Mark 24 NAG Library for SMP & Multicore News (PDF version)

NAG Library

Mark 24 NAG Library for SMP & Multicore News

+− Contents

+− 1 Introduction

1.1 New Functionality

1.2 New SMP parallelism and other optimizations

2 New Routines

3 Tuned Routines

4 Enhanced Routines

5 Withdrawn Routines

6 Routines Scheduled for Withdrawal

1 Introduction

1.1 New Functionality

At Mark 24 of the NAG Library for SMP & Multicore new functionality has been introduced in addition to improvements in existing areas. The Library now contains 1784 user-callable routines of which 139 are new at this mark.

New Chapter X07 (IEEE Arithmetic) has been introduced, providing routines relating to IEEE arithmetic such as determining or creating an infinite value or a NaN (Not a Number). There have also been extensions in functionality included in the areas of statistics, wavelets, ordinary differential equations, interpolation, surface fitting, optimization, matrix operations, linear algebra, operations research, and special functions.

Chapter C06 (Summation of Series) has Fast Fourier Transforms (FFTs) for two-dimensional and three-dimensional real data.

Chapter C09 (Wavelet Transforms) has three-dimensional discrete wavelet transforms.

Chapter D01 (Quadrature) has a comprehensive one-dimensional adaptive quadrature routine and a variant for badly behaved integrands.

Chapter D02 (Ordinary Differential Equations) has threadsafe versions of the suite implementing Runge–Kutta methods.

Chapter E01 (Interpolation) has the modified Shepard's method for interpolating in dimensions greater than

5

Chapter E02 (Curve and Surface Fitting) has a two-stage approximation method for two-dimensional scattered data.

Chapter E04 (Minimizing or Maximizing a Function) has non-negative least squares and an improved MPS data reader.

Chapter E05 (Global Optimization of a Function) has multi-start versions of general nonlinear programming and least squares routines.

Chapter F01 (Matrix Operations, Including Inversion) has greatly extended its range of matrix function routines including the calculation of condition numbers and the action on another matrix.

Chapter F02 (Eigenvalues and Eigenvectors) has a driver routine for calculating selected eigenvalues/vectors of real sparse general matrices.

Chapter F04 (Simultaneous Linear Equations) has norm estimators for rectangular matrices.

Chapter F11 (Large Scale Linear Systems) has a block diagonal (possibly overlapping) preconditioner and associated solver for real and complex nonsymmetric sparse matrices.

Chapter F12 (Large Scale Eigenproblems) has a driver for selected eigenvalues/vectors of general banded complex eigenproblems.

Chapter F16 (Further Linear Algebra Support Routines) has two additions from the BLAST set of routines.

Chapter G01 (Simple Calculations on Statistical Data) has routines for combining summary statistics from blocks of data, probabilities from a multivariate Student's

t

-distribution, and a large set of vectorized versions of routines for probabilities and density functions.

Chapter G02 (Correlation and Regression Analysis) has routines for weighted nearest correlation matrix and combining two sums of squares.

Chapter G03 (Multivariate Methods) has a Gaussian mixture model routine.

Chapter G05 (Random Number Generators) has Brownian bridge and random field routines.

Chapter G13 (Time Series Analysis) has moving averages for inhomogeneous time series.

Chapter H (Operations Research) has routines for computing best subsets.

Chapter S (Approximations of Special Functions) has added special functions: confluent hypergeometric, log beta and incomplete beta; additionally a large set of vectorized versions of existing special functions.

Chapter X07 (IEEE Arithmetic) has a set of IEEE routines including testing and setting Infs and NaNs.

1.2 New SMP parallelism and other optimizations

SMP parallelism has been added to new routines introduced at Mark 24 in the areas of discrete Fourier transforms, 3-D wavelets, multi-dimensional interpolation, cubic splines, global optimization, matrix functions, the mean and standard deviation of an arbitrary-sized data stream using a rolling window, Gaussian mixture model, Brownian bridge and univariate inhomogenous time series. An existing routine for Heston's model option pricing formula with Greeks has also been parallelized.

2 New Routines

The 139 new user-callable routines included in the NAG Library at Mark 24 are as follows.

Routine
Name
Purpose

C06PVF Two-dimensional real-to-complex discrete Fourier transform

C06PWF Two-dimensional complex-to-real discrete Fourier transform

C06PYF Three-dimensional real-to-complex discrete Fourier transform

C06PZF Three-dimensional complex-to-real discrete Fourier transform

C09ACF Three-dimensional wavelet filter initialization

C09FAF Three-dimensional discrete wavelet transform

C09FBF Three-dimensional inverse discrete wavelet transform

C09FCF Three-dimensional multi-level discrete wavelet transform

C09FDF Three-dimensional inverse multi-level discrete wavelet transform

D01RAF One-dimensional quadrature, adaptive, finite interval, multiple integrands, vectorized abscissae, reverse communication

D01RBF Diagnostic routine for D01RAF

D01RCF Determine required array dimensions for D01RAF

D01RGF One-dimensional quadrature, adaptive, finite interval, strategy due to Gonnet, allowing for badly behaved integrands

D01TBF Pre-computed weights and abscissae for Gaussian quadrature rules, restricted choice of rule

D01UAF One-dimensional Gaussian quadrature, choice of weight functions

D01ZKF Option setting routine

D01ZLF Option getting routine

D02PEF Ordinary differential equations, initial value problem, Runge–Kutta method, integration over range with output

D02PFF Ordinary differential equations, initial value problem, Runge–Kutta method, integration over one step

D02PQF Ordinary differential equations, initial value problem, setup for D02PEF and D02PFF

D02PRF Ordinary differential equations, initial value problem, resets end of range for D02PFF

D02PSF Ordinary differential equations, initial value problem, interpolation for D02PFF

D02PTF Ordinary differential equations, initial value problem, integration diagnostics for D02PEF and D02PFF

D02PUF Ordinary differential equations, initial value problem, error assessment diagnostics for D02PEF and D02PFF

E01ZMF Interpolating function, modified Shepard's method,

d

dimensions

E01ZNF Interpolated values, evaluate interpolant computed by E01ZMF, function and first derivatives,

d

dimensions

E02BFF Evaluation of fitted cubic spline, function and optionally derivatives at a vector of points

E02JDF Spline approximation to a set of scattered data using a two-stage approximation method

E02JEF Evaluation at a vector of points of a spline computed by E02JDF

E02JFF Evaluation at a mesh of points of a spline computed by E02JDF

E02ZKF Option setting routine

E02ZLF Option getting routine

E04MXF Reads MPS data file defining LP, QP, MILP or MIQP problem

E04PCF Computes the least squares solution to a set of linear equations subject to fixed upper and lower bounds on the variables. An option is provided to return a minimal length solution if a solution is not unique

E05UCF Global optimization using multi-start, nonlinear constraints

E05USF Global optimization of a sum of squares problem using multi-start, nonlinear constraints

F01EJF Real matrix logarithm

F01EKF Exponential, sine, cosine, sinh or cosh of a real matrix (Schur–Parlett algorithm)

F01ELF Function of a real matrix (using numerical differentiation)

F01EMF Function of a real matrix (using user-supplied derivatives)

F01FJF Complex matrix logarithm

F01FKF Exponential, sine, cosine, sinh or cosh of a complex matrix (Schur–Parlett algorithm)

F01FLF Function of a complex matrix (using numerical differentiation)

F01FMF Function of a complex matrix (using user-supplied derivatives)

F01GAF Action of a real matrix exponential on a real matrix

F01GBF Action of a real matrix exponential on a real matrix (reverse communication)

F01HAF Action of a complex matrix exponential on a complex matrix

F01HBF Action of a complex matrix exponential on a complex matrix (reverse communication)

F01JAF Condition number for the exponential, logarithm, sine, cosine, sinh or cosh of a real matrix

F01JBF Condition number for a function of a real matrix (using numerical differentiation)

F01JCF Condition number for a function of a real matrix (using user-supplied derivatives)

F01KAF Condition number for the exponential, logarithm, sine, cosine, sinh or cosh of a complex matrix

F01KBF Condition number for a function of a complex matrix (using numerical differentiation)

F01KCF Condition number for a function of a complex matrix (using user-supplied derivatives)

F02EKF Selected eigenvalues and eigenvectors of a real sparse general matrix

F04YDF Norm estimation (for use in condition estimation), real rectangular matrix

F04ZDF Norm estimation (for use in condition estimation), complex rectangular matrix

F11DFF Real sparse nonsymmetric linear system, incomplete

L U

factorization of local or overlapping diagonal blocks

F11DGF Solution of real sparse nonsymmetric linear system, RGMRES, CGS, Bi-CGSTAB or TFQMR method, incomplete

L U

block diagonal preconditioner computed by F11DFF

F11DTF Complex sparse nonsymmetric linear system, incomplete

L U

factorization of local or overlapping diagonal blocks

F11DUF Solution of complex sparse nonsymmetric linear system, RGMRES, CGS, Bi-CGSTAB or TFQMR method, incomplete

L U

block diagonal preconditioner computed by F11DTF

F12ATF Initialization routine for (F12AUF) computing selected eigenvalues and, optionally, eigenvectors of a complex banded (standard or generalized) eigenproblem.

F12AUF Selected eigenvalues and, optionally, eigenvectors of complex non-Hermitian banded eigenproblem, driver

F16ECF Real scaled vector accumulation

F16GCF Complex scaled vector accumulation

G01ATF Computes univariate summary information: mean, variance, skewness, kurtosis

G01AUF Combines multiple sets of summary information, for use after G01ATF

G01HDF Computes the probability for the multivariate Student's

t

-distribution

G01KKF Computes a vector of values for the probability density function of the gamma distribution

G01KQF Computes a vector of values for the probability density function of the Normal distribution

G01LBF Computes a vector of values for the probability density function of the multivariate Normal distribution

G01SAF Computes a vector of probabilities for the standard Normal distribution

G01SBF Computes a vector of probabilities for the Student's

t

-distribution

G01SCF Computes a vector of probabilities for

χ^{2}

distribution

G01SDF Computes a vector of probabilities for

F

-distribution

G01SEF Computes a vector of probabilities for the beta distribution

G01SFF Computes a vector of probabilities for the gamma distribution

G01SJF Computes a vector of probabilities for the binomial distribution

G01SKF Computes a vector of probabilities for the Poisson distribution

G01SLF Computes a vector of probabilities for the hypergeometric distribution

G01TAF Computes a vector of deviates for the standard Normal distribution

G01TBF Computes a vector of deviates for Student's

t

-distribution

G01TCF Computes a vector of deviates for

χ^{2}

distribution

G01TDF Computes a vector of deviates for

F

-distribution

G01TEF Computes a vector of deviates for the beta distribution

G01TFF Computes a vector of deviates for the gamma distribution

G01WAF Computes the mean and standard deviation using a rolling window

G02AJF Computes the nearest correlation matrix to a real square matrix, using element-wise weighting

G02BZF Combines two sums of squares matrices, for use after G02BUF

G03GAF Fits a Gaussian mixture model

G05XAF Initializes the Brownian bridge generator

G05XBF Generate paths for a free or non-free Wiener process using the Brownian bridge algorithm

G05XCF Initializes the generator which backs out the increments of sample paths generated by a Brownian bridge algorithm

G05XDF Backs out the increments from sample paths generated by a Brownian bridge algorithm

G05XEF Creates a Brownian bridge construction order out of a set of input times

G05ZMF Setup for simulating one-dimensional random fields, user-defined variogram

G05ZNF Setup for simulating one-dimensional random fields

G05ZPF Generates realisations of a one-dimensional random field

G05ZQF Setup for simulating two-dimensional random fields, user-defined variogram

G05ZRF Setup for simulating two-dimensional random fields, preset variogram

G05ZSF Generates realisations of a two-dimensional random field

G05ZTF Generates realisations of fractional Brownian motion

G13MEF Computes the iterated exponential moving average for a univariate inhomogeneous time series

G13MFF Computes the iterated exponential moving average for a univariate inhomogeneous time series, intermediate results are also returned

G13MGF Computes the exponential moving average for a univariate inhomogeneous time series

H05AAF Best

m

subsets of size

p

(reverse communication)

H05ABF Best

m

subsets of size

p

(direct communication)

S14CBF Logarithm of the beta function

\ln (B, a, b)

S14CCF Incomplete beta function

I_{x} (a, b)

and its complement

1 - I_{x}

S17AQF Bessel function vectorized

Y_{0} (x)

S17ARF Bessel function vectorized

Y_{1} (x)

S17ASF Bessel function vectorized

J_{0} (x)

S17ATF Bessel function vectorized

J_{1} (x)

S17AUF Airy function vectorized

Ai (x)

S17AVF Airy function vectorized

Bi (x)

S17AWF Airy function vectorized

{Ai}^{'} (x)

S17AXF Airy function vectorized

{Bi}^{'} (x)

S18AQF Modified Bessel function vectorized

K_{0} (x)

S18ARF Modified Bessel function vectorized

K_{1} (x)

S18ASF Modified Bessel function vectorized

I_{0} (x)

S18ATF Modified Bessel function vectorized

I_{1} (x)

S18CQF Scaled modified Bessel function vectorized

e^{x} K_{0} (x)

S18CRF Scaled modified Bessel function vectorized

e^{x} K_{1} (x)

S18CSF Scaled modified Bessel function vectorized

e^{- |x|} I_{0} (x)

S18CTF Scaled modified Bessel function vectorized

e^{- |x|} I_{1} (x)

S19ANF Kelvin function vectorized

ber x

S19APF Kelvin function vectorized

bei x

S19AQF Kelvin function vectorized

\ker x

S19ARF Kelvin function vectorized

kei x

S20AQF Fresnel integral vectorized

S (x)

S20ARF Fresnel integral vectorized

C (x)

S22BAF Real confluent hypergeometric function

F_{1}^{1} (a; b; x)

S22BBF Real confluent hypergeometric function

F_{1}^{1} (a; b; x)

in scaled form

X07AAF Determines whether its argument has a finite value

X07ABF Determines whether its argument is a NaN (Not A Number)

X07BAF Creates a signed infinite value.

X07BBF Creates a NaN (Not A Number)

X07CAF Gets current behaviour of floating point exceptions

X07CBF Sets behaviour of floating point exceptions

3 Tuned Routines

The following is a list of user-callable routines that have been parallelized, or otherwise optimized, since the last release. There are 28 of these routines at this release in the areas of Fourier and wavelet transforms, interpolation, cubic splines, global optimization, matrix functions, simple and multivariate statistics, random number generators (RNGs), time series analysis and option pricing. See the document ‘Tuned and Enhanced Routines in the NAG Library for SMP & Multicore’ for a full list of tuned routines.

Note: on some implementations, the equivalent vendor library routines may be substituted for some of the following list – consult the Users' Note for your implementation for further information.

Routine
Name
Purpose

C06PVF Two-dimensional real-to-complex discrete Fourier transform

C06PWF Two-dimensional complex-to-real discrete Fourier transform

C06PYF Three-dimensional real-to-complex discrete Fourier transform

C06PZF Three-dimensional complex-to-real discrete Fourier transform

C09FAF Three-dimensional discrete wavelet transform

C09FBF Three-dimensional inverse discrete wavelet transform

C09FCF Three-dimensional multi-level discrete wavelet transform

C09FDF Three-dimensional inverse multi-level discrete wavelet transform

E01ZMF Interpolating function, modified Shepard's method,

d

dimensions

E01ZNF Interpolated values, evaluate interpolant computed by E01ZMF, function and first derivatives,

d

dimensions

E02BFF Evaluation of fitted cubic spline, function and optionally derivatives at a vector of points

E05UCF Global optimization using multi-start, nonlinear constraints

E05USF Global optimization of a sum of squares problem using multi-start, nonlinear constraints

F01EJF Real matrix logarithm

F01EKF Exponential, sine, cosine, sinh or cosh of a real matrix (Schur–Parlett algorithm)

F01EMF Function of a real matrix (using user-supplied derivatives)

F01FJF Complex matrix logarithm

F01FKF Exponential, sine, cosine, sinh or cosh of a complex matrix (Schur–Parlett algorithm)

F01FMF Function of a complex matrix (using user-supplied derivatives)

G01ATF Computes univariate summary information: mean, variance, skewness, kurtosis

G01WAF Computes the mean and standard deviation using a rolling window

G03GAF Fits a Gaussian mixture model

G05XBF Generate paths for a free or non-free Wiener process using the Brownian bridge algorithm

G05XDF Backs out the increments from sample paths generated by a Brownian bridge algorithm

G13MEF Computes the iterated exponential moving average for a univariate inhomogeneous time series

G13MFF Computes the iterated exponential moving average for a univariate inhomogeneous time series, intermediate results are also returned

G13MGF Computes the exponential moving average for a univariate inhomogeneous time series

S30NBF Heston's model option pricing formula with Greeks

4 Enhanced Routines

These routines call one or more of the tuned routines as part of their core operations and may thereby exhibit improved performance and scalability. There are 24 newly enhanced routines at this Mark; these include the areas of curve and surface fitting, matrix functions, sparse linear algebra, correlation and regression analysis and random number generators (RNGs). See the document ‘Tuned and Enhanced Routines in the NAG Library for SMP & Multicore’ for a full list of enhanced routines.