Chapter Contents
NAG Toolbox

# Tuned and Enhanced Routines in the NAG Toolbox

## Introduction

Tuned routines are user-callable routines that have been parallelized, or otherwise optimized, in the NAG Toolbox to give improved performance. 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.

## Tuned LAPACK Functions

There are 77 tuned LAPACK routines within the Library.
 FunctionName Purpose nag_lapack_dgetrf (f07ad) LU$LU$ factorization of real m$m$ by n$n$ matrix nag_lapack_dgetrs (f07ae) Solution of real system of linear equations, multiple right-hand sides, matrix already factorized by nag_lapack_dgetrf (f07ad) nag_lapack_dgerfs (f07ah) Refined solution with error bounds of real system of linear equations, multiple right-hand sides nag_lapack_zgetrf (f07ar) LU$LU$ factorization of complex m$m$ by n$n$ matrix nag_lapack_zgetrs (f07as) Solution of complex system of linear equations, multiple right-hand sides, matrix already factorized by nag_lapack_zgetrf (f07ar) nag_lapack_zgerfs (f07av) Refined solution with error bounds of complex system of linear equations, multiple right-hand sides nag_lapack_dgbtrf (f07bd) LU$LU$ factorization of real m$m$ by n$n$ band matrix nag_lapack_dgbtrs (f07be) Solution of real band system of linear equations, multiple right-hand sides, matrix already factorized by nag_lapack_dgbtrf (f07bd) nag_lapack_dgbrfs (f07bh) Refined solution with error bounds of real band system of linear equations, multiple right-hand sides nag_lapack_zgbtrf (f07br) LU$LU$ factorization of complex m$m$ by n$n$ band matrix nag_lapack_zgbtrs (f07bs) Solution of complex band system of linear equations, multiple right-hand sides, matrix already factorized by nag_lapack_zgbtrf (f07br) nag_lapack_zgbrfs (f07bv) Refined solution with error bounds of complex band system of linear equations, multiple right-hand sides nag_lapack_dgtrfs (f07ch) Refined solution with error bounds of real tridiagonal system of linear equations, multiple right-hand sides nag_lapack_zgtrfs (f07cv) Refined solution with error bounds of complex tridiagonal system of linear equations, multiple right-hand sides nag_lapack_dpotrf (f07fd) Cholesky factorization of real symmetric positive definite matrix nag_lapack_dpotrs (f07fe) Solution of real symmetric positive definite system of linear equations, multiple right-hand sides, matrix already factorized by nag_lapack_dpotrf (f07fd) nag_lapack_dporfs (f07fh) Refined solution with error bounds of real symmetric positive definite system of linear equations, multiple right-hand sides nag_lapack_zpotrf (f07fr) Cholesky factorization of complex Hermitian positive definite matrix nag_lapack_zpotrs (f07fs) Solution of complex Hermitian positive definite system of linear equations, multiple right-hand sides, matrix already factorized by nag_lapack_zpotrf (f07fr) nag_lapack_zporfs (f07fv) Refined solution with error bounds of complex Hermitian positive definite system of linear equations, multiple right-hand sides nag_lapack_dpptrs (f07ge) Solution of real symmetric positive definite system of linear equations, multiple right-hand sides, matrix already factorized by nag_lapack_dpptrf (f07gd), packed storage nag_lapack_dpprfs (f07gh) Refined solution with error bounds of real symmetric positive definite system of linear equations, multiple right-hand sides, packed storage nag_lapack_zpptrs (f07gs) Solution of complex Hermitian positive definite system of linear equations, multiple right-hand sides, matrix already factorized by nag_lapack_zpptrf (f07gr), packed storage nag_lapack_zpprfs (f07gv) Refined solution with error bounds of complex Hermitian positive definite system of linear equations, multiple right-hand sides, packed storage nag_lapack_dpbtrs (f07he) Solution of real symmetric positive definite band system of linear equations, multiple right-hand sides, matrix already factorized by nag_lapack_dpbtrf (f07hd) nag_lapack_dpbrfs (f07hh) Refined solution with error bounds of real symmetric positive definite band system of linear equations, multiple right-hand sides nag_lapack_zpbtrs (f07hs) Solution of complex Hermitian positive definite band system of linear equations, multiple right-hand sides, matrix already factorized by nag_lapack_zpbtrf (f07hr) nag_lapack_zpbrfs (f07hv) Refined solution with error bounds of complex Hermitian positive definite band system of linear equations, multiple right-hand sides nag_lapack_dptrfs (f07jh) Refined solution with error bounds of real symmetric positive definite tridiagonal system of linear equations, multiple right-hand sides nag_lapack_zptrfs (f07jv) Refined solution with error bounds of complex Hermitian positive definite tridiagonal system of linear equations, multiple right-hand sides nag_lapack_dsyrfs (f07mh) Refined solution with error bounds of real symmetric indefinite system of linear equations, multiple right-hand sides nag_lapack_zherfs (f07mv) Refined solution with error bounds of complex Hermitian indefinite system of linear equations, multiple right-hand sides nag_lapack_zsyrfs (f07nv) Refined solution with error bounds of complex symmetric system of linear equations, multiple right-hand sides nag_lapack_dsprfs (f07ph) Refined solution with error bounds of real symmetric indefinite system of linear equations, multiple right-hand sides, packed storage nag_lapack_zhprfs (f07pv) Refined solution with error bounds of complex Hermitian indefinite system of linear equations, multiple right-hand sides, packed storage nag_lapack_zsprfs (f07qv) Refined solution with error bounds of complex symmetric system of linear equations, multiple right-hand sides, packed storage nag_lapack_dtrrfs (f07th) Error bounds for solution of real triangular system of linear equations, multiple right-hand sides nag_lapack_ztrrfs (f07tv) Error bounds for solution of complex triangular system of linear equations, multiple right-hand sides nag_lapack_dtptrs (f07ue) Solution of real triangular system of linear equations, multiple right-hand sides, packed storage nag_lapack_dtprfs (f07uh) Error bounds for solution of real triangular system of linear equations, multiple right-hand sides, packed storage nag_lapack_ztptrs (f07us) Solution of complex triangular system of linear equations, multiple right-hand sides, packed storage nag_lapack_ztprfs (f07uv) Error bounds for solution of complex triangular system of linear equations, multiple right-hand sides, packed storage nag_lapack_dtbtrs (f07ve) Solution of real band triangular system of linear equations, multiple right-hand sides nag_lapack_dtbrfs (f07vh) Error bounds for solution of real band triangular system of linear equations, multiple right-hand sides nag_lapack_ztbtrs (f07vs) Solution of complex band triangular system of linear equations, multiple right-hand sides nag_lapack_ztbrfs (f07vv) Error bounds for solution of complex band triangular system of linear equations, multiple right-hand sides nag_lapack_dgeqrf (f08ae) QR$QR$ factorization of real general rectangular matrix nag_lapack_dorgqr (f08af) Form all or part of orthogonal Q$Q$ from QR$QR$ factorization determined by nag_lapack_dgeqrf (f08ae), nag_lapack_dgeqpf (f08be) or nag_lapack_dgeqp3 (f08bf) nag_lapack_dormqr (f08ag) Apply orthogonal transformation determined by nag_lapack_dgeqrf (f08ae), nag_lapack_dgeqpf (f08be) or nag_lapack_dgeqp3 (f08bf) nag_lapack_zgeqrf (f08as) QR$QR$ factorization of complex general rectangular matrix nag_lapack_zungqr (f08at) Form all or part of unitary Q$Q$ from QR$QR$ factorization determined by nag_lapack_zgeqrf (f08as), nag_lapack_zgeqpf (f08bs) or nag_lapack_zgeqp3 (f08bt) nag_lapack_zunmqr (f08au) Apply unitary transformation determined by nag_lapack_zgeqrf (f08as), nag_lapack_zgeqpf (f08bs) or nag_lapack_zgeqp3 (f08bt) nag_lapack_dsytrd (f08fe) Orthogonal reduction of real symmetric matrix to symmetric tridiagonal form nag_lapack_dorgtr (f08ff) Generate orthogonal transformation matrix from reduction to tridiagonal form determined by nag_lapack_dsytrd (f08fe) nag_lapack_zhetrd (f08fs) Unitary reduction of complex Hermitian matrix to real symmetric tridiagonal form nag_lapack_zungtr (f08ft) Generate unitary transformation matrix from reduction to tridiagonal form determined by nag_lapack_zhetrd (f08fs) nag_lapack_dopgtr (f08gf) Generate orthogonal transformation matrix from reduction to tridiagonal form determined by nag_lapack_dsptrd (f08ge) nag_lapack_zupgtr (f08gt) Generate unitary transformation matrix from reduction to tridiagonal form determined by nag_lapack_zhptrd (f08gs) nag_lapack_dsbtrd (f08he) Orthogonal reduction of real symmetric band matrix to symmetric tridiagonal form nag_lapack_zhbtrd (f08hs) Unitary reduction of complex Hermitian band matrix to real symmetric tridiagonal form nag_lapack_dsteqr (f08je) All eigenvalues and eigenvectors of real symmetric tridiagonal matrix, reduced from real symmetric matrix using the implicit QL$QL$ or QR$QR$ algorithm nag_lapack_dstebz (f08jj) Selected eigenvalues of real symmetric tridiagonal matrix by bisection nag_lapack_dstein (f08jk) Selected eigenvectors of real symmetric tridiagonal matrix by inverse iteration, storing eigenvectors in real array nag_lapack_zsteqr (f08js) All eigenvalues and eigenvectors of real symmetric tridiagonal matrix, reduced from complex Hermitian matrix, using the implicit QL$QL$ or QR$QR$ algorithm nag_lapack_zstein (f08jx) Selected eigenvectors of real symmetric tridiagonal matrix by inverse iteration, storing eigenvectors in complex array nag_lapack_dgebrd (f08ke) Orthogonal reduction of real general rectangular matrix to bidiagonal form nag_lapack_zgebrd (f08ks) Unitary reduction of complex general rectangular matrix to bidiagonal form nag_lapack_dbdsqr (f08me) SVD of real bidiagonal matrix reduced from real general matrix nag_lapack_zbdsqr (f08ms) SVD of real bidiagonal matrix reduced from complex general matrix nag_lapack_dhsein (f08pk) Selected right and/or left eigenvectors of real upper Hessenberg matrix by inverse iteration nag_lapack_zhsein (f08px) Selected right and/or left eigenvectors of complex upper Hessenberg matrix by inverse iteration nag_lapack_dspgv (f08ta) Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, packed storage nag_lapack_dspgvx (f08tb) Computes selected eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, packed storage nag_lapack_dspgvd (f08tc) Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, packed storage (divide-and-conquer) nag_lapack_zhpgv (f08tn) Computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, packed storage nag_lapack_zhpgvx (f08tp) Computes selected eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, packed storage nag_lapack_zhpgvd (f08tq) Computes selected eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, packed storage (divide-and-conquer)

## Functions Enhanced by Calling Tuned LAPACK Functions

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.
 FunctionName Purpose nag_zeros_cubic_real (c02ak) All zeros of real cubic equation nag_zeros_quartic_real (c02al) All zeros of real quartic equation nag_zeros_cubic_complex (c02am) All zeros of complex cubic equation nag_zeros_quartic_complex (c02an) All zeros of complex quartic equation nag_roots_sys_func_easy (c05qb) Solution of a system of nonlinear equations using function values only (easy-to-use) nag_roots_sys_func_expert (c05qc) Solution of a system of nonlinear equations using function values only (comprehensive) nag_roots_sys_func_rcomm (c05qd) Solution of a system of nonlinear equations using function values only (reverse communication) nag_roots_sys_deriv_easy (c05rb) Solution of a system of nonlinear equations using first derivatives (easy-to-use) nag_roots_sys_deriv_expert (c05rc) Solution of a system of nonlinear equations using first derivatives (comprehensive) nag_roots_sys_deriv_rcomm (c05rd) Solution of a system of nonlinear equations using first derivatives (reverse communication) nag_ode_bvp_shoot_genpar_intern (d02ag) Ordinary differential equations, boundary value problem, shooting and matching technique, allowing interior matching point, general parameters to be determined nag_ode_bvp_shoot_bval (d02ha) Ordinary differential equations, boundary value problem, shooting and matching, boundary values to be determined nag_ode_bvp_shoot_genpar (d02hb) Ordinary differential equations, boundary value problem, shooting and matching, general parameters to be determined nag_ode_dae_dassl_gen (d02ne) Implicit ordinary differential equations/DAEs, initial value problem, DASSL method integrator nag_ode_bvp_shoot_genpar_algeq (d02sa) Ordinary differential equations, boundary value problem, shooting and matching technique, subject to extra algebraic equations, general parameters to be determined nag_ode_bvp_coll_nlin (d02tk) Ordinary differential equations, general nonlinear boundary value problem, collocation technique nag_ode_bvp_ps_lin_solve (d02ue) Solve linear constant coefficient boundary value problem on Chebyshev grid, Integral formulation nag_pde_1d_blackscholes_fd (d03nc) Finite difference solution of the Black–Scholes equations nag_inteq_abel2_weak (d05bd) Nonlinear convolution Volterra–Abel equation, second kind, weakly singular nag_inteq_abel1_weak (d05be) Nonlinear convolution Volterra–Abel equation, first kind, weakly singular nag_fit_2dspline_ts_sctr (e02jd) Spline approximation to a set of scattered data using a two-stage approximation method nag_opt_lsq_uncon_mod_func_comp (e04fc) Unconstrained minimum of a sum of squares, combined Gauss–Newton and modified Newton algorithm using function values only (comprehensive) nag_opt_lsq_uncon_mod_func_easy (e04fy) Unconstrained minimum of a sum of squares, combined Gauss–Newton and modified Newton algorithm using function values only (easy-to-use) nag_opt_lsq_uncon_quasi_deriv_comp (e04gb) Unconstrained minimum of a sum of squares, combined Gauss–Newton and quasi-Newton algorithm using first derivatives (comprehensive) nag_opt_lsq_uncon_mod_deriv_comp (e04gd) Unconstrained minimum of a sum of squares, combined Gauss–Newton and modified Newton algorithm using first derivatives (comprehensive) nag_opt_lsq_uncon_quasi_deriv_easy (e04gy) Unconstrained minimum of a sum of squares, combined Gauss–Newton and quasi-Newton algorithm, using first derivatives (easy-to-use) nag_opt_lsq_uncon_mod_deriv_easy (e04gz) Unconstrained minimum of a sum of squares, combined Gauss–Newton and modified Newton algorithm using first derivatives (easy-to-use) nag_opt_lsq_uncon_mod_deriv2_comp (e04he) Unconstrained minimum of a sum of squares, combined Gauss–Newton and modified Newton algorithm, using second derivatives (comprehensive) nag_opt_lsq_uncon_mod_deriv2_easy (e04hy) Unconstrained minimum of a sum of squares, combined Gauss–Newton and modified Newton algorithm, using second derivatives (easy-to-use) nag_opt_lsq_lincon_solve (e04nc) Convex QP problem or linearly-constrained linear least squares problem (dense) nag_opt_nlp1_solve (e04uc) Minimum, function of several variables, sequential QP method, nonlinear constraints, using function values and optionally first derivatives (comprehensive) nag_opt_nlp1_rcomm (e04uf) Minimum, function of several variables, sequential QP method, nonlinear constraints, using function values and optionally first derivatives (reverse communication, comprehensive) nag_opt_lsq_gencon_deriv (e04us) Minimum of a sum of squares, nonlinear constraints, sequential QP method, using function values and optionally first derivatives (comprehensive) nag_matop_real_symm_posdef_inv (f01ab) Inverse of real symmetric positive definite matrix using iterative refinement nag_matop_real_symm_posdef_inv_noref (f01ad) Inverse of real symmetric positive definite matrix nag_matop_real_gen_matrix_exp (f01ec) Real matrix exponential nag_matop_real_symm_matrix_exp (f01ed) Real symmetric matrix exponential nag_matop_real_symm_matrix_fun (f01ef) Function of a real symmetric matrix nag_matop_real_gen_matrix_fun_num (f01el) Function of a real matrix (using numerical differentiation) nag_matop_complex_gen_matrix_exp (f01fc) Complex matrix exponential nag_matop_complex_herm_matrix_exp (f01fd) Complex Hermitian matrix exponential nag_matop_complex_herm_matrix_fun (f01ff) Function of a complex Hermitian matrix nag_matop_complex_gen_matrix_fun_num (f01fl) Function of a complex matrix (using numerical differentiation) nag_matop_real_gen_matrix_cond_std (f01ja) Condition number for the exponential, logarithm, sine, cosine, sinh or cosh of a real matrix nag_matop_real_gen_matrix_cond_num (f01jb) Condition number for a function of a real matrix (using numerical differentiation) nag_matop_real_gen_matrix_cond_usd (f01jc) Condition number for a function of a real matrix (using user-supplied derivatives) nag_matop_complex_gen_matrix_cond_std (f01ka) Condition number for the exponential, logarithm, sine, cosine, sinh or cosh of a complex matrix nag_matop_complex_gen_matrix_cond_num (f01kb) Condition number for a function of a complex matrix (using numerical differentiation) nag_matop_complex_gen_matrix_cond_usd (f01kc) Condition number for a function of a complex matrix (using user-supplied derivatives) nag_eigen_real_gen_eigsys (f02ec) Selected eigenvalues and eigenvectors of real nonsymmetric matrix (Black Box) nag_eigen_real_symm_sparse_eigsys (f02fj) Selected eigenvalues and eigenvectors of sparse symmetric eigenproblem (Black Box) nag_eigen_complex_gen_eigsys (f02gc) Selected eigenvalues and eigenvectors of complex nonsymmetric matrix (Black Box) nag_eigen_real_gen_qu_svd (f02wd) QR$QR$ factorization, possibly followed by SVD nag_eigen_real_gen_partialsvd (f02wg) Computes leading terms in the singular value decomposition of a real general matrix; also computes corresponding left and right singular vectors nag_eigen_real_triang_svd (f02wu) SVD of real upper triangular matrix (Black Box) nag_eigen_complex_triang_svd (f02xu) SVD of complex upper triangular matrix (Black Box) nag_linsys_real_posdef_solve_ref (f04ab) Solution of real symmetric positive definite simultaneous linear equations with multiple right-hand sides using iterative refinement (Black Box) nag_linsys_real_posdef_solve_1rhs (f04as) Solution of real symmetric positive definite simultaneous linear equations, one right-hand side using iterative refinement (Black Box) nag_linsys_real_square_solve (f04ba) Computes the solution and error-bound to a real system of linear equations nag_linsys_real_band_solve (f04bb) Computes the solution and error-bound to a real banded system of linear equations nag_linsys_real_posdef_solve (f04bd) Computes the solution and error-bound to a real symmetric positive definite system of linear equations nag_linsys_real_posdef_packed_solve (f04be) Computes the solution and error-bound to a real symmetric positive definite system of linear equations, packed storage nag_linsys_real_posdef_band_solve (f04bf) Computes the solution and error-bound to a real symmetric positive definite banded system of linear equations nag_linsys_complex_square_solve (f04ca) Computes the solution and error-bound to a complex system of linear equations nag_linsys_complex_band_solve (f04cb) Computes the solution and error-bound to a complex banded system of linear equations nag_linsys_complex_posdef_solve (f04cd) Computes the solution and error-bound to a complex Hermitian positive definite system of linear equations nag_linsys_complex_posdef_packed_solve (f04ce) Computes the solution and error-bound to a complex Hermitian positive definite system of linear equations, packed storage nag_linsys_complex_posdef_band_solve (f04cf) Computes the solution and error-bound to a complex Hermitian positive definite banded system of linear equations nag_linsys_real_gen_solve (f04jg) Least squares (if rank = n$\text{}=n$) or minimal least squares (if rank < n$\text{}) solution of m$m$ real equations in n$n$ unknowns, m ≥ n$m\ge n$ nag_lapack_dgesv (f07aa) Computes the solution to a real system of linear equations nag_lapack_dgesvx (f07ab) Uses the LU$LU$ factorization to compute the solution, error-bound and condition estimate for a real system of linear equations nag_lapack_dsgesv (f07ac) Mixed precision real system solver nag_lapack_zgesv (f07an) Computes the solution to a complex system of linear equations nag_lapack_zgesvx (f07ap) Uses the LU$LU$ factorization to compute the solution, error-bound and condition estimate for a complex system of linear equations nag_lapack_zcgesv (f07aq) Mixed precision complex system solver nag_lapack_dgbsv (f07ba) Computes the solution to a real banded system of linear equations nag_lapack_dgbsvx (f07bb) Uses the LU$LU$ factorization to compute the solution, error-bound and condition estimate for a real banded system of linear equations nag_lapack_zgbsv (f07bn) Computes the solution to a complex banded system of linear equations nag_lapack_zgbsvx (f07bp) Uses the LU$LU$ factorization to compute the solution, error-bound and condition estimate for a complex banded system of linear equations nag_lapack_dgtsvx (f07cb) Uses the LU$LU$ factorization to compute the solution, error-bound and condition estimate for a real tridiagonal system of linear equations nag_lapack_zgtsvx (f07cp) Uses the LU$LU$ factorization to compute the solution, error-bound and condition estimate for a complex tridiagonal system of linear equations nag_lapack_dposv (f07fa) Computes the solution to a real symmetric positive definite system of linear equations nag_lapack_dposvx (f07fb) Uses the Cholesky factorization to compute the solution, error-bound and condition estimate for a real symmetric positive definite system of linear equations nag_lapack_dsposv (f07fc) Uses the Cholesky factorization to compute the solution for a real symmetric positive definite system of linear equations nag_lapack_zposv (f07fn) Computes the solution to a complex Hermitian positive definite system of linear equations nag_lapack_zposvx (f07fp) Uses the Cholesky factorization to compute the solution, error-bound and condition estimate for a complex Hermitian positive definite system of linear equations nag_lapack_zcposv (f07fq) Uses the Cholesky factorization to compute the solution for a complex Hermitian positive definite system of linear equations nag_lapack_dppsv (f07ga) Computes the solution to a real symmetric positive definite system of linear equations, packed storage nag_lapack_dppsvx (f07gb) 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 nag_lapack_zppsv (f07gn) Computes the solution to a complex Hermitian positive definite system of linear equations, packed storage nag_lapack_zppsvx (f07gp) 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 nag_lapack_dpbsv (f07ha) Computes the solution to a real symmetric positive definite banded system of linear equations nag_lapack_dpbsvx (f07hb) Uses the Cholesky factorization to compute the solution, error-bound and condition estimate for a real symmetric positive definite banded system of linear equations nag_lapack_zpbsv (f07hn) Computes the solution to a complex Hermitian positive definite banded system of linear equations nag_lapack_zpbsvx (f07hp) Uses the Cholesky factorization to compute the solution, error-bound and condition estimate for a complex Hermitian positive definite banded system of linear equations nag_lapack_dptsvx (f07jb) 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 nag_lapack_zptsvx (f07jp) 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 nag_lapack_dsysvx (f07mb) Uses the diagonal pivoting factorization to compute the solution to a real symmetric system of linear equations nag_lapack_zhesvx (f07mp) Uses the diagonal pivoting factorization to compute the solution to a complex Hermitian system of linear equations nag_lapack_zsysvx (f07np) Uses the diagonal pivoting factorization to compute the solution to a complex symmetric system of linear equations nag_lapack_dspsvx (f07pb) Uses the diagonal pivoting factorization to compute the solution to a real symmetric system of linear equations, packed storage. Error bounds and a condition estimate are also computed. nag_lapack_zhpsvx (f07pp) Uses the diagonal pivoting factorization to compute the solution to a complex Hermitian system of linear equations, packed storage nag_lapack_zspsvx (f07qp) Uses the diagonal pivoting factorization to compute the solution to a complex symmetric system of linear equations, packed storage nag_lapack_dpftrf (f07wd) Cholesky factorization of real symmetric positive definite matrix, Rectangular Full Packed format nag_lapack_zpftrf (f07wr) Cholesky factorization of complex Hermitian positive definite matrix, Rectangular Full Packed format nag_lapack_dgels (f08aa) Solves an overdetermined or underdetermined real linear system nag_lapack_zgels (f08an) Solves an overdetermined or underdetermined complex linear system nag_lapack_dgelsy (f08ba) Computes the minimum-norm solution to a real linear least squares problem nag_lapack_dgeqp3 (f08bf) QR$QR$ factorization of real general rectangular matrix with column pivoting, using BLAS-3 nag_lapack_zgelsy (f08bn) Computes the minimum-norm solution to a complex linear least squares problem nag_lapack_zgeqp3 (f08bt) QR$QR$ factorization of complex general rectangular matrix with column pivoting, using BLAS-3 nag_lapack_dsyev (f08fa) Computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix nag_lapack_dsyevx (f08fb) Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix nag_lapack_dsyevd (f08fc) Computes all eigenvalues and, optionally, all eigenvectors of real symmetric matrix (divide-and-conquer) nag_lapack_dsyevr (f08fd) Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix (Relatively Robust Representations) nag_lapack_dormtr (f08fg) Apply orthogonal transformation determined by nag_lapack_dsytrd (f08fe) nag_lapack_zheev (f08fn) Computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix nag_lapack_zheevx (f08fp) Computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix nag_lapack_zheevd (f08fq) Computes all eigenvalues and, optionally, all eigenvectors of complex Hermitian matrix (divide-and-conquer) nag_lapack_zheevr (f08fr) Computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix (Relatively Robust Representations) nag_lapack_zunmtr (f08fu) Apply unitary transformation matrix determined by nag_lapack_zhetrd (f08fs) nag_lapack_dspev (f08ga) Computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix, packed storage nag_lapack_dspevx (f08gb) Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix, packed storage nag_lapack_dspevd (f08gc) Computes all eigenvalues and, optionally, all eigenvectors of real symmetric matrix, packed storage (divide-and-conquer or Pal–Walker–Kahan variant of the QL$QL$ or QR$QR$ algorithm) nag_lapack_zhpev (f08gn) Computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix, packed storage nag_lapack_zhpevx (f08gp) Computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix, packed storage nag_lapack_zhpevd (f08gq) Computes all eigenvalues and, optionally, all eigenvectors of complex Hermitian matrix, packed storage (divide-and-conquer or Pal–Walker–Kahan variant of the QL$QL$ or QR$QR$ algorithm) nag_lapack_dsbev (f08ha) Computes all eigenvalues and, optionally, eigenvectors of a real symmetric band matrix nag_lapack_dsbevx (f08hb) Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric band matrix nag_lapack_dsbevd (f08hc) Computes all eigenvalues and, optionally, all eigenvectors of real symmetric band matrix (divide-and-conquer or Pal–Walker–Kahan variant of the QL$QL$ or QR$QR$ algorithm) nag_lapack_zhbev (f08hn) Computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix nag_lapack_zhbevx (f08hp) Computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix nag_lapack_zhbevd (f08hq) Computes all eigenvalues and, optionally, all eigenvectors of complex Hermitian band matrix (divide-and-conquer) nag_lapack_dstev (f08ja) Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix nag_lapack_dstevx (f08jb) Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix nag_lapack_dstevd (f08jc) Computes all eigenvalues and, optionally, all eigenvectors of real symmetric tridiagonal matrix (divide-and-conquer) nag_lapack_dstevr (f08jd) Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix (Relatively Robust Representations) nag_lapack_dpteqr (f08jg) Computes all eigenvalues and eigenvectors of real symmetric positive definite tridiagonal matrix, reduced from real symmetric positive definite matrix nag_lapack_dstedc (f08jh) Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix or a matrix reduced to this form (divide-and-conquer) nag_lapack_dstegr (f08jl) Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix or a symmetric matrix reduced to this form (Relatively Robust Representations) nag_lapack_zpteqr (f08ju) Computes all eigenvalues and eigenvectors of real symmetric positive definite tridiagonal matrix, reduced from complex Hermitian positive definite matrix nag_lapack_zstedc (f08jv) Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix or a complex Hermitian matrix reduced to this form (divide-and-conquer) nag_lapack_zstegr (f08jy) Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix or a complex Hermitian matrix reduced to this form (Relatively Robust Representations) nag_lapack_dgelss (f08ka) Computes the minimum-norm solution to a real linear least squares problem using singular value decomposition nag_lapack_dgesvd (f08kb) Computes the singular value decomposition of a real matrix, optionally computing the left and/or right singular vectors nag_lapack_dgelsd (f08kc) Computes the minimum-norm solution to a real linear least squares problem using singular value decomposition (divide-and-conquer) nag_lapack_dgesdd (f08kd) Computes the singular value decomposition of a real matrix, optionally computing the left and/or right singular vectors (divide-and-conquer) nag_lapack_dorgbr (f08kf) Generate orthogonal transformation matrices from reduction to bidiagonal form determined by nag_lapack_dgebrd (f08ke) nag_lapack_dormbr (f08kg) Apply orthogonal transformations from reduction to bidiagonal form determined by nag_lapack_dgebrd (f08ke) nag_lapack_dgejsv (f08kh) Computes the singular value decomposition of a real matrix, optionally computing the left and/or right singular vectors (preconditioned Jacobi) nag_lapack_zgelss (f08kn) Computes the minimum-norm solution to a complex linear least squares problem using singular value decomposition nag_lapack_zgesvd (f08kp) Computes the singular value decomposition of a complex matrix, optionally computing the left and/or right singular vectors nag_lapack_zgelsd (f08kq) Computes the minimum-norm solution to a complex linear least squares problem using singular value decomposition (divide-and-conquer) nag_lapack_zgesdd (f08kr) Computes the singular value decomposition of a complex matrix, optionally computing the left and/or right singular vectors (divide-and-conquer) nag_lapack_zungbr (f08kt) Generate unitary transformation matrices from reduction to bidiagonal form determined by nag_lapack_zgebrd (f08ks) nag_lapack_zunmbr (f08ku) Apply unitary transformations from reduction to bidiagonal form determined by nag_lapack_zgebrd (f08ks) nag_lapack_dbdsdc (f08md) Computes the singular value decomposition of a real bidiagonal matrix, optionally computing the singular vectors (divide-and-conquer) nag_lapack_dgeev (f08na) Computes all eigenvalues and, optionally, left and/or right eigenvectors of a real nonsymmetric matrix nag_lapack_dgeevx (f08nb) 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 nag_lapack_dorghr (f08nf) Generate orthogonal transformation matrix from reduction to Hessenberg form determined by nag_lapack_dgehrd (f08ne) nag_lapack_dormhr (f08ng) Apply orthogonal transformation matrix from reduction to Hessenberg form determined by nag_lapack_dgehrd (f08ne) nag_lapack_zgeev (f08nn) Computes all eigenvalues and, optionally, left and/or right eigenvectors of a complex nonsymmetric matrix nag_lapack_zgeevx (f08np) 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 nag_lapack_zunghr (f08nt) Generate unitary transformation matrix from reduction to Hessenberg form determined by nag_lapack_zgehrd (f08ns) nag_lapack_zunmhr (f08nu) Apply unitary transformation matrix from reduction to Hessenberg form determined by nag_lapack_zgehrd (f08ns) nag_lapack_dgees (f08pa) Computes for real square nonsymmetric matrix, the eigenvalues, the real Schur form, and, optionally, the matrix of Schur vectors nag_lapack_dgeesx (f08pb) 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 nag_lapack_dhseqr (f08pe) Computes the eigenvalues and Schur factorization of real upper Hessenberg matrix reduced from real general matrix nag_lapack_zgees (f08pn) Computes for complex square nonsymmetric matrix, the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors nag_lapack_zgeesx (f08pp) Computes for real square nonsymmetric matrix, the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors; also computes a reciprocal condition number for the average of the selected eigenvalues and for the right invariant subspace corresponding to these eigenvalues nag_lapack_zhseqr (f08ps) Computes the eigenvalues and Schur factorization of complex upper Hessenberg matrix reduced from complex general matrix nag_lapack_dsygv (f08sa) Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem nag_lapack_dsygvx (f08sb) Computes selected eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem nag_lapack_dsygvd (f08sc) Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem (divide-and-conquer) nag_lapack_zhegv (f08sn) Computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem nag_lapack_zhegvx (f08sp) Computes selected eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem nag_lapack_zhegvd (f08sq) Computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem (divide-and-conquer) nag_lapack_dsbgv (f08ua) Computes all the eigenvalues, and optionally, the eigenvectors of a real banded generalized symmetric-definite eigenproblem nag_lapack_dsbgvx (f08ub) Computes selected eigenvalues, and optionally, the eigenvectors of a real banded generalized symmetric-definite eigenproblem nag_lapack_dsbgvd (f08uc) Computes all the eigenvalues, and optionally, the eigenvectors of a real banded generalized symmetric-definite eigenproblem (divide-and-conquer) nag_lapack_zhbgv (f08un) Computes all the eigenvalues, and optionally, the eigenvectors of a complex banded generalized Hermitian-definite eigenproblem nag_lapack_zhbgvx (f08up) Computes selected eigenvalues, and optionally, the eigenvectors of a complex banded generalized Hermitian-definite eigenproblem nag_lapack_zhbgvd (f08uq) Computes all the eigenvalues, and optionally, the eigenvectors of a complex banded generalized Hermitian-definite eigenproblem (divide-and-conquer) nag_lapack_dggev (f08wa) Computes, for a real nonsymmetric matrix pair, the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors nag_lapack_dggevx (f08wb) 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 nag_lapack_zggev (f08wn) Computes, for a complex nonsymmetric matrix pair, the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors nag_lapack_zggevx (f08wp) 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 nag_lapack_dgges (f08xa) 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 nag_lapack_dggesx (f08xb) 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 nag_lapack_zgges (f08xn) 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 nag_lapack_zggesx (f08xp) 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 nag_lapack_dgglse (f08za) Solves the real linear equality-constrained least squares (LSE) problem nag_lapack_dggglm (f08zb) Solves a real general Gauss–Markov linear model (GLM) problem nag_lapack_dggqrf (f08ze) Computes a generalized QR$QR$ factorization of a real matrix pair nag_lapack_dggrqf (f08zf) Computes a generalized RQ$RQ$ factorization of a real matrix pair nag_lapack_zgglse (f08zn) Solves the complex linear equality-constrained least squares (LSE) problem nag_lapack_zggglm (f08zp) Solves a complex general Gauss–Markov linear model (GLM) problem nag_lapack_zggqrf (f08zs) Computes a generalized QR$QR$ factorization of a complex matrix pair nag_lapack_zggrqf (f08zt) Computes a generalized RQ$RQ$ factorization of a complex matrix pair nag_sparseig_complex_band_solve (f12au) Selected eigenvalues and, optionally, eigenvectors of complex non-Hermitian banded eigenproblem, driver nag_sparseig_real_symm_proc (f12fc) Selected eigenvalues and, optionally, eigenvectors of a real symmetric sparse eigenproblem, postprocessing for nag_sparseig_real_symm_iter (f12fb) nag_sparseig_real_symm_band_solve (f12fg) Selected eigenvalues and, optionally, eigenvectors of a real symmetric banded eigenproblem, driver nag_stat_prob_multi_normal (g01hb) Computes probabilities for the multivariate Normal distribution nag_stat_pdf_multi_normal_vector (g01lb) Computes a vector of values for the probability density function of the multivariate Normal distribution nag_correg_corrmat_nearest_bounded (g02ab) Computes the nearest correlation matrix to a real square matrix, augmented nag_correg_corrmat_nearest (g02aa) to incorporate weights and bounds nag_correg_corrmat_nearest_kfactor (g02ae) Computes the nearest correlation matrix with k$k$-factor structure to a real square matrix nag_nearest_correlation_h_weight (g02aj) Computes the nearest correlation matrix to a real square matrix, using element-wise weighting nag_correg_corrmat_partial (g02by) Computes partial correlation/variance-covariance matrix from correlation/variance-covariance matrix computed by nag_correg_corrmat (g02bx) nag_correg_linregm_coeffs_const (g02cg) Multiple linear regression, from correlation coefficients, with constant term nag_correg_linregm_coeffs_noconst (g02ch) Multiple linear regression, from correlation-like coefficients, without constant term nag_correg_linregm_fit (g02da) Fits a general (multiple) linear regression model nag_correg_linregm_update (g02dd) Estimates of linear parameters and general linear regression model from updated model nag_correg_linregm_rssq (g02ea) Computes residual sums of squares for all possible linear regressions for a set of independent variables nag_correg_linregm_fit_onestep (g02ee) Fits a linear regression model by forward selection nag_correg_glm_normal (g02ga) Fits a generalized linear model with Normal errors nag_correg_glm_binomial (g02gb) Fits a generalized linear model with binomial errors nag_correg_glm_poisson (g02gc) Fits a generalized linear model with Poisson errors nag_correg_glm_gamma (g02gd) Fits a generalized linear model with gamma errors nag_correg_robustm (g02ha) Robust regression, standard M$M$-estimates nag_correg_robustm_user (g02hd) Robust regression, compute regression with user-supplied functions and weights nag_correg_robustm_user_varmat (g02hf) Robust regression, variance-covariance matrix following nag_correg_robustm_user (g02hd) nag_correg_mixeff_reml (g02ja) Linear mixed effects regression using Restricted Maximum Likelihood (REML) nag_correg_mixeff_ml (g02jb) Linear mixed effects regression using Maximum Likelihood (ML) nag_correg_ridge_opt (g02ka) Ridge regression, optimizing a ridge regression parameter nag_correg_ridge (g02kb) Ridge regression using a number of supplied ridge regression parameters nag_correg_pls_svd (g02la) Partial least squares (PLS) regression using singular value decomposition nag_correg_pls_fit (g02lc) PLS parameter estimates following partial least squares regression by nag_correg_pls_svd (g02la) or nag_correg_pls_wold (g02lb) nag_correg_quantile_linreg_easy (g02qf) Linear quantile regression, simple interface, independent, identically distributed (IID) errors nag_correg_quantile_linreg (g02qg) Linear quantile regression, comprehensive interface nag_mv_prin_comp (g03aa) Performs principal component analysis nag_mv_canon_var (g03ac) Performs canonical variate analysis nag_mv_canon_corr (g03ad) Performs canonical correlation analysis nag_mv_rot_orthomax (g03ba) Computes orthogonal rotations for loading matrix, generalized orthomax criterion nag_mv_rot_procrustes (g03bc) Computes Procrustes rotations nag_mv_rot_promax (g03bd) ProMax rotations nag_mv_discrim (g03da) Computes test statistic for equality of within-group covariance matrices and matrices for discriminant analysis nag_mv_multidimscal_metric (g03fa) Performs principal coordinate analysis, classical metric scaling nag_anova_random (g04bb) Analysis of variance, randomized block or completely randomized design, treatment means and standard errors nag_anova_rowcol (g04bc) Analysis of variance, general row and column design, treatment means and standard errors nag_rand_times_mv_varma (g05pj) Generates a realisation of a multivariate time series from a VARMA model nag_nonpar_rank_regsn (g08ra) Regression using ranks, uncensored data nag_nonpar_rank_regsn_censored (g08rb) Regression using ranks, right-censored data nag_contab_condl_logistic (g11ca) Returns parameter estimates for the conditional analysis of stratified data nag_contab_binary (g11sa) Contingency table, latent variable model for binary data nag_surviv_logrank (g12ab) Computes rank statistics for comparing survival curves nag_surviv_coxmodel (g12ba) Fits Cox's proportional hazard model nag_tsa_uni_arima_estim (g13ae) Univariate time series, estimation, seasonal ARIMA model (comprehensive) nag_tsa_uni_arima_estim_easy (g13af) Univariate time series, estimation, seasonal ARIMA model (easy-to-use) nag_tsa_uni_arima_forcecast (g13aj) Univariate time series, state set and forecasts, from fully specified seasonal ARIMA model nag_tsa_uni_arima_resid (g13as) Univariate time series, diagnostic checking of residuals, following nag_tsa_uni_arima_estim (g13ae) or nag_tsa_uni_arima_estim_easy (g13af) nag_tsa_multi_filter_arima (g13ba) Multivariate time series, filtering (pre-whitening) by an ARIMA model nag_tsa_multi_filter_transf (g13bb) Multivariate time series, filtering by a transfer function model nag_tsa_multi_transf_prelim (g13bd) Multivariate time series, preliminary estimation of transfer function model nag_tsa_multi_inputmod_estim (g13be) Multivariate time series, estimation of multi-input model nag_tsa_multi_inputmod_forecast (g13bj) Multivariate time series, state set and forecasts from fully specified multi-input model nag_tsa_multi_autocorr_part (g13db) Multivariate time series, multiple squared partial autocorrelations nag_tsa_multi_varma_estimate (g13dd) Multivariate time series, estimation of VARMA model nag_tsa_multi_varma_forecast (g13dj) Multivariate time series, forecasts and their standard errors nag_tsa_multi_corrmat_partlag (g13dn) Multivariate time series, sample partial lag correlation matrices, χ2${\chi }^{2}$ statistics and significance levels nag_tsa_multi_regmat_partial (g13dp) Multivariate time series, partial autoregression matrices nag_tsa_multi_varma_diag (g13ds) Multivariate time series, diagnostic checking of residuals, following nag_tsa_multi_varma_estimate (g13dd) nag_tsa_uni_arma_roots (g13dx) Calculates the zeros of a vector autoregressive (or moving average) operator nag_tsa_uni_garch_asym1_estim (g13fa) Univariate time series, parameter estimation for either a symmetric GARCH process or a GARCH process with asymmetry of the form (εt − 1 + γ)2${\left({\epsilon }_{t-1}+\gamma \right)}^{2}$ nag_tsa_uni_garch_asym2_estim (g13fc) Univariate time series, parameter estimation for a GARCH process with asymmetry of the form (|εt − 1| + γεt − 1)2${\left(|{\epsilon }_{t-1}|+\gamma {\epsilon }_{t-1}\right)}^{2}$ nag_tsa_uni_garch_gjr_estim (g13fe) Univariate time series, parameter estimation for an asymmetric Glosten, Jagannathan and Runkle (GJR) GARCH process nag_tsa_uni_garch_exp_estim (g13fg) Univariate time series, parameter estimation for an exponential GARCH (EGARCH) process

## Tuned NAG-specific Functions

These NAG-specific functions have been parallelized, or otherwise optimized, in the NAG Toolbox compared to the equivalent function in the NAG Fortran Library. There are 149 of these routines within the Library.
 FunctionName Purpose nag_sum_convcorr_real (c06fk) Circular convolution or correlation of two real vectors, extra workspace for greater speed nag_sum_fft_real_1d_multi_rfmt (c06fp) Multiple one-dimensional real discrete Fourier transforms nag_sum_fft_hermitian_1d_multi_rfmt (c06fq) Multiple one-dimensional Hermitian discrete Fourier transforms nag_sum_fft_complex_3d_sep (c06fx) Three-dimensional complex discrete Fourier transform nag_sum_fft_realherm_1d (c06pa) Single one-dimensional real and Hermitian complex discrete Fourier transform, using complex storage format for Hermitian sequences nag_sum_fft_complex_multid_1d (c06pf) One-dimensional complex discrete Fourier transform of multidimensional data (using complex data type) nag_sum_fft_complex_multid (c06pj) Multidimensional complex discrete Fourier transform of multidimensional data (using complex data type) nag_sum_convcorr_complex (c06pk) Circular convolution or correlation of two complex vectors nag_sum_fft_realherm_1d_multi_row (c06pp) Multiple one-dimensional real and Hermitian complex discrete Fourier transforms, using row ordered complex storage format for Hermitian sequences nag_sum_fft_realherm_1d_multi_col (c06pq) Multiple one-dimensional real and Hermitian complex discrete Fourier transforms, using column ordered complex storage format for Hermitian sequences nag_sum_fft_complex_1d_multi_row (c06pr) Multiple one-dimensional complex discrete Fourier transforms using complex data type nag_sum_fft_complex_1d_multi_col (c06ps) Multiple one-dimensional complex discrete Fourier transforms using complex data type and sequences stored as columns nag_sum_fft_complex_2d (c06pu) Two-dimensional complex discrete Fourier transform, complex data type nag_sum_fft_real_2d (c06pv) Two-dimensional real-to-complex discrete Fourier transform nag_sum_fft_hermitian_2d (c06pw) Two-dimensional complex-to-real discrete Fourier transform nag_sum_fft_complex_3d (c06px) Three-dimensional complex discrete Fourier transform, complex data type nag_sum_fft_real_3d (c06py) Three-dimensional real-to-complex discrete Fourier transform nag_sum_fft_hermitian_3d (c06pz) Three-dimensional complex-to-real discrete Fourier transform nag_sum_fft_real_sine_simple (c06ra) Discrete sine transform (easy-to-use) nag_sum_fft_real_cosine_simple (c06rb) Discrete cosine transform (easy-to-use) nag_sum_fft_real_qtrsine_simple (c06rc) Discrete quarter-wave sine transform (easy-to-use) nag_sum_fft_real_qtrcosine_simple (c06rd) Discrete quarter-wave cosine transform (easy-to-use) nag_wav_2d_sngl_fwd (c09ea) Two-dimensional discrete wavelet transform nag_wav_2d_sngl_inv (c09eb) Two-dimensional inverse discrete wavelet transform nag_wav_2d_multi_fwd (c09ec) Two-dimensional multi-level discrete wavelet transform nag_wav_2d_multi_inv (c09ed) Two-dimensional inverse multi-level discrete wavelet transform nag_wav_3d_sngl_fwd (c09fa) Three-dimensional discrete wavelet transform nag_wav_3d_sngl_inv (c09fb) Three-dimensional inverse discrete wavelet transform nag_wav_3d_multi_fwd (c09fc) Three-dimensional multi-level discrete wavelet transform nag_wav_3d_mxolap_multi_inv (c09fd) Three-dimensional inverse multi-level discrete wavelet transform nag_quad_2d_fin (d01da) Two-dimensional quadrature, finite region nag_quad_md_adapt (d01fc) Multidimensional adaptive quadrature over hyper-rectangle nag_quad_1d_data (d01ga) One-dimensional quadrature, integration of function defined by data values, Gill–Miller method nag_pde_3d_ellip_helmholtz (d03fa) Elliptic PDE, Helmholtz equation, three-dimensional Cartesian coordinates nag_pde_2d_gen_order2_rectangle (d03ra) General system of second-order PDEs, method of lines, finite differences, remeshing, two space variables, rectangular region nag_pde_2d_gen_order2_rectilinear (d03rb) General system of second-order PDEs, method of lines, finite differences, remeshing, two space variables, rectilinear region nag_interp_2d_scat_shep (e01sg) Interpolating functions, modified Shepard's method, two variables nag_interp_2d_scat_shep_eval (e01sh) Interpolated values, evaluate interpolant computed by nag_interp_2d_scat_shep (e01sg), function and first derivatives, two variables nag_interp_3d_scat_shep (e01tg) Interpolating functions, modified Shepard's method, three variables nag_interp_3d_scat_shep_eval (e01th) Interpolated values, evaluate interpolant computed by nag_interp_3d_scat_shep (e01tg), function and first derivatives, three variables nag_interp_4d_scat_shep (e01tk) Interpolating functions, modified Shepard's method, four variables nag_interp_4d_scat_shep_eval (e01tl) Interpolated values, evaluate interpolant computed by nag_interp_4d_scat_shep (e01tk), function and first derivatives, four variables nag_interp_5d_scat_shep (e01tm) Interpolating functions, modified Shepard's method, five variables nag_interp_5d_scat_shep_eval (e01tn) Interpolated values, evaluate interpolant computed by nag_interp_5d_scat_shep (e01tm), function and first derivatives, five variables nag_interp_nd_scat_shep (e01zm) Interpolating function, modified Shepard's method, d$d$ dimensions nag_interp_nd_scat_shep_eval (e01zn) Interpolated values, evaluate interpolant computed by nag_interp_nd_scat_shep (e01zm), function and first derivatives, d$d$ dimensions nag_fit_1dspline_deriv_vector (e02bf) Evaluation of fitted cubic spline, function and optionally derivatives at a vector of points nag_fit_2dcheb_lines (e02ca) Least squares surface fit by polynomials, data on lines parallel to one independent coordinate axis nag_fit_2dcheb_eval (e02cb) Evaluation of fitted polynomial in two variables nag_fit_2dspline_evalm (e02df) Evaluation of fitted bicubic spline at a mesh of points nag_glopt_bnd_pso (e05sa) Global optimization using particle swarm algorithm (PSO), bound constraints only nag_glopt_nlp_pso (e05sb) Global optimization using particle swarm algorithm (PSO), comprehensive nag_glopt_nlp_multistart_sqp (e05uc) Global optimization using multi-start, nonlinear constraints nag_glopt_nlp_multistart_sqp_lsq (e05us) Global optimization of a sum of squares problem using multi-start, nonlinear constraints nag_matop_real_addsub (f01ct) Sum or difference of two real matrices, optional scaling and transposition nag_matop_complex_addsub (f01cw) Sum or difference of two complex matrices, optional scaling and transposition nag_matop_real_gen_matrix_log (f01ej) Real matrix logarithm nag_matop_real_gen_matrix_fun_std (f01ek) Exponential, sine, cosine, sinh or cosh of a real matrix (Schur–Parlett algorithm) nag_matop_real_gen_matrix_fun_usd (f01em) Function of a real matrix (using user-supplied derivatives) nag_matop_complex_gen_matrix_log (f01fj) Complex matrix logarithm nag_matop_complex_gen_matrix_fun_std (f01fk) Exponential, sine, cosine, sinh or cosh of a complex matrix (Schur–Parlett algorithm) nag_matop_complex_gen_matrix_fun_usd (f01fm) Function of a complex matrix (using user-supplied derivatives) nag_orthog_real_gram_schmidt (f05aa) Gram–Schmidt orthogonalization of n$n$ vectors of order m$m$ nag_sparse_real_gen_basic_solver (f11be) Real sparse nonsymmetric linear systems, preconditioned RGMRES, CGS, Bi-CGSTAB or TFQMR method nag_sparse_complex_gen_basic_solver (f11bs) Complex sparse non-Hermitian linear systems, preconditioned RGMRES, CGS, Bi-CGSTAB or TFQMR method nag_sparse_real_symm_basic_solver (f11ge) Real sparse symmetric linear systems, preconditioned conjugate gradient or Lanczos method or the MINRES algorithm nag_sparse_complex_herm_basic_solver (f11gs) Complex sparse Hermitian linear systems, preconditioned conjugate gradient or Lanczos nag_sparse_direct_real_gen_lu (f11me) LU$LU$ factorization of real sparse matrix nag_sparse_direct_real_gen_solve (f11mf) Solution of real sparse simultaneous linear equations (coefficient matrix already factorized) nag_sparse_direct_real_gen_refine (f11mh) Refined solution with error bounds of real system of linear equations, multiple right-hand sides nag_sparse_direct_real_gen_matmul (f11mk) Real sparse nonsymmetric matrix-matrix multiply, compressed column storage nag_sparse_real_gen_matvec (f11xa) Real sparse nonsymmetric matrix vector multiply nag_sparse_real_symm_matvec (f11xe) Real sparse symmetric matrix vector multiply nag_sparse_complex_gen_matvec (f11xn) Complex sparse non-Hermitian matrix vector multiply nag_sparse_complex_herm_matvec (f11xs) Complex sparse Hermitian matrix vector multiply nag_sparseig_real_iter (f12ab) Selected eigenvalues and, optionally, eigenvectors of a real nonsymmetric sparse eigenproblem, reverse communication nag_sparseig_real_band_solve (f12ag) Selected eigenvalues and, optionally, eigenvectors of a real nonsymmetric banded eigenproblem, driver nag_sparseig_complex_iter (f12ap) Selected eigenvalues and, optionally, eigenvectors of a complex sparse eigenproblem, reverse communication nag_sparseig_real_symm_iter (f12fb) Selected eigenvalues and, optionally, eigenvectors of a real symmetric sparse eigenproblem, reverse communication nag_stat_summary_onevar (g01at) Computes univariate summary information: mean, variance, skewness, kurtosis nag_stat_moving_average (g01wa) Computes the mean and standard deviation using a rolling window nag_correg_corrmat_nearest (g02aa) Computes the nearest correlation matrix to a real square matrix, using the method of Qi and Sun nag_correg_coeffs_pearson (g02ba) Pearson product-moment correlation coefficients, all variables, no missing values nag_correg_coeffs_zero (g02bd) Correlation-like coefficients (about zero), all variables, no missing values nag_correg_coeffs_kspearman_overwrite (g02bn) Kendall/Spearman non-parametric rank correlation coefficients, no missing values, overwriting input data nag_correg_coeffs_kspearman_miss_case_overwrite (g02bp) Kendall/Spearman non-parametric rank correlation coefficients, casewise treatment of missing values, overwriting input data nag_correg_coeffs_kspearman (g02bq) Kendall/Spearman non-parametric rank correlation coefficients, no missing values, preserving input data nag_correg_coeffs_kspearman_miss_case (g02br) Kendall/Spearman non-parametric rank correlation coefficients, casewise treatment of missing values, preserving input data nag_correg_mixeff_hier_reml (g02jd) Hierarchical mixed effects regression using Restricted Maximum Likelihood (REML) nag_correg_mixeff_hier_ml (g02je) Hierarchical mixed effects regression using Maximum Likelihood (ML) nag_mv_factor (g03ca) Computes maximum likelihood estimates of the parameters of a factor analysis model, factor loadings, communalities and residual correlations nag_mv_distance_mat (g03ea) Computes distance matrix nag_mv_cluster_hier (g03ec) Hierarchical cluster analysis nag_mv_gaussian_mixture (g03ga) Fits a Gaussian mixture model nag_rand_copula_students_t (g05rc) Generates a matrix of pseudorandom numbers from a Student's t$t$-copula nag_rand_copula_normal (g05rd) Generates a matrix of pseudorandom numbers from a Gaussian copula nag_rand_copula_clayton_bivar (g05re) Generates a matrix of pseudorandom numbers from a bivariate Clayton/Cook–Johnson copula nag_rand_copula_frank_bivar (g05rf) Generates a matrix of pseudorandom numbers from a bivariate Frank copula nag_rand_copula_plackett_bivar (g05rg) Generates a matrix of pseudorandom numbers from a bivariate Plackett copula nag_rand_copula_clayton (g05rh) Generates a matrix of pseudorandom numbers from a multivariate Clayton/Cook–Johnson copula nag_rand_copula_frank (g05rj) Generates a matrix of pseudorandom numbers from a multivariate Frank copula nag_rand_copula_gumbel (g05rk) Generates a matrix of pseudorandom numbers from a Gumbel–Hougaard copula nag_rand_multivar_students_t (g05ry) Generates a matrix of pseudorandom numbers from a multivariate Student's t$t$-distribution nag_rand_dist_uniform01 (g05sa) Generates a vector of pseudorandom numbers from a uniform distribution over (0,1]$\left(0,1\right]$ nag_rand_dist_beta (g05sb) Generates a vector of pseudorandom numbers from a beta distribution nag_rand_dist_cauchy (g05sc) Generates a vector of pseudorandom numbers from a Cauchy distribution nag_rand_dist_chisq (g05sd) Generates a vector of pseudorandom numbers from a χ2${\chi }^{2}$ distribution nag_rand_dist_dirichlet (g05se) Generates a vector of pseudorandom numbers from a Dirichlet distribution nag_rand_dist_exp (g05sf) Generates a vector of pseudorandom numbers from an exponential distribution nag_rand_dist_expmix (g05sg) Generates a vector of pseudorandom numbers from an exponential mix distribution nag_rand_dist_f (g05sh) Generates a vector of pseudorandom numbers from an F$F$-distribution nag_rand_dist_gamma (g05sj) Generates a vector of pseudorandom numbers from a gamma distribution nag_rand_dist_normal (g05sk) Generates a vector of pseudorandom numbers from a Normal distribution nag_rand_dist_logistic (g05sl) Generates a vector of pseudorandom numbers from a logistic distribution nag_rand_dist_lognormal (g05sm) Generates a vector of pseudorandom numbers from a log-normal distribution nag_rand_dist_students_t (g05sn) Generates a vector of pseudorandom numbers from a Student's t$t$-distribution nag_rand_dist_triangular (g05sp) Generates a vector of pseudorandom numbers from a triangular distribution nag_rand_dist_uniform (g05sq) Generates a vector of pseudorandom numbers from a uniform distribution over [a,b]$\left[a,b\right]$ nag_rand_dist_vonmises (g05sr) Generates a vector of pseudorandom numbers from a von Mises distribution nag_rand_dist_weibull (g05ss) Generates a vector of pseudorandom numbers from a Weibull distribution nag_rand_bb (g05xb) Generate paths for a free or non-free Wiener process using the Brownian bridge algorithm nag_rand_bb_inc (g05xd) Backs out the increments from sample paths generated by a Brownian bridge algorithm nag_rand_quasi_normal (g05yj) Generates a Normal quasi-random number sequence nag_rand_quasi_lognormal (g05yk) Generates a log-normal quasi-random number sequence nag_rand_quasi_uniform (g05ym) Generates a uniform quasi-random number sequence nag_tsa_multi_kalman_sqrt_var (g13ea) Combined measurement and time update, one iteration of Kalman filter, time-varying, square root covariance filter nag_tsa_multi_kalman_sqrt_invar (g13eb) Combined measurement and time update, one iteration of Kalman filter, time-invariant, square root covariance filter nag_tsa_inhom_iema (g13me) Computes the iterated exponential moving average for a univariate inhomogeneous time series nag_tsa_inhom_iema_all (g13mf) Computes the iterated exponential moving average for a univariate inhomogeneous time series, intermediate results are also returned nag_tsa_inhom_ma (g13mg) Computes the exponential moving average for a univariate inhomogeneous time series nag_sort_realvec_sort (m01ca) Sort a vector, real numbers nag_sort_intvec_sort (m01cb) Sort a vector, integer numbers nag_sort_charvec_sort (m01cc) Sort a vector, character data nag_specfun_opt_bsm_price (s30aa) Black–Scholes–Merton option pricing formula nag_specfun_opt_bsm_greeks (s30ab) Black–Scholes–Merton option pricing formula with Greeks nag_specfun_opt_lookback_fls_price (s30ba) Floating-strike lookback option pricing formula nag_specfun_opt_lookback_fls_greeks (s30bb) Floating-strike lookback option pricing formula with Greeks nag_specfun_opt_binary_con_price (s30ca) Binary option, cash-or-nothing pricing formula nag_specfun_opt_binary_con_greeks (s30cb) Binary option, cash-or-nothing pricing formula with Greeks nag_specfun_opt_binary_aon_price (s30cc) Binary option, asset-or-nothing pricing formula nag_specfun_opt_binary_aon_greeks (s30cd) Binary option, asset-or-nothing pricing formula with Greeks nag_specfun_opt_barrier_std_price (s30fa) Standard barrier option pricing formula nag_specfun_opt_jumpdiff_merton_price (s30ja) Jump-diffusion, Merton's model, option pricing formula nag_specfun_opt_jumpdiff_merton_greeks (s30jb) Jump-diffusion, Merton's model, option pricing formula with Greeks nag_specfun_opt_heston_price (s30na) Heston's model option pricing formula nag_specfun_opt_heston_greeks (s30nb) Heston's model option pricing formula with Greeks nag_specfun_opt_amer_bs_price (s30qc) American option, Bjerksund and Stensland pricing formula nag_specfun_opt_asian_geom_price (s30sa) Asian option, geometric continuous average rate pricing formula nag_specfun_opt_asian_geom_greeks (s30sb) Asian option, geometric continuous average rate pricing formula with Greeks

## Functions Enhanced by Calling Tuned NAG-specific Functions

These functions call one or more of the tuned NAG-specific functions as part of their core operations and may thereby exhibit improved performance and scalability. There are 165 of these routines within the Library.
 FunctionName Purpose nag_roots_sparsys_func_expert (c05qs) Solution of a sparse system of nonlinear equations using function values only (easy-to-use) nag_quad_md_mcarlo (d01gb) Multidimensional quadrature over hyper-rectangle, Monte–Carlo method nag_quad_md_numth (d01gc) Multidimensional quadrature, general product region, number-theoretic method nag_quad_md_numth_vec (d01gd) Multidimensional quadrature, general product region, number-theoretic method, variant of nag_quad_md_numth (d01gc) efficient on vector machines nag_quad_md_simplex (d01pa) Multidimensional quadrature over an n$n$-simplex nag_ode_ivp_bdf_zero_simple (d02ej) Ordinary differential equations, stiff initial value problem, backward differentiation formulae method, until function of solution is zero, intermediate output (simple driver) nag_ode_ivp_stiff_exp_fulljac (d02nb) Explicit ordinary differential equations, stiff initial value problem, full Jacobian (comprehensive) nag_ode_ivp_stiff_exp_bandjac (d02nc) Explicit ordinary differential equations, stiff initial value problem, banded Jacobian (comprehensive) nag_ode_ivp_stiff_exp_sparjac (d02nd) Explicit ordinary differential equations, stiff initial value problem, sparse Jacobian (comprehensive) nag_ode_ivp_stiff_imp_fulljac (d02ng) Implicit/algebraic ordinary differential equations, stiff initial value problem, full Jacobian (comprehensive) nag_ode_ivp_stiff_imp_bandjac (d02nh) Implicit/algebraic ordinary differential equations, stiff initial value problem, banded Jacobian (comprehensive) nag_ode_ivp_stiff_imp_sparjac (d02nj) Implicit/algebraic ordinary differential equations, stiff initial value problem, sparse Jacobian (comprehensive) nag_ode_ivp_stiff_exp_revcom (d02nm) Explicit ordinary differential equations, stiff initial value problem (reverse communication, comprehensive) nag_ode_ivp_stiff_imp_revcom (d02nn) Implicit/algebraic ordinary differential equations, stiff initial value problem (reverse communication, comprehensive) nag_ode_bvp_ps_lin_coeffs (d02ua) Coefficients of Chebyshev interpolating polynomial from function values on Chebyshev grid nag_ode_bvp_ps_lin_cgl_vals (d02ub) Function or low-order-derivative values on Chebyshev grid from coefficients of Chebyshev interpolating polynomial nag_pde_1d_parab_fd (d03pc) General system of parabolic PDEs, method of lines, finite differences, one space variable nag_pde_1d_parab_coll (d03pd) General system of parabolic PDEs, method of lines, Chebyshev C0${C}^{0}$ collocation, one space variable nag_pde_1d_parab_keller (d03pe) General system of first-order PDEs, method of lines, Keller box discretization, one space variable nag_pde_1d_parab_convdiff (d03pf) 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 nag_pde_1d_parab_dae_fd (d03ph) General system of parabolic PDEs, coupled DAEs, method of lines, finite differences, one space variable nag_pde_1d_parab_dae_coll (d03pj) General system of parabolic PDEs, coupled DAEs, method of lines, Chebyshev C0${C}^{0}$ collocation, one space variable nag_pde_1d_parab_dae_keller (d03pk) General system of first-order PDEs, coupled DAEs, method of lines, Keller box discretization, one space variable nag_pde_1d_parab_convdiff_dae (d03pl) 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 nag_pde_1d_parab_remesh_fd (d03pp) General system of parabolic PDEs, coupled DAEs, method of lines, finite differences, remeshing, one space variable nag_pde_1d_parab_remesh_keller (d03pr) General system of first-order PDEs, coupled DAEs, method of lines, Keller box discretization, remeshing, one space variable nag_pde_1d_parab_convdiff_remesh (d03ps) General system of convection-diffusion PDEs, coupled DAEs, method of lines, upwind scheme, remeshing, one space variable nag_inteq_fredholm2_split (d05aa) Linear nonsingular Fredholm integral equation, second kind, split kernel nag_inteq_fredholm2_smooth (d05ab) Linear nonsingular Fredholm integral equation, second kind, smooth kernel nag_mesh_2d_sparsity (d06cb) Generates a sparsity pattern of a Finite Element matrix associated with a given mesh nag_mesh_2d_renumber (d06cc) Renumbers a given mesh using Gibbs method nag_fit_pade_app (e02ra) Padé approximants nag_opt_lsq_uncon_mod_func_comp (e04fc) Unconstrained minimum of a sum of squares, combined Gauss–Newton and modified Newton algorithm using function values only (comprehensive) nag_opt_lsq_uncon_mod_func_easy (e04fy) Unconstrained minimum of a sum of squares, combined Gauss–Newton and modified Newton algorithm using function values only (easy-to-use) nag_opt_lsq_uncon_quasi_deriv_comp (e04gb) Unconstrained minimum of a sum of squares, combined Gauss–Newton and quasi-Newton algorithm using first derivatives (comprehensive) nag_opt_lsq_uncon_mod_deriv_comp (e04gd) Unconstrained minimum of a sum of squares, combined Gauss–Newton and modified Newton algorithm using first derivatives (comprehensive) nag_opt_lsq_uncon_quasi_deriv_easy (e04gy) Unconstrained minimum of a sum of squares, combined Gauss–Newton and quasi-Newton algorithm, using first derivatives (easy-to-use) nag_opt_lsq_uncon_mod_deriv_easy (e04gz) Unconstrained minimum of a sum of squares, combined Gauss–Newton and modified Newton algorithm using first derivatives (easy-to-use) nag_opt_lsq_uncon_mod_deriv2_comp (e04he) Unconstrained minimum of a sum of squares, combined Gauss–Newton and modified Newton algorithm, using second derivatives (comprehensive) nag_opt_lsq_uncon_mod_deriv2_easy (e04hy) Unconstrained minimum of a sum of squares, combined Gauss–Newton and modified Newton algorithm, using second derivatives (easy-to-use) nag_opt_lsq_lincon_solve (e04nc) Convex QP problem or linearly-constrained linear least squares problem (dense) nag_opt_nlp1_solve (e04uc) Minimum, function of several variables, sequential QP method, nonlinear constraints, using function values and optionally first derivatives (comprehensive) nag_opt_nlp1_rcomm (e04uf) Minimum, function of several variables, sequential QP method, nonlinear constraints, using function values and optionally first derivatives (reverse communication, comprehensive) nag_opt_nlp1_sparse_solve (e04ug) NLP problem (sparse) nag_opt_lsq_gencon_deriv (e04us) Minimum of a sum of squares, nonlinear constraints, sequential QP method, using function values and optionally first derivatives (comprehensive) nag_opt_lsq_uncon_covariance (e04yc) Covariance matrix for nonlinear least squares problem (unconstrained) nag_matop_real_symm_posdef_inv (f01ab) Inverse of real symmetric positive definite matrix using iterative refinement nag_matop_real_gen_matrix_fun_num (f01el) Function of a real matrix (using numerical differentiation) nag_matop_complex_gen_matrix_fun_num (f01fl) Function of a complex matrix (using numerical differentiation) nag_matop_real_gen_matrix_actexp (f01ga) Action of a real matrix exponential on a real matrix nag_matop_real_gen_matrix_actexp_rcomm (f01gb) Action of a real matrix exponential on a real matrix (reverse communication) nag_matop_complex_gen_matrix_actexp (f01ha) Action of a complex matrix exponential on a complex matrix nag_matop_complex_gen_matrix_actexp_rcomm (f01hb) Action of a complex matrix exponential on a complex matrix (reverse communication) nag_matop_real_gen_matrix_cond_std (f01ja) Condition number for the exponential, logarithm, sine, cosine, sinh or cosh of a real matrix nag_matop_real_gen_matrix_cond_num (f01jb) Condition number for a function of a real matrix (using numerical differentiation) nag_matop_real_gen_matrix_cond_usd (f01jc) Condition number for a function of a real matrix (using user-supplied derivatives) nag_matop_complex_gen_matrix_cond_std (f01ka) Condition number for the exponential, logarithm, sine, cosine, sinh or cosh of a complex matrix nag_matop_complex_gen_matrix_cond_num (f01kb) Condition number for a function of a complex matrix (using numerical differentiation) nag_matop_complex_gen_matrix_cond_usd (f01kc) Condition number for a function of a complex matrix (using user-supplied derivatives) nag_eigen_real_gen_sparse_arnoldi (f02ek) Selected eigenvalues and eigenvectors of a real sparse general matrix nag_eigen_real_symm_sparse_eigsys (f02fj) Selected eigenvalues and eigenvectors of sparse symmetric eigenproblem (Black Box) nag_eigen_real_gen_qu_svd (f02wd) QR$QR$ factorization, possibly followed by SVD nag_eigen_real_gen_partialsvd (f02wg) Computes leading terms in the singular value decomposition of a real general matrix; also computes corresponding left and right singular vectors nag_eigen_real_triang_svd (f02wu) SVD of real upper triangular matrix (Black Box) nag_eigen_complex_triang_svd (f02xu) SVD of complex upper triangular matrix (Black Box) nag_linsys_real_posdef_solve_ref (f04ab) Solution of real symmetric positive definite simultaneous linear equations with multiple right-hand sides using iterative refinement (Black Box) nag_linsys_real_square_solve_ref (f04ae) Solution of real simultaneous linear equations with multiple right-hand sides using iterative refinement (Black Box) nag_linsys_real_posdef_solve_1rhs (f04as) Solution of real symmetric positive definite simultaneous linear equations, one right-hand side using iterative refinement (Black Box) nag_linsys_real_square_solve_1rhs (f04at) Solution of real simultaneous linear equations, one right-hand side using iterative refinement (Black Box) nag_linsys_real_gen_solve (f04jg) Least squares (if rank = n$\text{}=n$) or minimal least squares (if rank < n$\text{}) solution of m$m$ real equations in n$n$ unknowns, m ≥ n$m\ge n$ nag_sparse_real_gen_solve_ilu (f11dc) Solution of real sparse nonsymmetric linear system, RGMRES, CGS, Bi-CGSTAB or TFQMR method, preconditioner computed by nag_sparse_real_gen_precon_ilu (f11da) nag_sparse_real_gen_solve_jacssor (f11de) Solution of real sparse nonsymmetric linear system, RGMRES, CGS, Bi-CGSTAB, or TFQMR method, Jacobi or SSOR preconditioner (Black Box) nag_sparse_real_gen_solve_bdilu (f11dg) Solution of real sparse nonsymmetric linear system, RGMRES, CGS, Bi-CGSTAB or TFQMR method, incomplete LU$LU$ block diagonal preconditioner computed by nag_sparse_real_gen_precon_bdilu (f11df) nag_sparse_real_gen_precon_jacobi (f11dk) Real sparse nonsymmetric linear systems, line Jacobi preconditioner nag_sparse_complex_gen_solve_ilu (f11dq) Solution of complex sparse non-Hermitian linear system, RGMRES, CGS, Bi-CGSTAB or TFQMR method, preconditioner computed by nag_sparse_complex_gen_precon_ilu (f11dn) (Black Box) nag_sparse_complex_gen_solve_jacssor (f11ds) Solution of complex sparse non-Hermitian linear system, RGMRES, CGS, Bi-CGSTAB or TFQMR method, Jacobi or SSOR preconditioner Black Box nag_sparse_complex_gen_solve_bdilu (f11du) Solution of complex sparse nonsymmetric linear system, RGMRES, CGS, Bi-CGSTAB or TFQMR method, incomplete LU$LU$ block diagonal preconditioner computed by nag_sparse_complex_gen_precon_bdilu (f11dt) nag_sparse_complex_gen_precon_jacobi (f11dx) Complex sparse nonsymmetric linear systems, line Jacobi preconditioner nag_sparse_real_symm_solve_ichol (f11jc) Solution of real sparse symmetric linear system, conjugate gradient/Lanczos method, preconditioner computed by nag_sparse_real_symm_precon_ichol (f11ja) (Black Box) nag_sparse_real_symm_solve_jacssor (f11je) Solution of real sparse symmetric linear system, conjugate gradient/Lanczos method, Jacobi or SSOR preconditioner (Black Box) nag_sparse_complex_herm_solve_ilu (f11jq) Solution of complex sparse Hermitian linear system, conjugate gradient/Lanczos method, preconditioner computed by nag_sparse_complex_herm_precon_ilu (f11jn) (Black Box) nag_sparse_complex_herm_solve_jacssor (f11js) Solution of complex sparse Hermitian linear system, conjugate gradient/Lanczos method, Jacobi or SSOR preconditioner (Black Box) nag_sparse_direct_real_gen_setup (f11md) Real sparse nonsymmetric linear systems, setup for nag_sparse_direct_real_gen_lu (f11me) nag_sparseig_complex_band_solve (f12au) Selected eigenvalues and, optionally, eigenvectors of complex non-Hermitian banded eigenproblem, driver nag_sparseig_real_symm_band_solve (f12fg) Selected eigenvalues and, optionally, eigenvectors of a real symmetric banded eigenproblem, driver nag_stat_plot_scatter_2var (g01ag) Lineprinter scatterplot of two variables nag_stat_plot_scatter_normal (g01ah) Lineprinter scatterplot of one variable against Normal scores nag_stat_quantiles_stream_fixed (g01an) Calculates approximate quantiles from a data stream of known size nag_stat_quantiles_stream_arbitrary (g01ap) Calculates approximate quantiles from a data stream of unknown size nag_stat_plot_stem_leaf (g01ar) Constructs a stem and leaf plot nag_stat_prob_studentized_range (g01em) Computes probability for the Studentized range statistic nag_stat_prob_multi_normal (g01hb) Computes probabilities for the multivariate Normal distribution nag_stat_prob_chisq_lincomb (g01jd) Computes lower tail probability for a linear combination of (central) χ2${\chi }^{2}$ variables nag_correg_corrmat_nearest_bounded (g02ab) Computes the nearest correlation matrix to a real square matrix, augmented nag_correg_corrmat_nearest (g02aa) to incorporate weights and bounds nag_correg_corrmat_nearest_kfactor (g02ae) Computes the nearest correlation matrix with k$k$-factor structure to a real square matrix nag_nearest_correlation_h_weight (g02aj) Computes the nearest correlation matrix to a real square matrix, using element-wise weighting nag_correg_linregm_coeffs_const (g02cg) Multiple linear regression, from correlation coefficients, with constant term nag_correg_linregm_coeffs_noconst (g02ch) Multiple linear regression, from correlation-like coefficients, without constant term nag_correg_linregm_fit (g02da) Fits a general (multiple) linear regression model nag_correg_linregm_update (g02dd) Estimates of linear parameters and general linear regression model from updated model nag_correg_linregm_var_add (g02de) Add a new independent variable to a general linear regression model nag_correg_linregm_fit_newvar (g02dg) Fits a general linear regression model to new dependent variable nag_correg_linregm_constrain (g02dk) Estimates and standard errors of parameters of a general linear regression model for given constraints nag_correg_linregm_fit_onestep (g02ee) Fits a linear regression model by forward selection nag_correg_glm_normal (g02ga) Fits a generalized linear model with Normal errors nag_correg_glm_binomial (g02gb) Fits a generalized linear model with binomial errors nag_correg_glm_poisson (g02gc) Fits a generalized linear model with Poisson errors nag_correg_glm_gamma (g02gd) Fits a generalized linear model with gamma errors nag_correg_glm_constrain (g02gk) Estimates and standard errors of parameters of a general linear model for given constraints nag_correg_robustm (g02ha) Robust regression, standard M$M$-estimates nag_correg_robustm_user (g02hd) Robust regression, compute regression with user-supplied functions and weights nag_correg_robustm_user_varmat (g02hf) Robust regression, variance-covariance matrix following nag_correg_robustm_user (g02hd) nag_correg_robustm_corr_huber (g02hk) Calculates a robust estimation of a correlation matrix, Huber's weight function nag_correg_mixeff_reml (g02ja) Linear mixed effects regression using Restricted Maximum Likelihood (REML) nag_correg_mixeff_ml (g02jb) Linear mixed effects regression using Maximum Likelihood (ML) nag_correg_ridge_opt (g02ka) Ridge regression, optimizing a ridge regression parameter nag_correg_ridge (g02kb) Ridge regression using a number of supplied ridge regression parameters nag_mv_canon_var (g03ac) Performs canonical variate analysis nag_mv_canon_corr (g03ad) Performs canonical correlation analysis nag_anova_dummyvars (g04ea) Computes orthogonal polynomials or dummy variables for factor/classification variable nag_rand_times_garch_asym1 (g05pd) Generates a realisation of a time series from a GARCH process with asymmetry of the form (εt − 1 + γ)2${\left({\epsilon }_{t-1}+\gamma \right)}^{2}$ nag_rand_times_garch_asym2 (g05pe) Generates a realisation of a time series from a GARCH process with asymmetry of the form (|εt − 1| + γεt − 1)2${\left(|{\epsilon }_{t-1}|+\gamma {\epsilon }_{t-1}\right)}^{2}$ nag_rand_times_garch_gjr (g05pf) Generates a realisation of a time series from an asymmetric Glosten, Jagannathan and Runkle (GJR) GARCH process nag_rand_times_garch_exp (g05pg) Generates a realisation of a time series from an exponential GARCH (EGARCH) process nag_rand_times_mv_varma (g05pj) Generates a realisation of a multivariate time series from a VARMA model nag_rand_matrix_corr (g05py) Generates a random correlation matrix nag_rand_multivar_normal (g05rz) Generates a matrix of pseudorandom numbers from a multivariate Normal distribution nag_rand_field_1d_generate (g05zp) Generates realisations of a one-dimensional random field nag_rand_field_2d_user_setup (g05zq) Setup for simulating two-dimensional random fields, user-defined variogram nag_rand_field_2d_predef_setup (g05zr) Setup for simulating two-dimensional random fields, preset variogram nag_rand_field_2d_generate (g05zs) Generates realisations of a two-dimensional random field nag_rand_field_fracbm_generate (g05zt) Generates realisations of fractional Brownian motion nag_univar_estim_weibull (g07be) Computes maximum likelihood estimates for parameters of the Weibull distribution nag_univar_estim_genpareto (g07bf) Estimates parameter values of the generalized Pareto distribution nag_univar_robust_1var_median (g07da) Robust estimation, median, median absolute deviation, robust standard deviation nag_univar_robust_1var_mestim (g07db) Robust estimation, M$M$-estimates for location and scale parameters, standard weight functions nag_univar_robust_1var_mestim_wgt (g07dc) Robust estimation, M$M$-estimates for location and scale parameters, user-defined weight functions nag_univar_robust_1var_trimmed (g07dd) Computes a trimmed and winsorized mean of a single sample with estimates of their variance nag_univar_robust_1var_ci (g07ea) Robust confidence intervals, one-sample nag_univar_robust_2var_ci (g07eb) Robust confidence intervals, two-sample nag_nonpar_test_wilcoxon (g08ag) Performs the Wilcoxon one-sample (matched pairs) signed rank test nag_nonpar_prob_mwu_ties (g08ak) Computes the exact probabilities for the Mann–Whitney U$U$ statistic, ties in pooled sample nag_nonpar_test_ks_1sample (g08cb) Performs the one-sample Kolmogorov–Smirnov test for standard distributions nag_nonpar_test_ks_1sample_user (g08cc) Performs the one-sample Kolmogorov–Smirnov test for a user-supplied distribution nag_nonpar_test_ks_2sample (g08cd) Performs the two-sample Kolmogorov–Smirnov test nag_nonpar_rank_regsn (g08ra) Regression using ranks, uncensored data nag_nonpar_rank_regsn_censored (g08rb) Regression using ranks, right-censored data nag_contab_tabulate_percentile (g11bb) Computes multiway table from set of classification factors using given percentile/quantile nag_contab_tabulate_margin (g11bc) Computes marginal tables for multiway table computed by nag_contab_tabulate_stat (g11ba) or nag_contab_tabulate_percentile (g11bb) nag_contab_binary (g11sa) Contingency table, latent variable model for binary data nag_surviv_logrank (g12ab) Computes rank statistics for comparing survival curves nag_tsa_uni_arima_prelim (g13ad) Univariate time series, preliminary estimation, seasonal ARIMA model nag_tsa_uni_arima_estim (g13ae) Univariate time series, estimation, seasonal ARIMA model (comprehensive) nag_tsa_uni_arima_estim_easy (g13af) Univariate time series, estimation, seasonal ARIMA model (easy-to-use) nag_tsa_uni_arima_forcecast (g13aj) Univariate time series, state set and forecasts, from fully specified seasonal ARIMA model nag_tsa_multi_xcorr (g13bc) Multivariate time series, cross-correlations nag_tsa_multi_inputmod_estim (g13be) Multivariate time series, estimation of multi-input model nag_tsa_multi_inputmod_forecast (g13bj) Multivariate time series, state set and forecasts from fully specified multi-input model nag_tsa_multi_autocorr_part (g13db) Multivariate time series, multiple squared partial autocorrelations nag_tsa_multi_varma_estimate (g13dd) Multivariate time series, estimation of VARMA model nag_tsa_multi_corrmat_partlag (g13dn) Multivariate time series, sample partial lag correlation matrices, χ2${\chi }^{2}$ statistics and significance levels nag_tsa_uni_garch_asym1_estim (g13fa) Univariate time series, parameter estimation for either a symmetric GARCH process or a GARCH process with asymmetry of the form (εt − 1 + γ)2${\left({\epsilon }_{t-1}+\gamma \right)}^{2}$ nag_tsa_uni_garch_asym2_estim (g13fc) Univariate time series, parameter estimation for a GARCH process with asymmetry of the form (|εt − 1| + γεt − 1)2${\left(|{\epsilon }_{t-1}|+\gamma {\epsilon }_{t-1}\right)}^{2}$ nag_tsa_uni_garch_gjr_estim (g13fe) Univariate time series, parameter estimation for an asymmetric Glosten, Jagannathan and Runkle (GJR) GARCH process nag_tsa_uni_garch_exp_estim (g13fg) Univariate time series, parameter estimation for an exponential GARCH (EGARCH) process

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NAG Toolbox

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