetric or Complex Hermitian Band Matrix . . . . . . . . . . . .
STRMV, DTRMV, CTRMV, ZTRMV, STPMV, DTPMV, CTPMV, and
ZTPMV—Matrix-Vector Product for a Triangular Matrix, Its Transpose, or
Its Conjugate Transpose . . . . . . . . . . . . . . . . . . . . . . . . . . . .
STBMV, DTBMV, CTBMV, and ZTBMV—Matrix-Vector Product for a
Triangular Band Matrix, Its Transpose, or Its Conjugate Transpose . .
Sparse Matrix-Vector Subprograms . . . . . . . . . . . . . . . . . . . . . . . .
DSMMX—Matrix-Vector Product for a Sparse Matrix in
Compressed-Matrix Storage Mode . . . . . . . . . . . . . . . . . . . . . .
DSMTM—Transpose a Sparse Matrix in Compressed-Matrix Storage
Mode
 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
DSDMX—Matrix-Vector Product for a Sparse Matrix or Its Transpose in
Compressed-Diagonal Storage Mode . . . . . . . . . . . . . . . . . . . . .
313
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338
345
350
356
362
363
366
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Chapter 9. Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . .
Overview of the Matrix Operation Subroutines
 . . . . . . . . . . . . . . . . .
Use Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Specifying Normal, Transposed, or Conjugate Transposed Input Matrices
Transposing or Conjugate Transposing: . . . . . . . . . . . . . . . . . . . .
Performance and Accuracy Considerations . . . . . . . . . . . . . . . . . . . .
In General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
For Large Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
For Combined Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Matrix Operation Subroutines . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SGEADD, DGEADD, CGEADD, and ZGEADD—Matrix Addition for
General Matrices or Their Transposes . . . . . . . . . . . . . . . . . . . .
SGESUB, DGESUB, CGESUB, and ZGESUB—Matrix Subtraction for
General Matrices or Their Transposes . . . . . . . . . . . . . . . . . . . .
SGEMUL, DGEMUL, CGEMUL, and ZGEMUL—Matrix Multiplication for
General Matrices, Their Transposes, or Conjugate Transposes . . . . .
SGEMMS, DGEMMS, CGEMMS, and ZGEMMS—Matrix Multiplication for
General Matrices, Their Transposes, or Conjugate Transposes Using
Winograd's Variation of Strassen's Algorithm . . . . . . . . . . . . . . .
SGEMM, DGEMM, CGEMM, and ZGEMM—Combined Matrix
Multiplication and Addition for General Matrices, Their Transposes, or
Conjugate Transposes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SSYMM, DSYMM, CSYMM, ZSYMM, CHEMM, and
ZHEMM—Matrix-Matrix Product Where One Matrix is Real or Complex
Symmetric or Complex Hermitian . . . . . . . . . . . . . . . . . . . . . . .
STRMM, DTRMM, CTRMM, and ZTRMM—Triangular Matrix-Matrix
Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 ix
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x
 EngineeringSSYRK, DSYRK, CSYRK, ZSYRK, CHERK, and ZHERK—Rank-K
Update of a Real or Complex Symmetric or a Complex Hermitian Matrix
SSYR2K, DSYR2K, CSYR2K, ZSYR2K, CHER2K, and
ZHER2K—Rank-2K Update of a Real or Complex Symmetric or a
Complex Hermitian Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . .
SGETMI, DGETMI, CGETMI, and ZGETMI—General Matrix Transpose
(In-Place) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SGETMO, DGETMO, CGETMO, and ZGETMO—General Matrix
Transpose (Out-of-Place) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
433
440
448
451
Chapter 10. Linear Algebraic Equations . . . . . . . . . . . . . . . . . . . .
Overview of the Linear Algebraic Equation Subroutines
 . . . . . . . . . . . .
Dense Linear Algebraic Equation Subroutines
 . . . . . . . . . . . . . . . .
Banded Linear Algebraic Equation Subroutines . . . . . . . . . . . . . . . .
Sparse Linear Algebraic Equation Subroutines . . . . . . . . . . . . . . . .
Linear Least Squares Subroutines . . . . . . . . . . . . . . . . . . . . . . .
Dense and Banded Linear Algebraic Equation Considerations . . . . . . . . .
Use Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Performance and Accuracy Considerations . . . . . . . . . . . . . . . . . .
Sparse Matrix Direct Solver Considerations
 . . . . . . . . . . . . . . . . . . .
Use Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Performance and Accuracy Considerations . . . . . . . . . . . . . . . . . .
Sparse Matrix Skyline Solver Considerations . . . . . . . . . . . . . . . . . . .
Use Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Performance and Accuracy Considerations . . . . . . . . . . . . . . . . . .
Sparse Matrix Iterative Solver Considerations . . . . . . . . . . . . . . . . . .
Use Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Performance and Accuracy Considerations . . . . . . . . . . . . . . . . . .
Linear Least Squares Considerations . . . . . . . . . . . . . . . . . . . . . . .
Use Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Performance and Accuracy Considerations . . . . . . . . . . . . . . . . . .
Dense Linear Algebraic Equation Subroutines . . . . . . . . . . . . . . . . . .
SGEF, DGEF, CGEF, and ZGEF—General Matrix Factorization . . . . .
SGES, DGES, CGES, and ZGES—General Matrix, Its Transpose, or Its
Conjugate Transpose Solve . . . . . . . . . . . . . . . . . . . . . . . . . .
SGESM, DGESM, CGESM, and ZGESM—General Matrix, Its Transpose,
or Its Conjugate Transpose Multiple Right-Hand Side Solve
 . . . . . .
SGETRF, DGETRF, CGETRF and ZGETRF—General Matrix
Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SGETRS, DGETRS, CGETRS, and ZGETRS—General Matrix Multiple
Right-Hand Side Solve . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SGEFCD and DGEFCD—General Matrix Factorization, Condition Number
Reciprocal, and Determinant . . . . . . . . . . . . . . . . . . . . . . . . . .
SPPF, DPPF, SPOF, DPOF, CPOF, and ZPOF—Positive Definite Real
Symmetric or Complex Hermitian Matrix Factorization . . . . . . . . . .
SPPS and DPPS—Positive Definite Real Symmetric Matrix Solve . . . .
SPOSM, DPOSM, CPOSM, and ZPOSM—Positive Definite Real
Symmetric or Complex Hermitian Matrix Multiple Right-Hand Side Solve
SPPFCD, DPPFCD, SPOFCD, and DPOFCD—Positive Definite Real
Symmetric Matrix Factorization, Condition Number Reciprocal, and
Determinant
 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
DBSSV—Symmetric Indefinite Matrix Factorization and Multiple
Right-Hand Side Solve . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
DBSTRF—Symmetric Indefinite Matrix Factorization . . . . . . . . . . . . .
455
455
455
456
457
458
458
458
459
459
459
459
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and Scientific Subroutine Library Version 3 Guide and Reference
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 DBSTRS—Symmetric Indefinite Matrix Multiple Right-Hand Side Solve .
SGEICD and DGEICD—General Matrix Inverse, Condition Number
Reciprocal, and Determinant . . . . . . . . . . . . . . . . . . . . . . . . . .
SPPICD, DPPICD, SPOICD, and DPOICD—Positive Definite Real
Symmetric Matrix Inverse, Condition Number Reciprocal, and
Determinant
 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
STRSV, DTRSV, CTRSV, ZTRSV, STPSV, DTPSV, CTPSV, and
ZTPSV—Solution of a Triangular System of Equations with a Single
Right-Hand Side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
STRSM, DTRSM, CTRSM, and ZTRSM—Solution of Triangular Systems
of Equations with Multiple Right-Hand Sides . . . . . . . . . . . . . . . .
STRI, DTRI, STPI, and DTPI—Triangular Matrix Inverse . . . . . . . . . .
Banded Linear Algebraic Equation Subroutines . . . . . . . . . . . . . . . . .
SGBF and DGBF—General Band Matrix Factorization
 . . . . . . . . . . .
SGBS and DGBS—General Band Matrix Solve . . . . . . . . . . . . . . . .
SPBF, DPBF, SPBCHF, and DPBCHF—Positive Definite Symmetric Band
Matrix Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SPBS, DPBS, SPBCHS, and DPBCHS—Positive Definite Symmetric
Band Matrix Solve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SGTF and DGTF—General Tridiagonal Matrix Factorization . . . . . . .
SGTS and DGTS—General Tridiagonal Matrix Solve . . . . . . . . . . . .
SGTNP, DGTNP, CGTNP, and ZGTNP—General Tridiagonal Matrix
Combined Factorization and Solve with No Pivoting . . . . . . . . . . . .
SGTNPF, DGTNPF, CGTNPF, and ZGTNPF—General Tridiagonal Matrix
Factorization with No Pivoting . . . . . . . . . . . . . . . . . . . . . . . . .
SGTNPS, DGTNPS, CGTNPS, and ZGTNPS—General Tridiagonal Matrix
Solve with No Pivoting . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SPTF and DPTF—Positive Definite Symmetric Tridiagonal Matrix
Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SPTS and DPTS—Positive Definite Symmetric Tridiagonal Matrix Solve
STBSV, DTBSV, CTBSV, and ZTBSV—Triangular Band Equation Solve
Sparse Linear Algebraic Equation Subroutines . . . . . . . . . . . . . . . . . .
DGSF—General Sparse Matrix Factorization Using Storage by Indices,
Rows, or Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
DGSS—General Sparse Matrix or Its Transpose Solve Using Storage by
Indices, Rows, or Columns . . . . . . . . . . . . . . . . . . . . . . . . . . .
DGKFS—General Sparse Matrix or Its Transpose Factorization,
Determinant, and Solve Using Skyline Storage Mode . . . . . . . . . .
DSKFS—Symmetric Sparse Matrix Factorization, Determinant, and Solve
Using Skyline Storage Mode . . . . . . . . . . . . . . . . . . . . . . . . . .
DSRIS—Iterative Linear System Solver for a General or Symmetric
Sparse Matrix Stored by Rows . . . . . . . . . . . . . . . . . . . . . . . . .
DSMCG—Sparse Positive Definite or Negative Definite Symmetric Matrix
Iterative Solve Using Compressed-Matrix Storage Mode . . . . . . . . .
DSDCG—Sparse Positive Definite or Negative Definite Symmetric Matrix
Iterative Solve Using Compressed-Diagonal Storage Mode . . . . . . .
DSMGCG—General Sparse Matrix Iterative Solve Using
Compressed-Matrix Storage Mode . . . . . . . . . . . . . . . . . . . . . .
DSDGCG—General Sparse Matrix Iterative Solve Using
Compressed-Diagonal Storage Mode . . . . . . . . . . . . . . . . . . . . .
Linear Least Squares Subroutines . . . . . . . . . . . . . . . . . . . . . . . . .
SGESVF and DGESVF—Singular Value Decomposition for a General
Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
525
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 xi
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SGESVS and DGESVS—Linear Least Squares Solution for a General
Matrix Using the Singular Value Decomposition . . . . . . . . . . . . .
 . .
DGEQRF—General Matrix QR Factorization . . . . . . . . . . . . . . .
 . .
SGELLS and DGELLS—Linear Least Squares Solution for a General
Matrix Using a QR Decomposition with Column Pivoting
 . . . . . .
 . .
698
703
708
Chapter 11. Eigensystem Analysis . . . . . . . . . . . . . . . . . . . . . . .
 715
Overview of the Eigensystem Analysis Subroutines . . . . . . . . . . . . . . .
 715
Performance and Accuracy Considerations . . . . . . . . . . . . . . . . . . . .
 715
Eigensystem Analysis Subroutines . . . . . . . . . . . . . . . . . . . . . . . . .
 717
SGEEV, DGEEV, CGEEV, and ZGEEV—Eigenvalues and, Optionally, All
or Selected Eigenvectors of a General Matrix . . . . . . . . . . . . . . .
 718
SSPEV, DSPEV, CHPEV, and ZHPEV—Eigenvalues and, Optionally, the
Eigenvectors of a Real Symmetric Matrix or a Complex Hermitian Matrix
 729
SSPSV, DSPSV, CHPSV, and ZHPSV—Extreme Eigenvalues and,
Optionally, the Eigenvectors of a Real Symmetric Matrix or a Complex
Hermitian Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 738
SGEGV and DGEGV—Eigenvalues and, Optionally, the Eigenvectors of a
Generalized Real Eigensystem, Az=wBz, where A and B Are Real
General Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 746
SSYGV and DSYGV—Eigenvalues and, Optionally, the Eigenvectors of a
Generalized Real Symmetric Eigensystem, Az=wBz, where A Is Real
Symmetric and B Is Real Symmetric Positive Definite . . . . . . . . . .
 752
Chapter 12. Fourier Transforms, Convolutions and Correlations, and
Related Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 .
Overview of the Signal Processing Subroutines . . . . . . . . . . . . . . . .
 .
Fourier Transforms Subroutines . . . . . . . . . . . . . . . . . . . . . . . .
 .
Convolution and Correlation Subroutines
 . . . . . . . . . . . . . . . . . .
 .
Related-Computation Subroutines
 . . . . . . . . . . . . . . . . . . . . . .
 .
Fourier Transforms, Convolutions, and Correlations Considerations
 . . . .
 .
Use Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 .
Initializing Auxiliary Working Storage . . . . . . . . . . . . . . . . . . . . .
 .
Determining the Amount of Auxiliary Working Storage That You Need .
 .
Performance and Accuracy Considerations . . . . . . . . . . . . . . . . . . .
 .
When Running on the Workstation Processors . . . . . . . . . . . . . . .
 .
Defining Arrays
 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 .
Fourier Transform Considerations . . . . . . . . . . . . . . . . . . . . . . .
 .
How the Fourier Transform Subroutines Achieve High Performance . .
 .
Convolution and Correlation Considerations . . . . . . . . . . . . . . . . .
 .
Related Computation Considerations . . . . . . . . . . . . . . . . . . . . . .
 .
Accuracy Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 .
Fourier Transform Subroutines . . . . . . . . . . . . . . . . . . . . . . . . . .
 .
SCFT and DCFT—Complex Fourier Transform . . . . . . . . . . . . . . .
 .
SRCFT and DRCFT—Real-to-Complex Fourier Transform . . . . . . . .
 .
SCRFT and DCRFT—Complex-to-Real Fourier Transform . . . . . . . .
 .
SCOSF and DCOSF—Cosine Transform . . . . . . . . . . . . . . . . . .
 .
SSINF and DSINF—Sine Transform . . . . . . . . . . . . . . . . . . . . .
 .
SCFT2 and DCFT2—Complex Fourier Transform in Two Dimensions
 .
SRCFT2 and DRCFT2—Real-to-Complex Fourier Transform in Two
Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 .
SCRFT2 and DCRFT2—Complex-to-Real Fourier Transform in Two
Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 .
SCFT3 and DCFT3—Complex Fourier Transform in Three Dimensions
 .
759
759
759
760
760
761
761
763
763
764
764
764
764
765
765
767
767
768
769
777
785
793
800
807
814
821
829
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 Engineering and Scientific Subroutine Library Version 3 Guide and Reference
SRCFT3 and DRCFT3—Real-to-Complex Fourier Transform in Three
Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SCRFT3 and DCRFT3—Complex-to-Real Fourier Transform in Three
Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Convolution and Correlation Subroutines . . . . . . . . . . . . . . . . . . . . .
SCON and SCOR—Convolution or Correlation of One Sequence with
One or More Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SCOND and SCORD—Convolution or Correlation of One Sequence with
Another Sequence Using a Direct Method . . . . . . . . . . . . . . . . . .
SCONF and SCORF—Convolution or Correlation of One Sequence with
One or More Sequences Using the Mixed-Radix Fourier Method . . . .
SDCON, DDCON, SDCOR, and DDCOR—Convolution or Correlation with
Decimated Output Using a Direct Method . . . . . . . . . . . . . . . . . .
SACOR—Autocorrelation of One or More Sequences . . . . . . . . . . . .
SACORF—Autocorrelation of One or More Sequences Using the
Mixed-Radix Fourier Method . . . . . . . . . . . . . . . . . . . . . . . . . .
Related-Computation Subroutines . . . . . . . . . . . . . . . . . . . . . . . . .
SPOLY and DPOLY—Polynomial Evaluation . . . . . . . . . . . . . . . . .
SIZC and DIZC—I-th Zero Crossing . . . . . . . . . . . . . . . . . . . . . .
STREC and DTREC—Time-Varying Recursive Filter
 . . . . . . . . . . . .
SQINT and DQINT—Quadratic Interpolation
 . . . . . . . . . . . . . . . . .
SWLEV, DWLEV, CWLEV, and ZWLEV—Wiener-Levinson Filter
Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
835
841
847
848
854
860
869
873
877
882
883
886
889
892
896
Chapter 13. Sorting and Searching . . . . . . . . . . . . . . . . . . . . . . .
Overview of the Sorting and Searching Subroutines . . . . . . . . . . . . . .
Use Considerations
 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Performance and Accuracy Considerations . . . . . . . . . . . . . . . . . . .
Sorting and Searching Subroutines . . . . . . . . . . . . . . . . . . . . . . . .
ISORT, SSORT, and DSORT—Sort the Elements of a Sequence . . . .
ISORTX, SSORTX, and DSORTX—Sort the Elements of a Sequence and
Note the Original Element Positions
 . . . . . . . . . . . . . . . . . . . . .
ISORTS, SSORTS, and DSORTS—Sort the Elements of a Sequence
Using a Stable Sort and Note the Original Element Positions . . . . . .
IBSRCH, SBSRCH, and DBSRCH—Binary Search for Elements of a
Sequence X in a Sorted Sequence Y . . . . . . . . . . . . . . . . . . . . .
ISSRCH, SSSRCH, and DSSRCH—Sequential Search for Elements of a
Sequence X in the Sequence Y . . . . . . . . . . . . . . . . . . . . . . . .
901
901
901
901
903
904
906
909
912
916
Chapter 14. Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . .
 . . .
Overview of the Interpolation Subroutines . . . . . . . . . . . . . . . . .
 . . .
Use Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 . . .
Performance and Accuracy Considerations . . . . . . . . . . . . . . . .
 . . .
Interpolation Subroutines . . . . . . . . . . . . . . . . . . . . . . . . . . .
 . . .
SPINT and DPINT—Polynomial Interpolation . . . . . . . . . . . . . .
 . . .
STPINT and DTPINT—Local Polynomial Interpolation . . . . . . . . .
 . . .
SCSINT and DCSINT—Cubic Spline Interpolation . . . . . . . . . . .
 . . .
SCSIN2 and DCSIN2—Two-Dimensional Cubic Spline Interpolation
 . . .
921
921
921
921
922
923
928
931
937
Chapter 15. Numerical Quadrature . . . . . . . .
 . . . . . . . . . . . . . . .
Overview of the Numerical Quadrature Subroutines
 . . . . . . . . . . . . . .
Use Considerations . . . . . . . . . . . . . . . . . .
 . . . . . . . . . . . . . . .
Choosing the Method . . . . . . . . . . . . . . . .
 . . . . . . . . . . . . . . .
Performance and Accuracy Considerations . . . .
 . . . . . . . . . . . . . . .
941
941
941
941
941
Contents
 xiii
Programming Considerations for the SUBF Subroutine . . . . . . . . . . . . .
Designing SUBF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Coding and Setting Up SUBF in Your Program . . . . . . . . . . . . . . . .
Numerical Quadrature Subroutines
 . . . . . . . . . . . . . . . . . . . . . . . .
SPTNQ and DPTNQ—Numerical Quadrature Performed on a Set of
Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SGLNQ and DGLNQ—Numerical Quadrature Performed on a Function
Using Gauss-Legendre Quadrature . . . . . . . . . . . . . . . . . . . . . .
SGLNQ2 and DGLNQ2—Numerical Quadrature Performed on a Function
Over a Rectangle Using Two-Dimensional Gauss-Legendre Quadrature
SGLGQ and DGLGQ—Numerical Quadrature Performed on a Function
Using Gauss-Laguerre Quadrature . . . . . . . . . . . . . . . . . . . . . .
SGRAQ and DGRAQ—Numerical Quadrature Performed on a Function
Using Gauss-Rational Quadrature . . . . . . . . . . . . . . . . . . . . . . .
SGHMQ and DGHMQ—Numerical Quadrature Performed on a Function
Using Gauss-Hermite Quadrature . . . . . . . . . . . . . . . . . . . . . . .
942
942
943
944
945
948
951
957
960
964
Chapter 16. Random Number Generation . . . . . . . . . . . . . . . . . .
 .
Overview of the Random Number Generation Subroutines . . . . . . . . . .
 .
Use Considerations
 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 .
Random Number Generation Subroutines
 . . . . . . . . . . . . . . . . . . .
 .
SURAND and DURAND—Generate a Vector of Uniformly Distributed
Random Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 .
SNRAND and DNRAND—Generate a Vector of Normally Distributed
Random Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 .
SURXOR and DURXOR—Generate a Vector of Long Period Uniformly
Distributed Random Numbers . . . . . . . . . . . . . . . . . . . . . . . .
 .
967
967
967
967
968
971
975
Chapter 17. Utilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 979
Overview of the Utility Subroutines . . . . . . . . . . . . . . . . . . . . . . . . 979
Use Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 979
Determining the Level of ESSL Installed . . . . . . . . . . . . . . . . . . . . 979
Finding the Optimal Stride(s) for Your Fourier Transforms
 . . . . . . . . .
 979
Converting Sparse Matrix Storage . . . . . . . . . . . . . . . . . . . . . . . 980
Utility Subroutines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 981
EINFO—ESSL Error Information-Handler Subroutine
 . . . . . . . . . . . .
 982
ERRSAV—ESSL ERRSAV Subroutine for ESSL . . . . . . . . . . . . . . . 985
ERRSET—ESSL ERRSET Subroutine for ESSL . . . . . . . . . . . . . . . 986
ERRSTR—ESSL ERRSTR Subroutine for ESSL . . . . . . . . . . . . . . . 988
IESSL—Determine the Level of ESSL Installed . . . . . . . . . . . . . . . . 989
STRIDE—Determine the Stride Value for Optimal Performance in
Specified Fourier Transform Subroutines . . . . . . . . . . . . . . . . . . . 991
DSRSM—Convert a Sparse Matrix from Storage-by-Rows to
Compressed-Matrix Storage Mode . . . . . . . . . . . . . . . . . . . . . . 1001
DGKTRN—For a General Sparse Matrix, Convert Between Diagonal-Out
and Profile-In Skyline Storage Mode . . . . . . . . . . . . . . . . . . . . . 1005
DSKTRN—For a Symmetric Sparse Matrix, Convert Between
Diagonal-Out and Profile-In Skyline Storage Mode . . . . . . . . . . . . . 1011
Part 3. Appendixes
 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 1017
Appendix A. Basic Linear Algebra Subprograms (BLAS)
 . . . . . . . .
 APA-1
Level 1 BLAS . . . . . . . . . . . . . . . . . . . . . . . . . . .
 . . . . . . . . .
 APA-1
xiv
 Engineering and Scientific Subroutine Library Version 3 Guide and Reference
Level 2 BLAS
 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 APA-1
Level 3 BLAS
 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 APA-2
Appendix B. LAPACK
 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 APB-1
LAPACK . . . . . . . .
 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 APB-1
Notices . . . . . . . . .
 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 NOT-1
Trademarks . . . . . . .
 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 NOT-2
Software Update Protocol
 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 NOT-2
Programming Interfaces
 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 NOT-2
Glossary
 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 GLOS-1
Bibliography . . . . . . . . . . . . . . . . .
 . . . . . . . . . . . . . . . . . . .
 BIB-1
References . . . . . . . . . . . . . . . . . .
 . . . . . . . . . . . . . . . . . . .
 BIB-1
ESSL Publications . . . . . . . . . . . . . .
 . . . . . . . . . . . . . . . . . . .
 BIB-5
Evaluation and Planning . . . . . . . . .
 . . . . . . . . . . . . . . . . . . .
 BIB-5
Installation . . . . . . . . . . . . . . . . .
 . . . . . . . . . . . . . . . . . . .
 BIB-5
Application Programming . . . . . . . . .
 . . . . . . . . . . . . . . . . . . .
 BIB-5
Related Publications . . . . . . . . . . . . .
 . . . . . . . . . . . . . . . . . . .
 BIB-5
AIX for the RS/6000
 . . . . . . . . . . .
 . . . . . . . . . . . . . . . . . . .
 BIB-5
AIX Version 4 Release 3 for the RS/6000
 . . . . . . . . . . . . . . . . . .
 BIB-5
XL Fortran . . . . . . . . . . . . . . . . .
 . . . . . . . . . . . . . . . . . . .
 BIB-5
PL/I . . . . . . . . . . . . . . . . . . . . .
 . . . . . . . . . . . . . . . . . . .
 BIB-5
Workstation Processors
 . . . . . . . . .
 . . . . . . . . . . . . . . . . . . .
 BIB-5
IBM 3838 Array Processor . . . . . . . .
 . . . . . . . . . . . . . . . . . . .
 BIB-6
Index
 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 INDEX-1