Tuned and Enhanced Routines in the NAG Library for SMP & Multicore (PDF version)
NAG Library Manual

NAG LibraryTuned and Enhanced Routines in the NAG Library for SMP & Multicore

1  Introduction

Tuned routines are user-callable routines that have been parallelized, or otherwise optimized, in the NAG Library for SMP & Multicore to give improved performance over the equivalent routines in the NAG Fortran Library or in the standard Netlib version of LAPACK. Enhanced routines are defined to be those user-callable routines which internally call one or more of the tuned routines, and hence may also exhibit improved performance and scalability. There are a total of 226 tuned routines and a total of 361 enhanced routines within the Library.
The NAG Library for SMP & Multicore is designed to be used in conjunction with the appropriate vendor library on each platform, as it relies upon the vendor library for optimized BLAS and FFT routines. The vendor libraries generally include LAPACK as well, and the vendor may also have parallelized or otherwise optimized some of these LAPACK routines. For each implementation, the performance of the LAPACK routines listed in Section 2 has been investigated, and the best combination of NAG Library for SMP & Multicore and vendor library versions is selected. Thus, in a given implementation, not all of the routines listed in Section 2 will actually be the NAG Library for SMP & Multicore version – consult the Users' Note for your implementation for further information.

2  Tuned LAPACK Routines

There are 77 tuned LAPACK routines within the Library.
 RoutineName Purpose F07ADF $LU$ factorization of real $m$ by $n$ matrix F07AEF Solution of real system of linear equations, multiple right-hand sides, matrix already factorized by F07ADF (DGETRF) F07AHF Refined solution with error bounds of real system of linear equations, multiple right-hand sides F07ARF $LU$ factorization of complex $m$ by $n$ matrix F07ASF Solution of complex system of linear equations, multiple right-hand sides, matrix already factorized by F07ARF (ZGETRF) F07AVF Refined solution with error bounds of complex system of linear equations, multiple right-hand sides F07BDF $LU$ factorization of real $m$ by $n$ band matrix F07BEF Solution of real band system of linear equations, multiple right-hand sides, matrix already factorized by F07BDF (DGBTRF) F07BHF Refined solution with error bounds of real band system of linear equations, multiple right-hand sides F07BRF $LU$ factorization of complex $m$ by $n$ band matrix F07BSF Solution of complex band system of linear equations, multiple right-hand sides, matrix already factorized by F07BRF (ZGBTRF) F07BVF Refined solution with error bounds of complex band system of linear equations, multiple right-hand sides F07CHF Refined solution with error bounds of real tridiagonal system of linear equations, multiple right-hand sides F07CVF Refined solution with error bounds of complex tridiagonal system of linear equations, multiple right-hand sides F07FDF Cholesky factorization of real symmetric positive definite matrix F07FEF Solution of real symmetric positive definite system of linear equations, multiple right-hand sides, matrix already factorized by F07FDF (DPOTRF) F07FHF Refined solution with error bounds of real symmetric positive definite system of linear equations, multiple right-hand sides F07FRF Cholesky factorization of complex Hermitian positive definite matrix F07FSF Solution of complex Hermitian positive definite system of linear equations, multiple right-hand sides, matrix already factorized by F07FRF (ZPOTRF) F07FVF Refined solution with error bounds of complex Hermitian positive definite system of linear equations, multiple right-hand sides F07GEF Solution of real symmetric positive definite system of linear equations, multiple right-hand sides, matrix already factorized by F07GDF (DPPTRF), packed storage F07GHF Refined solution with error bounds of real symmetric positive definite system of linear equations, multiple right-hand sides, packed storage F07GSF Solution of complex Hermitian positive definite system of linear equations, multiple right-hand sides, matrix already factorized by F07GRF (ZPPTRF), packed storage F07GVF Refined solution with error bounds of complex Hermitian positive definite system of linear equations, multiple right-hand sides, packed storage F07HEF Solution of real symmetric positive definite band system of linear equations, multiple right-hand sides, matrix already factorized by F07HDF (DPBTRF) F07HHF Refined solution with error bounds of real symmetric positive definite band system of linear equations, multiple right-hand sides F07HSF Solution of complex Hermitian positive definite band system of linear equations, multiple right-hand sides, matrix already factorized by F07HRF (ZPBTRF) F07HVF Refined solution with error bounds of complex Hermitian positive definite band system of linear equations, multiple right-hand sides F07JHF Refined solution with error bounds of real symmetric positive definite tridiagonal system of linear equations, multiple right-hand sides F07JVF Refined solution with error bounds of complex Hermitian positive definite tridiagonal system of linear equations, multiple right-hand sides F07MHF Refined solution with error bounds of real symmetric indefinite system of linear equations, multiple right-hand sides F07MVF Refined solution with error bounds of complex Hermitian indefinite system of linear equations, multiple right-hand sides F07NVF Refined solution with error bounds of complex symmetric system of linear equations, multiple right-hand sides F07PHF Refined solution with error bounds of real symmetric indefinite system of linear equations, multiple right-hand sides, packed storage F07PVF Refined solution with error bounds of complex Hermitian indefinite system of linear equations, multiple right-hand sides, packed storage F07QVF Refined solution with error bounds of complex symmetric system of linear equations, multiple right-hand sides, packed storage F07THF Error bounds for solution of real triangular system of linear equations, multiple right-hand sides F07TVF Error bounds for solution of complex triangular system of linear equations, multiple right-hand sides F07UEF Solution of real triangular system of linear equations, multiple right-hand sides, packed storage F07UHF Error bounds for solution of real triangular system of linear equations, multiple right-hand sides, packed storage F07USF Solution of complex triangular system of linear equations, multiple right-hand sides, packed storage F07UVF Error bounds for solution of complex triangular system of linear equations, multiple right-hand sides, packed storage F07VEF Solution of real band triangular system of linear equations, multiple right-hand sides F07VHF Error bounds for solution of real band triangular system of linear equations, multiple right-hand sides F07VSF Solution of complex band triangular system of linear equations, multiple right-hand sides F07VVF Error bounds for solution of complex band triangular system of linear equations, multiple right-hand sides F08AEF $QR$ factorization of real general rectangular matrix F08AFF Form all or part of orthogonal $Q$ from $QR$ factorization determined by F08AEF (DGEQRF), F08BEF (DGEQPF) or F08BFF (DGEQP3) F08AGF Apply orthogonal transformation determined by F08AEF (DGEQRF), F08BEF (DGEQPF) or F08BFF (DGEQP3) F08ASF $QR$ factorization of complex general rectangular matrix F08ATF Form all or part of unitary $Q$ from $QR$ factorization determined by F08ASF (ZGEQRF), F08BSF (ZGEQPF) or F08BTF (ZGEQP3) F08AUF Apply unitary transformation determined by F08ASF (ZGEQRF), F08BSF (ZGEQPF) or F08BTF (ZGEQP3) F08FEF Orthogonal reduction of real symmetric matrix to symmetric tridiagonal form F08FFF Generate orthogonal transformation matrix from reduction to tridiagonal form determined by F08FEF (DSYTRD) F08FSF Unitary reduction of complex Hermitian matrix to real symmetric tridiagonal form F08FTF Generate unitary transformation matrix from reduction to tridiagonal form determined by F08FSF (ZHETRD) F08GFF Generate orthogonal transformation matrix from reduction to tridiagonal form determined by F08GEF (DSPTRD) F08GTF Generate unitary transformation matrix from reduction to tridiagonal form determined by F08GSF (ZHPTRD) F08HEF Orthogonal reduction of real symmetric band matrix to symmetric tridiagonal form F08HSF Unitary reduction of complex Hermitian band matrix to real symmetric tridiagonal form F08JEF All eigenvalues and eigenvectors of real symmetric tridiagonal matrix, reduced from real symmetric matrix using the implicit $QL$ or $QR$ algorithm F08JJF Selected eigenvalues of real symmetric tridiagonal matrix by bisection F08JKF Selected eigenvectors of real symmetric tridiagonal matrix by inverse iteration, storing eigenvectors in real array F08JSF All eigenvalues and eigenvectors of real symmetric tridiagonal matrix, reduced from complex Hermitian matrix, using the implicit $QL$ or $QR$ algorithm F08JXF Selected eigenvectors of real symmetric tridiagonal matrix by inverse iteration, storing eigenvectors in complex array F08KEF Orthogonal reduction of real general rectangular matrix to bidiagonal form F08KSF Unitary reduction of complex general rectangular matrix to bidiagonal form F08MEF SVD of real bidiagonal matrix reduced from real general matrix F08MSF SVD of real bidiagonal matrix reduced from complex general matrix F08PKF Selected right and/or left eigenvectors of real upper Hessenberg matrix by inverse iteration F08PXF Selected right and/or left eigenvectors of complex upper Hessenberg matrix by inverse iteration F08TAF Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, packed storage F08TBF Computes selected eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, packed storage F08TCF Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, packed storage (divide-and-conquer) F08TNF Computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, packed storage F08TPF Computes selected eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, packed storage F08TQF Computes selected eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, packed storage (divide-and-conquer)

3  Routines Enhanced by Calling Tuned LAPACK Routines

These routines call one or more of the tuned LAPACK routines as part of their core operations and may thereby exhibit improved performance and scalability. There are 266 of these routines within the Library.
 RoutineName Purpose C02AKF All zeros of real cubic equation C02ALF All zeros of real quartic equation C02AMF All zeros of complex cubic equation C02ANF All zeros of complex quartic equation C05QBF Solution of a system of nonlinear equations using function values only (easy-to-use) C05QCF Solution of a system of nonlinear equations using function values only (comprehensive) C05QDF Solution of a system of nonlinear equations using function values only (reverse communication) C05RBF Solution of a system of nonlinear equations using first derivatives (easy-to-use) C05RCF Solution of a system of nonlinear equations using first derivatives (comprehensive) C05RDF Solution of a system of nonlinear equations using first derivatives (reverse communication) D02AGF Ordinary differential equations, boundary value problem, shooting and matching technique, allowing interior matching point, general parameters to be determined D02HAF Ordinary differential equations, boundary value problem, shooting and matching, boundary values to be determined D02HBF Ordinary differential equations, boundary value problem, shooting and matching, general parameters to be determined D02NEF Implicit ordinary differential equations/DAEs, initial value problem, DASSL method integrator D02SAF Ordinary differential equations, boundary value problem, shooting and matching technique, subject to extra algebraic equations, general parameters to be determined D02TKF Ordinary differential equations, general nonlinear boundary value problem, collocation technique D02UEF Solve linear constant coefficient boundary value problem on Chebyshev grid, Integral formulation D03NCF Finite difference solution of the Black–Scholes equations D05BDF Nonlinear convolution Volterra–Abel equation, second kind, weakly singular D05BEF Nonlinear convolution Volterra–Abel equation, first kind, weakly singular E02JDF Spline approximation to a set of scattered data using a two-stage approximation method E04FCF Unconstrained minimum of a sum of squares, combined Gauss–Newton and modified Newton algorithm using function values only (comprehensive) E04FYF Unconstrained minimum of a sum of squares, combined Gauss–Newton and modified Newton algorithm using function values only (easy-to-use) E04GBF Unconstrained minimum of a sum of squares, combined Gauss–Newton and quasi-Newton algorithm using first derivatives (comprehensive) E04GDF Unconstrained minimum of a sum of squares, combined Gauss–Newton and modified Newton algorithm using first derivatives (comprehensive) E04GYF Unconstrained minimum of a sum of squares, combined Gauss–Newton and quasi-Newton algorithm, using first derivatives (easy-to-use) E04GZF Unconstrained minimum of a sum of squares, combined Gauss–Newton and modified Newton algorithm using first derivatives (easy-to-use) E04HEF Unconstrained minimum of a sum of squares, combined Gauss–Newton and modified Newton algorithm, using second derivatives (comprehensive) E04HYF Unconstrained minimum of a sum of squares, combined Gauss–Newton and modified Newton algorithm, using second derivatives (easy-to-use) E04NCF Convex QP problem or linearly-constrained linear least squares problem (dense) E04UCF Minimum, function of several variables, sequential QP method, nonlinear constraints, using function values and optionally first derivatives (comprehensive) E04UFF Minimum, function of several variables, sequential QP method, nonlinear constraints, using function values and optionally first derivatives (reverse communication, comprehensive) E04USF Minimum of a sum of squares, nonlinear constraints, sequential QP method, using function values and optionally first derivatives (comprehensive) F01ABF Inverse of real symmetric positive definite matrix using iterative refinement F01ADF Inverse of real symmetric positive definite matrix F01ECF Real matrix exponential F01EDF Real symmetric matrix exponential F01EFF Function of a real symmetric matrix F01ELF Function of a real matrix (using numerical differentiation) F01FCF Complex matrix exponential F01FDF Complex Hermitian matrix exponential F01FFF Function of a complex Hermitian matrix F01FLF Function of a complex matrix (using numerical differentiation) 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) F02ECF Selected eigenvalues and eigenvectors of real nonsymmetric matrix (Black Box) F02FJF Selected eigenvalues and eigenvectors of sparse symmetric eigenproblem (Black Box) F02GCF Selected eigenvalues and eigenvectors of complex nonsymmetric matrix (Black Box) F02WDF $QR$ factorization, possibly followed by SVD F02WGF Computes leading terms in the singular value decomposition of a real general matrix; also computes corresponding left and right singular vectors F02WUF SVD of real upper triangular matrix (Black Box) F02XUF SVD of complex upper triangular matrix (Black Box) F04ABF Solution of real symmetric positive definite simultaneous linear equations with multiple right-hand sides using iterative refinement (Black Box) F04ASF Solution of real symmetric positive definite simultaneous linear equations, one right-hand side using iterative refinement (Black Box) F04BAF Computes the solution and error-bound to a real system of linear equations F04BBF Computes the solution and error-bound to a real banded system of linear equations F04BDF Computes the solution and error-bound to a real symmetric positive definite system of linear equations F04BEF Computes the solution and error-bound to a real symmetric positive definite system of linear equations, packed storage F04BFF Computes the solution and error-bound to a real symmetric positive definite banded system of linear equations F04CAF Computes the solution and error-bound to a complex system of linear equations F04CBF Computes the solution and error-bound to a complex banded system of linear equations F04CDF Computes the solution and error-bound to a complex Hermitian positive definite system of linear equations F04CEF Computes the solution and error-bound to a complex Hermitian positive definite system of linear equations, packed storage F04CFF Computes the solution and error-bound to a complex Hermitian positive definite banded system of linear equations F04JGF Least squares (if rank $\text{}=n$) or minimal least squares (if rank $\text{}) solution of $m$ real equations in $n$ unknowns, $m\ge n$ F07AAF Computes the solution to a real system of linear equations F07ABF Uses the $LU$ factorization to compute the solution, error-bound and condition estimate for a real system of linear equations F07ACF Mixed precision real system solver F07ANF Computes the solution to a complex system of linear equations F07APF Uses the $LU$ factorization to compute the solution, error-bound and condition estimate for a complex system of linear equations F07AQF Mixed precision complex system solver F07BAF Computes the solution to a real banded system of linear equations F07BBF Uses the $LU$ factorization to compute the solution, error-bound and condition estimate for a real banded system of linear equations F07BNF Computes the solution to a complex banded system of linear equations F07BPF Uses the $LU$ factorization to compute the solution, error-bound and condition estimate for a complex banded system of linear equations F07CBF Uses the $LU$ factorization to compute the solution, error-bound and condition estimate for a real tridiagonal system of linear equations F07CPF Uses the $LU$ factorization to compute the solution, error-bound and condition estimate for a complex tridiagonal system of linear equations F07FAF Computes the solution to a real symmetric positive definite system of linear equations F07FBF Uses the Cholesky factorization to compute the solution, error-bound and condition estimate for a real symmetric positive definite system of linear equations F07FCF Uses the Cholesky factorization to compute the solution for a real symmetric positive definite system of linear equations F07FNF Computes the solution to a complex Hermitian positive definite system of linear equations F07FPF Uses the Cholesky factorization to compute the solution, error-bound and condition estimate for a complex Hermitian positive definite system of linear equations F07FQF Uses the Cholesky factorization to compute the solution for a complex Hermitian positive definite system of linear equations F07GAF Computes the solution to a real symmetric positive definite system of linear equations, packed storage F07GBF Uses the Cholesky factorization to compute the solution, error-bound and condition estimate for a real symmetric positive definite system of linear equations, packed storage F07GNF Computes the solution to a complex Hermitian positive definite system of linear equations, packed storage F07GPF Uses the Cholesky factorization to compute the solution, error-bound and condition estimate for a complex Hermitian positive definite system of linear equations, packed storage F07HAF Computes the solution to a real symmetric positive definite banded system of linear equations F07HBF Uses the Cholesky factorization to compute the solution, error-bound and condition estimate for a real symmetric positive definite banded system of linear equations F07HNF Computes the solution to a complex Hermitian positive definite banded system of linear equations F07HPF Uses the Cholesky factorization to compute the solution, error-bound and condition estimate for a complex Hermitian positive definite banded system of linear equations F07JBF Uses the modified Cholesky factorization to compute the solution, error-bound and condition estimate for a real symmetric positive definite tridiagonal system of linear equations F07JPF Uses the modified Cholesky factorization to compute the solution, error-bound and condition estimate for a complex Hermitian positive definite tridiagonal system of linear equations F07MBF Uses the diagonal pivoting factorization to compute the solution to a real symmetric system of linear equations F07MPF Uses the diagonal pivoting factorization to compute the solution to a complex Hermitian system of linear equations F07NPF Uses the diagonal pivoting factorization to compute the solution to a complex symmetric system of linear equations F07PBF Uses the diagonal pivoting factorization to compute the solution to a real symmetric system of linear equations, packed storage F07PPF Uses the diagonal pivoting factorization to compute the solution to a complex Hermitian system of linear equations, packed storage F07QPF Uses the diagonal pivoting factorization to compute the solution to a complex symmetric system of linear equations, packed storage F07WDF Cholesky factorization of real symmetric positive definite matrix, Rectangular Full Packed format F07WRF Cholesky factorization of complex Hermitian positive definite matrix, Rectangular Full Packed format F08AAF Solves an overdetermined or underdetermined real linear system F08ANF Solves an overdetermined or underdetermined complex linear system F08BAF Computes the minimum-norm solution to a real linear least squares problem F08BFF $QR$ factorization of real general rectangular matrix with column pivoting, using BLAS-3 F08BNF Computes the minimum-norm solution to a complex linear least squares problem F08BTF $QR$ factorization of complex general rectangular matrix with column pivoting, using BLAS-3 F08FAF Computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix F08FBF Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix F08FCF Computes all eigenvalues and, optionally, all eigenvectors of real symmetric matrix (divide-and-conquer) F08FDF Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix (Relatively Robust Representations) F08FGF Apply orthogonal transformation determined by F08FEF (DSYTRD) F08FNF Computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix F08FPF Computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix F08FQF Computes all eigenvalues and, optionally, all eigenvectors of complex Hermitian matrix (divide-and-conquer) F08FRF Computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix (Relatively Robust Representations) F08FUF Apply unitary transformation matrix determined by F08FSF (ZHETRD) F08GAF Computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix, packed storage F08GBF Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix, packed storage F08GCF Computes all eigenvalues and, optionally, all eigenvectors of real symmetric matrix, packed storage (divide-and-conquer) F08GNF Computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix, packed storage F08GPF Computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix, packed storage F08GQF Computes all eigenvalues and, optionally, all eigenvectors of complex Hermitian matrix, packed storage (divide-and-conquer) F08HAF Computes all eigenvalues and, optionally, eigenvectors of a real symmetric band matrix F08HBF Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric band matrix F08HCF Computes all eigenvalues and, optionally, all eigenvectors of real symmetric band matrix (divide-and-conquer) F08HNF Computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix F08HPF Computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix F08HQF Computes all eigenvalues and, optionally, all eigenvectors of complex Hermitian band matrix (divide-and-conquer) F08JAF Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix F08JBF Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix F08JCF Computes all eigenvalues and, optionally, all eigenvectors of real symmetric tridiagonal matrix (divide-and-conquer) F08JDF Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix (Relatively Robust Representations) F08JGF Computes all eigenvalues and eigenvectors of real symmetric positive definite tridiagonal matrix, reduced from real symmetric positive definite matrix F08JHF Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix or a matrix reduced to this form (divide-and-conquer) F08JLF Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix or a symmetric matrix reduced to this form (Relatively Robust Representations) F08JUF Computes all eigenvalues and eigenvectors of real symmetric positive definite tridiagonal matrix, reduced from complex Hermitian positive definite matrix F08JVF Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix or a complex Hermitian matrix reduced to this form (divide-and-conquer) F08JYF Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix or a complex Hermitian matrix reduced to this form (Relatively Robust Representations) F08KAF Computes the minimum-norm solution to a real linear least squares problem using singular value decomposition F08KBF Computes the singular value decomposition of a real matrix, optionally computing the left and/or right singular vectors F08KCF Computes the minimum-norm solution to a real linear least squares problem using singular value decomposition (divide-and-conquer) F08KDF Computes the singular value decomposition of a real matrix, optionally computing the left and/or right singular vectors (divide-and-conquer) F08KFF Generate orthogonal transformation matrices from reduction to bidiagonal form determined by F08KEF (DGEBRD) F08KGF Apply orthogonal transformations from reduction to bidiagonal form determined by F08KEF (DGEBRD) F08KHF Computes the singular value decomposition of a real matrix, optionally computing the left and/or right singular vectors (preconditioned Jacobi) F08KNF Computes the minimum-norm solution to a complex linear least squares problem using singular value decomposition F08KPF Computes the singular value decomposition of a complex matrix, optionally computing the left and/or right singular vectors F08KQF Computes the minimum-norm solution to a complex linear least squares problem using singular value decomposition (divide-and-conquer) F08KRF Computes the singular value decomposition of a complex matrix, optionally computing the left and/or right singular vectors (divide-and-conquer) F08KTF Generate unitary transformation matrices from reduction to bidiagonal form determined by F08KSF (ZGEBRD) F08KUF Apply unitary transformations from reduction to bidiagonal form determined by F08KSF (ZGEBRD) F08MDF Computes the singular value decomposition of a real bidiagonal matrix, optionally computing the singular vectors (divide-and-conquer) F08NAF Computes all eigenvalues and, optionally, left and/or right eigenvectors of a real nonsymmetric matrix F08NBF Computes all eigenvalues and, optionally, left and/or right eigenvectors of a real nonsymmetric matrix; also, optionally, the balancing transformation, the reciprocal condition numbers for the eigenvalues and for the right eigenvectors F08NFF Generate orthogonal transformation matrix from reduction to Hessenberg form determined by F08NEF (DGEHRD) F08NGF Apply orthogonal transformation matrix from reduction to Hessenberg form determined by F08NEF (DGEHRD) F08NNF Computes all eigenvalues and, optionally, left and/or right eigenvectors of a complex nonsymmetric matrix F08NPF Computes all eigenvalues and, optionally, left and/or right eigenvectors of a complex nonsymmetric matrix; also, optionally, the balancing transformation, the reciprocal condition numbers for the eigenvalues and for the right eigenvectors F08NTF Generate unitary transformation matrix from reduction to Hessenberg form determined by F08NSF (ZGEHRD) F08NUF Apply unitary transformation matrix from reduction to Hessenberg form determined by F08NSF (ZGEHRD) F08PAF Computes for real square nonsymmetric matrix, the eigenvalues, the real Schur form, and, optionally, the matrix of Schur vectors F08PBF Computes for real square nonsymmetric matrix, the eigenvalues, the real Schur form, and, optionally, the matrix of Schur vectors; also, optionally, computes reciprocal condition numbers for selected eigenvalues F08PEF Computes the eigenvalues and Schur factorization of real upper Hessenberg matrix reduced from real general matrix F08PNF Computes for complex square nonsymmetric matrix, the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors F08PPF Computes for real square nonsymmetric matrix, the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors; also, optionally, computes reciprocal condition numbers for selected eigenvalues F08PSF Computes the eigenvalues and Schur factorization of complex upper Hessenberg matrix reduced from complex general matrix F08SAF Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem F08SBF Computes selected eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem F08SCF Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem (divide-and-conquer) F08SNF Computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem F08SPF Computes selected eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem F08SQF Computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem (divide-and-conquer) F08UAF Computes all the eigenvalues, and optionally, the eigenvectors of a real banded generalized symmetric-definite eigenproblem F08UBF Computes selected eigenvalues, and optionally, the eigenvectors of a real banded generalized symmetric-definite eigenproblem F08UCF Computes all the eigenvalues, and optionally, the eigenvectors of a real banded generalized symmetric-definite eigenproblem (divide-and-conquer) F08UNF Computes all the eigenvalues, and optionally, the eigenvectors of a complex banded generalized Hermitian-definite eigenproblem F08UPF Computes selected eigenvalues, and optionally, the eigenvectors of a complex banded generalized Hermitian-definite eigenproblem F08UQF Computes all the eigenvalues, and optionally, the eigenvectors of a complex banded generalized Hermitian-definite eigenproblem (divide-and-conquer) F08WAF Computes, for a real nonsymmetric matrix pair, the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors F08WBF Computes, for a real nonsymmetric matrix pair, the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors; also, optionally, the balancing transformation, the reciprocal condition numbers for the eigenvalues and for the right eigenvectors F08WNF Computes, for a complex nonsymmetric matrix pair, the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors F08WPF Computes, for a complex nonsymmetric matrix pair, the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors; also, optionally, the balancing transformation, the reciprocal condition numbers for the eigenvalues and for the right eigenvectors F08XAF Computes, for a real nonsymmetric matrix pair, the generalized eigenvalues, the generalized real Schur form and, optionally, the left and/or right matrices of Schur vectors F08XBF Computes, for a real nonsymmetric matrix pair, the generalized eigenvalues, the generalized real Schur form and, optionally, the left and/or right matrices of Schur vectors; also, optionally, computes reciprocal condition numbers for selected eigenvalues F08XNF Computes, for a complex nonsymmetric matrix pair, the generalized eigenvalues, the generalized complex Schur form and, optionally, the left and/or right matrices of Schur vectors F08XPF Computes, for a complex nonsymmetric matrix pair, the generalized eigenvalues, the generalized complex Schur form and, optionally, the left and/or right matrices of Schur vectors; also, optionally, computes reciprocal condition numbers for selected eigenvalues F08ZAF Solves the real linear equality-constrained least squares (LSE) problem F08ZBF Solves a real general Gauss–Markov linear model (GLM) problem F08ZEF Computes a generalized $QR$ factorization of a real matrix pair F08ZFF Computes a generalized $RQ$ factorization of a real matrix pair F08ZNF Solves the complex linear equality-constrained least squares (LSE) problem F08ZPF Solves a complex general Gauss–Markov linear model (GLM) problem F08ZSF Computes a generalized $QR$ factorization of a complex matrix pair F08ZTF Computes a generalized $RQ$ factorization of a complex matrix pair F12AUF Selected eigenvalues and, optionally, eigenvectors of complex non-Hermitian banded eigenproblem, driver F12FCF Selected eigenvalues and, optionally, eigenvectors of a real symmetric sparse eigenproblem, postprocessing for F12FBF F12FGF Selected eigenvalues and, optionally, eigenvectors of a real symmetric banded eigenproblem, driver G01HBF Computes probabilities for the multivariate Normal distribution G01LBF Computes a vector of values for the probability density function of the multivariate Normal distribution G02ABF Computes the nearest correlation matrix to a real square matrix, augmented G02AAF to incorporate weights and bounds G02AEF Computes the nearest correlation matrix with $k$-factor structure to a real square matrix G02AJF Computes the nearest correlation matrix to a real square matrix, using element-wise weighting G02BYF Computes partial correlation/variance-covariance matrix from correlation/variance-covariance matrix computed by G02BXF G02CGF Multiple linear regression, from correlation coefficients, with constant term G02CHF Multiple linear regression, from correlation-like coefficients, without constant term G02DAF Fits a general (multiple) linear regression model G02DDF Estimates of linear parameters and general linear regression model from updated model G02EAF Computes residual sums of squares for all possible linear regressions for a set of independent variables G02EEF Fits a linear regression model by forward selection G02GAF Fits a generalized linear model with Normal errors G02GBF Fits a generalized linear model with binomial errors G02GCF Fits a generalized linear model with Poisson errors G02GDF Fits a generalized linear model with gamma errors G02HAF Robust regression, standard $M$-estimates G02HDF Robust regression, compute regression with user-supplied functions and weights G02HFF Robust regression, variance-covariance matrix following G02HDF G02JAF Linear mixed effects regression using Restricted Maximum Likelihood (REML) G02JBF Linear mixed effects regression using Maximum Likelihood (ML) G02KAF Ridge regression, optimizing a ridge regression parameter G02KBF Ridge regression using a number of supplied ridge regression parameters G02LAF Partial least squares (PLS) regression using singular value decomposition G02LCF PLS parameter estimates following partial least squares regression by G02LAF or G02LBF G02QFF Linear quantile regression, simple interface, independent, identically distributed (IID) errors G02QGF Linear quantile regression, comprehensive interface G03AAF Performs principal component analysis G03ACF Performs canonical variate analysis G03ADF Performs canonical correlation analysis G03BAF Computes orthogonal rotations for loading matrix, generalized orthomax criterion G03BCF Computes Procrustes rotations G03BDF ProMax rotations G03DAF Computes test statistic for equality of within-group covariance matrices and matrices for discriminant analysis G03FAF Performs principal coordinate analysis, classical metric scaling G04BBF Analysis of variance, randomized block or completely randomized design, treatment means and standard errors G04BCF Analysis of variance, general row and column design, treatment means and standard errors G05PJF Generates a realisation of a multivariate time series from a VARMA model G08RAF Regression using ranks, uncensored data G08RBF Regression using ranks, right-censored data G11CAF Returns parameter estimates for the conditional analysis of stratified data G11SAF Contingency table, latent variable model for binary data G12ABF Computes rank statistics for comparing survival curves G12BAF Fits Cox's proportional hazard model G13AEF Univariate time series, estimation, seasonal ARIMA model (comprehensive) G13AFF Univariate time series, estimation, seasonal ARIMA model (easy-to-use) G13AJF Univariate time series, state set and forecasts, from fully specified seasonal ARIMA model G13ASF Univariate time series, diagnostic checking of residuals, following G13AEF or G13AFF G13BAF Multivariate time series, filtering (pre-whitening) by an ARIMA model G13BBF Multivariate time series, filtering by a transfer function model G13BDF Multivariate time series, preliminary estimation of transfer function model G13BEF Multivariate time series, estimation of multi-input model G13BJF Multivariate time series, state set and forecasts from fully specified multi-input model G13DBF Multivariate time series, multiple squared partial autocorrelations G13DDF Multivariate time series, estimation of VARMA model G13DJF Multivariate time series, forecasts and their standard errors G13DNF Multivariate time series, sample partial lag correlation matrices, ${\chi }^{2}$ statistics and significance levels G13DPF Multivariate time series, partial autoregression matrices G13DSF Multivariate time series, diagnostic checking of residuals, following G13DDF G13DXF Calculates the zeros of a vector autoregressive (or moving average) operator G13FAF Univariate time series, parameter estimation for either a symmetric GARCH process or a GARCH process with asymmetry of the form ${\left({\epsilon }_{t-1}+\gamma \right)}^{2}$ G13FCF Univariate time series, parameter estimation for a GARCH process with asymmetry of the form ${\left(\left|{\epsilon }_{t-1}\right|+\gamma {\epsilon }_{t-1}\right)}^{2}$ G13FEF Univariate time series, parameter estimation for an asymmetric Glosten, Jagannathan and Runkle (GJR) GARCH process G13FGF Univariate time series, parameter estimation for an exponential GARCH (EGARCH) process

4  Tuned NAG-specific Routines

These NAG-specific routines have been parallelized, or otherwise optimized, in the NAG Library for SMP & Multicore compared to the equivalent routine in the NAG Fortran Library. There are 149 of these routines within the Library.
 RoutineName Purpose C06FKF Circular convolution or correlation of two real vectors, extra workspace for greater speed C06FPF Multiple one-dimensional real discrete Fourier transforms C06FQF Multiple one-dimensional Hermitian discrete Fourier transforms C06FXF Three-dimensional complex discrete Fourier transform C06PAF Single one-dimensional real and Hermitian complex discrete Fourier transform, using complex storage format for Hermitian sequences C06PFF One-dimensional complex discrete Fourier transform of multidimensional data (using complex data type) C06PJF Multidimensional complex discrete Fourier transform of multidimensional data (using complex data type) C06PKF Circular convolution or correlation of two complex vectors C06PPF Multiple one-dimensional real and Hermitian complex discrete Fourier transforms, using row ordered complex storage format for Hermitian sequences C06PQF Multiple one-dimensional real and Hermitian complex discrete Fourier transforms, using column ordered complex storage format for Hermitian sequences C06PRF Multiple one-dimensional complex discrete Fourier transforms using complex data type C06PSF Multiple one-dimensional complex discrete Fourier transforms using complex data type and sequences stored as columns C06PUF Two-dimensional complex discrete Fourier transform, complex data type C06PVF Two-dimensional real-to-complex discrete Fourier transform C06PWF Two-dimensional complex-to-real discrete Fourier transform C06PXF Three-dimensional complex discrete Fourier transform, complex data type C06PYF Three-dimensional real-to-complex discrete Fourier transform C06PZF Three-dimensional complex-to-real discrete Fourier transform C06RAF Discrete sine transform (easy-to-use) C06RBF Discrete cosine transform (easy-to-use) C06RCF Discrete quarter-wave sine transform (easy-to-use) C06RDF Discrete quarter-wave cosine transform (easy-to-use) C09EAF Two-dimensional discrete wavelet transform C09EBF Two-dimensional inverse discrete wavelet transform C09ECF Two-dimensional multi-level discrete wavelet transform C09EDF Two-dimensional inverse multi-level discrete wavelet 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 D01DAF Two-dimensional quadrature, finite region D01FCF Multidimensional adaptive quadrature over hyper-rectangle D01GAF One-dimensional quadrature, integration of function defined by data values, Gill–Miller method D03FAF Elliptic PDE, Helmholtz equation, three-dimensional Cartesian coordinates D03RAF General system of second-order PDEs, method of lines, finite differences, remeshing, two space variables, rectangular region D03RBF General system of second-order PDEs, method of lines, finite differences, remeshing, two space variables, rectilinear region E01SGF Interpolating functions, modified Shepard's method, two variables E01SHF Interpolated values, evaluate interpolant computed by E01SGF, function and first derivatives, two variables E01TGF Interpolating functions, modified Shepard's method, three variables E01THF Interpolated values, evaluate interpolant computed by E01TGF, function and first derivatives, three variables E01TKF Interpolating functions, modified Shepard's method, four variables E01TLF Interpolated values, evaluate interpolant computed by E01TKF, function and first derivatives, four variables E01TMF Interpolating functions, modified Shepard's method, five variables E01TNF Interpolated values, evaluate interpolant computed by E01TMF, function and first derivatives, five variables 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 E02CAF Least squares surface fit by polynomials, data on lines parallel to one independent coordinate axis E02CBF Evaluation of fitted polynomial in two variables E02DFF Evaluation of fitted bicubic spline at a mesh of points E05SAF Global optimization using particle swarm algorithm (PSO), bound constraints only E05SBF Global optimization using particle swarm algorithm (PSO), comprehensive E05UCF Global optimization using multi-start, nonlinear constraints E05USF Global optimization of a sum of squares problem using multi-start, nonlinear constraints F01CTF Sum or difference of two real matrices, optional scaling and transposition F01CWF Sum or difference of two complex matrices, optional scaling and transposition 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) F05AAF Gram–Schmidt orthogonalization of $n$ vectors of order $m$ F11BEF Real sparse nonsymmetric linear systems, preconditioned RGMRES, CGS, Bi-CGSTAB or TFQMR method F11BSF Complex sparse non-Hermitian linear systems, preconditioned RGMRES, CGS, Bi-CGSTAB or TFQMR method F11GEF Real sparse symmetric linear systems, preconditioned conjugate gradient or Lanczos method or the MINRES algorithm F11GSF Complex sparse Hermitian linear systems, preconditioned conjugate gradient or Lanczos F11MEF $LU$ factorization of real sparse matrix F11MFF Solution of real sparse simultaneous linear equations (coefficient matrix already factorized) F11MHF Refined solution with error bounds of real system of linear equations, multiple right-hand sides F11MKF Real sparse nonsymmetric matrix-matrix multiply, compressed column storage F11XAF Real sparse nonsymmetric matrix vector multiply F11XEF Real sparse symmetric matrix vector multiply F11XNF Complex sparse non-Hermitian matrix vector multiply F11XSF Complex sparse Hermitian matrix vector multiply F12ABF Selected eigenvalues and, optionally, eigenvectors of a real nonsymmetric sparse eigenproblem, reverse communication F12AGF Selected eigenvalues and, optionally, eigenvectors of a real nonsymmetric banded eigenproblem, driver F12APF Selected eigenvalues and, optionally, eigenvectors of a complex sparse eigenproblem, reverse communication F12FBF Selected eigenvalues and, optionally, eigenvectors of a real symmetric sparse eigenproblem, reverse communication G01ATF Computes univariate summary information: mean, variance, skewness, kurtosis G01WAF Computes the mean and standard deviation using a rolling window G02AAF Computes the nearest correlation matrix to a real square matrix, using the method of Qi and Sun G02BAF Pearson product-moment correlation coefficients, all variables, no missing values G02BDF Correlation-like coefficients (about zero), all variables, no missing values G02BNF Kendall/Spearman non-parametric rank correlation coefficients, no missing values, overwriting input data G02BPF Kendall/Spearman non-parametric rank correlation coefficients, casewise treatment of missing values, overwriting input data G02BQF Kendall/Spearman non-parametric rank correlation coefficients, no missing values, preserving input data G02BRF Kendall/Spearman non-parametric rank correlation coefficients, casewise treatment of missing values, preserving input data G02JDF Hierarchical mixed effects regression using Restricted Maximum Likelihood (REML) G02JEF Hierarchical mixed effects regression using Maximum Likelihood (ML) G03CAF Computes maximum likelihood estimates of the parameters of a factor analysis model, factor loadings, communalities and residual correlations G03EAF Computes distance matrix G03ECF Hierarchical cluster analysis G03GAF Fits a Gaussian mixture model G05RCF Generates a matrix of pseudorandom numbers from a Student's $t$-copula G05RDF Generates a matrix of pseudorandom numbers from a Gaussian copula G05REF Generates a matrix of pseudorandom numbers from a bivariate Clayton/Cook–Johnson copula G05RFF Generates a matrix of pseudorandom numbers from a bivariate Frank copula G05RGF Generates a matrix of pseudorandom numbers from a bivariate Plackett copula G05RHF Generates a matrix of pseudorandom numbers from a multivariate Clayton/Cook–Johnson copula G05RJF Generates a matrix of pseudorandom numbers from a multivariate Frank copula G05RKF Generates a matrix of pseudorandom numbers from a Gumbel–Hougaard copula G05RYF Generates a matrix of pseudorandom numbers from a multivariate Student's $t$-distribution G05SAF Generates a vector of pseudorandom numbers from a uniform distribution over $\left(0,1\right]$ G05SBF Generates a vector of pseudorandom numbers from a beta distribution G05SCF Generates a vector of pseudorandom numbers from a Cauchy distribution G05SDF Generates a vector of pseudorandom numbers from a ${\chi }^{2}$ distribution G05SEF Generates a vector of pseudorandom numbers from a Dirichlet distribution G05SFF Generates a vector of pseudorandom numbers from an exponential distribution G05SGF Generates a vector of pseudorandom numbers from an exponential mix distribution G05SHF Generates a vector of pseudorandom numbers from an $F$-distribution G05SJF Generates a vector of pseudorandom numbers from a gamma distribution G05SKF Generates a vector of pseudorandom numbers from a Normal distribution G05SLF Generates a vector of pseudorandom numbers from a logistic distribution G05SMF Generates a vector of pseudorandom numbers from a log-normal distribution G05SNF Generates a vector of pseudorandom numbers from a Student's $t$-distribution G05SPF Generates a vector of pseudorandom numbers from a triangular distribution G05SQF Generates a vector of pseudorandom numbers from a uniform distribution over $\left[a,b\right]$ G05SRF Generates a vector of pseudorandom numbers from a von Mises distribution G05SSF Generates a vector of pseudorandom numbers from a Weibull distribution 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 G05YJF Generates a Normal quasi-random number sequence G05YKF Generates a log-normal quasi-random number sequence G05YMF Generates a uniform quasi-random number sequence G13EAF Combined measurement and time update, one iteration of Kalman filter, time-varying, square root covariance filter G13EBF Combined measurement and time update, one iteration of Kalman filter, time-invariant, square root covariance filter 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 M01CAF Sort a vector, real numbers M01CBF Sort a vector, integer numbers M01CCF Sort a vector, character data S30AAF Black–Scholes–Merton option pricing formula S30ABF Black–Scholes–Merton option pricing formula with Greeks S30BAF Floating-strike lookback option pricing formula S30BBF Floating-strike lookback option pricing formula with Greeks S30CAF Binary option, cash-or-nothing pricing formula S30CBF Binary option, cash-or-nothing pricing formula with Greeks S30CCF Binary option, asset-or-nothing pricing formula S30CDF Binary option, asset-or-nothing pricing formula with Greeks S30FAF Standard barrier option pricing formula S30JAF Jump-diffusion, Merton's model, option pricing formula S30JBF Jump-diffusion, Merton's model, option pricing formula with Greeks S30NAF Heston's model option pricing formula S30NBF Heston's model option pricing formula with Greeks S30QCF American option, Bjerksund and Stensland pricing formula S30SAF Asian option, geometric continuous average rate pricing formula S30SBF Asian option, geometric continuous average rate pricing formula with Greeks

5  Routines Enhanced by Calling Tuned NAG-specific Routines

These routines call one or more of the tuned NAG-specific routines as part of their core operations and may thereby exhibit improved performance and scalability. There are 165 of these routines within the Library.
 RoutineName Purpose C05QSF Solution of a sparse system of nonlinear equations using function values only (easy-to-use) D01GBF Multidimensional quadrature over hyper-rectangle, Monte–Carlo method D01GCF Multidimensional quadrature, general product region, number-theoretic method D01GDF Multidimensional quadrature, general product region, number-theoretic method, variant of D01GCF efficient on vector machines D01PAF Multidimensional quadrature over an $n$-simplex D02EJF Ordinary differential equations, stiff initial value problem, backward differentiation formulae method, until function of solution is zero, intermediate output (simple driver) D02NBF Explicit ordinary differential equations, stiff initial value problem, full Jacobian (comprehensive) D02NCF Explicit ordinary differential equations, stiff initial value problem, banded Jacobian (comprehensive) D02NDF Explicit ordinary differential equations, stiff initial value problem, sparse Jacobian (comprehensive) D02NGF Implicit/algebraic ordinary differential equations, stiff initial value problem, full Jacobian (comprehensive) D02NHF Implicit/algebraic ordinary differential equations, stiff initial value problem, banded Jacobian (comprehensive) D02NJF Implicit/algebraic ordinary differential equations, stiff initial value problem, sparse Jacobian (comprehensive) D02NMF Explicit ordinary differential equations, stiff initial value problem (reverse communication, comprehensive) D02NNF Implicit/algebraic ordinary differential equations, stiff initial value problem (reverse communication, comprehensive) D02UAF Coefficients of Chebyshev interpolating polynomial from function values on Chebyshev grid D02UBF Function or low-order-derivative values on Chebyshev grid from coefficients of Chebyshev interpolating polynomial D03PCF General system of parabolic PDEs, method of lines, finite differences, one space variable D03PDF General system of parabolic PDEs, method of lines, Chebyshev ${C}^{0}$ collocation, one space variable D03PEF General system of first-order PDEs, method of lines, Keller box discretization, one space variable D03PFF General system of convection-diffusion PDEs with source terms in conservative form, method of lines, upwind scheme using numerical flux function based on Riemann solver, one space variable D03PHF General system of parabolic PDEs, coupled DAEs, method of lines, finite differences, one space variable D03PJF General system of parabolic PDEs, coupled DAEs, method of lines, Chebyshev ${C}^{0}$ collocation, one space variable D03PKF General system of first-order PDEs, coupled DAEs, method of lines, Keller box discretization, one space variable D03PLF General system of convection-diffusion PDEs with source terms in conservative form, coupled DAEs, method of lines, upwind scheme using numerical flux function based on Riemann solver, one space variable D03PPF General system of parabolic PDEs, coupled DAEs, method of lines, finite differences, remeshing, one space variable D03PRF General system of first-order PDEs, coupled DAEs, method of lines, Keller box discretization, remeshing, one space variable D03PSF General system of convection-diffusion PDEs, coupled DAEs, method of lines, upwind scheme, remeshing, one space variable D05AAF Linear nonsingular Fredholm integral equation, second kind, split kernel D05ABF Linear nonsingular Fredholm integral equation, second kind, smooth kernel D06CBF Generates a sparsity pattern of a Finite Element matrix associated with a given mesh D06CCF Renumbers a given mesh using Gibbs method E02RAF Padé approximants E04FCF Unconstrained minimum of a sum of squares, combined Gauss–Newton and modified Newton algorithm using function values only (comprehensive) E04FYF Unconstrained minimum of a sum of squares, combined Gauss–Newton and modified Newton algorithm using function values only (easy-to-use) E04GBF Unconstrained minimum of a sum of squares, combined Gauss–Newton and quasi-Newton algorithm using first derivatives (comprehensive) E04GDF Unconstrained minimum of a sum of squares, combined Gauss–Newton and modified Newton algorithm using first derivatives (comprehensive) E04GYF Unconstrained minimum of a sum of squares, combined Gauss–Newton and quasi-Newton algorithm, using first derivatives (easy-to-use) E04GZF Unconstrained minimum of a sum of squares, combined Gauss–Newton and modified Newton algorithm using first derivatives (easy-to-use) E04HEF Unconstrained minimum of a sum of squares, combined Gauss–Newton and modified Newton algorithm, using second derivatives (comprehensive) E04HYF Unconstrained minimum of a sum of squares, combined Gauss–Newton and modified Newton algorithm, using second derivatives (easy-to-use) E04NCF Convex QP problem or linearly-constrained linear least squares problem (dense) E04UCF Minimum, function of several variables, sequential QP method, nonlinear constraints, using function values and optionally first derivatives (comprehensive) E04UFF Minimum, function of several variables, sequential QP method, nonlinear constraints, using function values and optionally first derivatives (reverse communication, comprehensive) E04UGF NLP problem (sparse) E04USF Minimum of a sum of squares, nonlinear constraints, sequential QP method, using function values and optionally first derivatives (comprehensive) E04YCF Covariance matrix for nonlinear least squares problem (unconstrained) F01ABF Inverse of real symmetric positive definite matrix using iterative refinement F01ELF Function of a real matrix (using numerical differentiation) F01FLF Function of a complex matrix (using numerical differentiation) 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 F02FJF Selected eigenvalues and eigenvectors of sparse symmetric eigenproblem (Black Box) F02WDF $QR$ factorization, possibly followed by SVD F02WGF Computes leading terms in the singular value decomposition of a real general matrix; also computes corresponding left and right singular vectors F02WUF SVD of real upper triangular matrix (Black Box) F02XUF SVD of complex upper triangular matrix (Black Box) F04ABF Solution of real symmetric positive definite simultaneous linear equations with multiple right-hand sides using iterative refinement (Black Box) F04AEF Solution of real simultaneous linear equations with multiple right-hand sides using iterative refinement (Black Box) F04ASF Solution of real symmetric positive definite simultaneous linear equations, one right-hand side using iterative refinement (Black Box) F04ATF Solution of real simultaneous linear equations, one right-hand side using iterative refinement (Black Box) F04JGF Least squares (if rank $\text{}=n$) or minimal least squares (if rank $\text{}) solution of $m$ real equations in $n$ unknowns, $m\ge n$ F11DCF Solution of real sparse nonsymmetric linear system, RGMRES, CGS, Bi-CGSTAB or TFQMR method, preconditioner computed by F11DAF F11DEF Solution of real sparse nonsymmetric linear system, RGMRES, CGS, Bi-CGSTAB, or TFQMR method, Jacobi or SSOR preconditioner (Black Box) F11DGF Solution of real sparse nonsymmetric linear system, RGMRES, CGS, Bi-CGSTAB or TFQMR method, incomplete $LU$ block diagonal preconditioner computed by F11DFF F11DKF Real sparse nonsymmetric linear systems, line Jacobi preconditioner F11DQF Solution of complex sparse non-Hermitian linear system, RGMRES, CGS, Bi-CGSTAB or TFQMR method, preconditioner computed by F11DNF (Black Box) F11DSF Solution of complex sparse non-Hermitian linear system, RGMRES, CGS, Bi-CGSTAB or TFQMR method, Jacobi or SSOR preconditioner Black Box F11DUF Solution of complex sparse nonsymmetric linear system, RGMRES, CGS, Bi-CGSTAB or TFQMR method, incomplete $LU$ block diagonal preconditioner computed by F11DTF F11DXF Complex sparse nonsymmetric linear systems, line Jacobi preconditioner F11JCF Solution of real sparse symmetric linear system, conjugate gradient/Lanczos method, preconditioner computed by F11JAF (Black Box) F11JEF Solution of real sparse symmetric linear system, conjugate gradient/Lanczos method, Jacobi or SSOR preconditioner (Black Box) F11JQF Solution of complex sparse Hermitian linear system, conjugate gradient/Lanczos method, preconditioner computed by F11JNF (Black Box) F11JSF Solution of complex sparse Hermitian linear system, conjugate gradient/Lanczos method, Jacobi or SSOR preconditioner (Black Box) F11MDF Real sparse nonsymmetric linear systems, setup for F11MEF F12AUF Selected eigenvalues and, optionally, eigenvectors of complex non-Hermitian banded eigenproblem, driver F12FGF Selected eigenvalues and, optionally, eigenvectors of a real symmetric banded eigenproblem, driver G01AGF Lineprinter scatterplot of two variables G01AHF Lineprinter scatterplot of one variable against Normal scores G01ANF Calculates approximate quantiles from a data stream of known size G01APF Calculates approximate quantiles from a data stream of unknown size G01ARF Constructs a stem and leaf plot G01EMF Computes probability for the Studentized range statistic G01HBF Computes probabilities for the multivariate Normal distribution G01JDF Computes lower tail probability for a linear combination of (central) ${\chi }^{2}$ variables G02ABF Computes the nearest correlation matrix to a real square matrix, augmented G02AAF to incorporate weights and bounds G02AEF Computes the nearest correlation matrix with $k$-factor structure to a real square matrix G02AJF Computes the nearest correlation matrix to a real square matrix, using element-wise weighting G02CGF Multiple linear regression, from correlation coefficients, with constant term G02CHF Multiple linear regression, from correlation-like coefficients, without constant term G02DAF Fits a general (multiple) linear regression model G02DDF Estimates of linear parameters and general linear regression model from updated model G02DEF Add a new independent variable to a general linear regression model G02DGF Fits a general linear regression model to new dependent variable G02DKF Estimates and standard errors of parameters of a general linear regression model for given constraints G02EEF Fits a linear regression model by forward selection G02GAF Fits a generalized linear model with Normal errors G02GBF Fits a generalized linear model with binomial errors G02GCF Fits a generalized linear model with Poisson errors G02GDF Fits a generalized linear model with gamma errors G02GKF Estimates and standard errors of parameters of a general linear model for given constraints G02HAF Robust regression, standard $M$-estimates G02HDF Robust regression, compute regression with user-supplied functions and weights G02HFF Robust regression, variance-covariance matrix following G02HDF G02HKF Calculates a robust estimation of a correlation matrix, Huber's weight function G02JAF Linear mixed effects regression using Restricted Maximum Likelihood (REML) G02JBF Linear mixed effects regression using Maximum Likelihood (ML) G02KAF Ridge regression, optimizing a ridge regression parameter G02KBF Ridge regression using a number of supplied ridge regression parameters G03ACF Performs canonical variate analysis G03ADF Performs canonical correlation analysis G04EAF Computes orthogonal polynomials or dummy variables for factor/classification variable G05PDF Generates a realisation of a time series from a GARCH process with asymmetry of the form ${\left({\epsilon }_{t-1}+\gamma \right)}^{2}$ G05PEF Generates a realisation of a time series from a GARCH process with asymmetry of the form ${\left(\left|{\epsilon }_{t-1}\right|+\gamma {\epsilon }_{t-1}\right)}^{2}$ G05PFF Generates a realisation of a time series from an asymmetric Glosten, Jagannathan and Runkle (GJR) GARCH process G05PGF Generates a realisation of a time series from an exponential GARCH (EGARCH) process G05PJF Generates a realisation of a multivariate time series from a VARMA model G05PYF Generates a random correlation matrix G05RZF Generates a matrix of pseudorandom numbers from a multivariate Normal distribution 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 G07BEF Computes maximum likelihood estimates for parameters of the Weibull distribution G07BFF Estimates parameter values of the generalized Pareto distribution G07DAF Robust estimation, median, median absolute deviation, robust standard deviation G07DBF Robust estimation, $M$-estimates for location and scale parameters, standard weight functions G07DCF Robust estimation, $M$-estimates for location and scale parameters, user-defined weight functions G07DDF Computes a trimmed and winsorized mean of a single sample with estimates of their variance G07EAF Robust confidence intervals, one-sample G07EBF Robust confidence intervals, two-sample G08AGF Performs the Wilcoxon one-sample (matched pairs) signed rank test G08AKF Computes the exact probabilities for the Mann–Whitney $U$ statistic, ties in pooled sample G08CBF Performs the one-sample Kolmogorov–Smirnov test for standard distributions G08CCF Performs the one-sample Kolmogorov–Smirnov test for a user-supplied distribution G08CDF Performs the two-sample Kolmogorov–Smirnov test G08RAF Regression using ranks, uncensored data G08RBF Regression using ranks, right-censored data G11BBF Computes multiway table from set of classification factors using given percentile/quantile G11BCF Computes marginal tables for multiway table computed by G11BAF or G11BBF G11SAF Contingency table, latent variable model for binary data G12ABF Computes rank statistics for comparing survival curves G13ADF Univariate time series, preliminary estimation, seasonal ARIMA model G13AEF Univariate time series, estimation, seasonal ARIMA model (comprehensive) G13AFF Univariate time series, estimation, seasonal ARIMA model (easy-to-use) G13AJF Univariate time series, state set and forecasts, from fully specified seasonal ARIMA model G13BCF Multivariate time series, cross-correlations G13BEF Multivariate time series, estimation of multi-input model G13BJF Multivariate time series, state set and forecasts from fully specified multi-input model G13DBF Multivariate time series, multiple squared partial autocorrelations G13DDF Multivariate time series, estimation of VARMA model G13DNF Multivariate time series, sample partial lag correlation matrices, ${\chi }^{2}$ statistics and significance levels G13FAF Univariate time series, parameter estimation for either a symmetric GARCH process or a GARCH process with asymmetry of the form ${\left({\epsilon }_{t-1}+\gamma \right)}^{2}$ G13FCF Univariate time series, parameter estimation for a GARCH process with asymmetry of the form ${\left(\left|{\epsilon }_{t-1}\right|+\gamma {\epsilon }_{t-1}\right)}^{2}$ G13FEF Univariate time series, parameter estimation for an asymmetric Glosten, Jagannathan and Runkle (GJR) GARCH process G13FGF Univariate time series, parameter estimation for an exponential GARCH (EGARCH) process

Tuned and Enhanced Routines in the NAG Library for SMP & Multicore (PDF version)
NAG Library Manual