\begin{page}{manpageXXonline}{NAG On-line Documentation: online} \beginscroll \begin{verbatim} DOC INTRO(3NAG) Foundation Library (12/10/92) DOC INTRO(3NAG) Introduction to NAG On-Line Documentation The on-line documentation for the NAG Foundation Library has been generated automatically from the same base material used to create the printed Reference Manual. To make the documentation readable on the widest range of machines, only the basic set of ascii characters has been used. Certain mathematical symbols have been constructed using plain ascii characters: integral signs / | / summation signs -- > -- square root signs ---- / \/ Large brackets are constructed using vertical stacks of the equivalent ascii character: ( ) [ ] { } | ( ) [ ] { } | ( ) [ ] { } | Fractions are represented as: a --- x+1 Greek letters are represented by their names enclosed in round brackets: (alpha) (beta) (gamma) ..... (Alpha) (Beta) (Gamma) ..... Some characters are accented using: ^ ~ X X X Other mathematical symbols are represented as follows: * times <=> left-right arrow <- left arrow ~ similar to ~= similar or equal to == equivalent to >= greater than or equal to <= less than or equal to >> much greater than << much less than >~ greater than or similar to /= not equal to dd partial derivative +- plus or minus (nabla) Nabla \end{verbatim} \endscroll \end{page} \begin{page}{manpageXXsummary}{NAG On-line Documentation: summary} \beginscroll \begin{verbatim} SUMMARY(3NAG) Foundation Library (12/10/92) SUMMARY(3NAG) Introduction List of Routines List of Routines The NAG Foundation Library contains three categories of routines which can be called by users. They are listed separately in the three sections below. Fully Documented Routines 254 routines, for each of which an individual routine document is provided. These are regarded as the primary contents of the Foundation Library. Fundamental Support Routines 83 comparatively simple routines which are documented in compact form in the relevant Chapter Introductions (F06, X01, X02). Routines from the NAG Fortran Library An additional 167 routines from the NAG Fortran Library, which are used as auxiliaries in the Foundation Library. They are not documented in this publication, but can be called if you are already familiar with their use in the Fortran Library. Only their names are given here. Note: all the routines in the above categories have names ending in 'F'. Occasionally this publication may refer to routines whose names end in some other letter (e.g. 'Z', 'Y', 'X'). These are auxiliary routines whose names may be passed as parameters to a Foundation Library routine; you only need to know their names, not how to call them directly. Fully Documented Routines The Foundation Library contains 254 user-callable routines, for each of which an individual routine document is provided, in the following chapters: C02 -- Zeros of Polynomials C02AFF All zeros of complex polynomial, modified Laguerre method C02AGF All zeros of real polynomial, modified Laguerre method C05 -- Roots of One or More Transcendental Equations C05ADF Zero of continuous function in given interval, Bus and Dekker algorithm C05NBF Solution of system of nonlinear equations using function values only C05PBF Solution of system of nonlinear equations using 1st derivatives C05ZAF Check user's routine for calculating 1st derivatives C06 -- Summation of Series C06EAF Single 1-D real discrete Fourier transform, no extra workspace C06EBF Single 1-D Hermitian discrete Fourier transform, no extra workspace C06ECF Single 1-D complex discrete Fourier transform, no extra workspace C06EKF Circular convolution or correlation of two real vectors, no extra workspace C06FPF Multiple 1-D real discrete Fourier transforms C06FQF Multiple 1-D Hermitian discrete Fourier transforms C06FRF Multiple 1-D complex discrete Fourier transforms C06FUF 2-D complex discrete Fourier transform C06GBF Complex conjugate of Hermitian sequence C06GCF Complex conjugate of complex sequence C06GQF Complex conjugate of multiple Hermitian sequences C06GSF Convert Hermitian sequences to general complex sequences D01 -- Quadrature D01AJF 1-D quadrature, adaptive, finite interval, strategy due to Piessens and de Doncker, allowing for badly-behaved integrands D01AKF 1-D quadrature, adaptive, finite interval, method suitable for oscillating functions D01ALF 1-D quadrature, adaptive, finite interval, allowing for singularities at user-specified break-points D01AMF 1-D quadrature, adaptive, infinite or semi-infinite interval D01ANF 1-D quadrature, adaptive, finite interval, weight function cos((omega)x) or sin((omega)x) D01APF 1-D quadrature, adaptive, finite interval, weight function with end-point singularities of algebraico- logarithmic type D01AQF 1-D quadrature, adaptive, finite interval, weight function 1/(x-c), Cauchy principal value (Hilbert transform) D01ASF 1-D quadrature, adaptive, semi-infinite interval, weight function cos((omega)x) or sin((omega)x) D01BBF Weights and abscissae for Gaussian quadrature rules D01FCF Multi-dimensional adaptive quadrature over hyper- rectangle D01GAF 1-D quadrature, integration of function defined by data values, Gill-Miller method D01GBF Multi-dimensional quadrature over hyper-rectangle, Monte Carlo method D02 -- Ordinary Differential Equations D02BBF ODEs, IVP, Runge-Kutta-Merson method, over a range, intermediate output D02BHF ODEs, IVP, Runge-Kutta-Merson method, until function of solution is zero D02CJF ODEs, IVP, Adams method, until function of solution is zero, intermediate output D02EJF ODEs, stiff IVP, BDF method, until function of solution is zero, intermediate output D02GAF ODEs, boundary value problem, finite difference technique with deferred correction, simple nonlinear problem D02GBF ODEs, boundary value problem, finite difference technique with deferred correction, general linear problem D02KEF 2nd order Sturm-Liouville problem, regular/singular system, finite/infinite range, eigenvalue and eigenfunction, user-specified break-points D02RAF ODEs, general nonlinear boundary value problem, finite difference technique with deferred correction, continuation facility D03 -- Partial Differential Equations D03EDF Elliptic PDE, solution of finite difference equations by a multigrid technique D03EEF Discretize a 2nd order elliptic PDE on a rectangle D03FAF Elliptic PDE, Helmholtz equation, 3-D Cartesian co- ordinates E01 -- Interpolation E01BAF Interpolating functions, cubic spline interpolant, one variable E01BEF Interpolating functions, monotonicity-preserving, piecewise cubic Hermite, one variable E01BFF Interpolated values, interpolant computed by E01BEF, function only, one variable E01BGF Interpolated values, interpolant computed by E01BEF, function and 1st derivative, one variable E01BHF Interpolated values, interpolant computed by E01BEF, definite integral, one variable E01DAF Interpolating functions, fitting bicubic spline, data on rectangular grid E01SAF Interpolating functions, method of Renka and Cline, two variables E01SBF Interpolated values, evaluate interpolant computed by E01SAF, two variables E01SEF Interpolating functions, modified Shepard's method, two variables E01SFF Interpolated values, evaluate interpolant computed by E01SEF, two variables E02 -- Curve and Surface Fitting E02ADF Least-squares curve fit, by polynomials, arbitrary data points E02AEF Evaluation of fitted polynomial in one variable from Chebyshev series form (simplified parameter list) E02AGF Least-squares polynomial fit, values and derivatives may be constrained, arbitrary data points, E02AHF Derivative of fitted polynomial in Chebyshev series form E02AJF Integral of fitted polynomial in Chebyshev series form E02AKF Evaluation of fitted polynomial in one variable, from Chebyshev series form E02BAF Least-squares curve cubic spline fit (including interpolation) E02BBF Evaluation of fitted cubic spline, function only E02BCF Evaluation of fitted cubic spline, function and derivatives E02BDF Evaluation of fitted cubic spline, definite integral E02BEF Least-squares cubic spline curve fit, automatic knot placement E02DAF Least-squares surface fit, bicubic splines E02DCF Least-squares surface fit by bicubic splines with automatic knot placement, data on rectangular grid E02DDF Least-squares surface fit by bicubic splines with automatic knot placement, scattered data E02DEF Evaluation of a fitted bicubic spline at a vector of points E02DFF Evaluation of a fitted bicubic spline at a mesh of points E02GAF L -approximation by general linear function 1 E02ZAF Sort 2-D data into panels for fitting bicubic splines E04 -- Minimizing or Maximizing a Function E04DGF Unconstrained minimum, pre-conditioned conjugate gradient algorithm, function of several variables using 1st derivatives E04DJF Read optional parameter values for E04DGF from external file E04DKF Supply optional parameter values to E04DGF E04FDF Unconstrained minimum of a sum of squares, combined Gauss-Newton and modified Newton algorithm using function values only E04GCF Unconstrained minimum of a sum of squares, combined Gauss-Newton and quasi-Newton algorithm, using 1st derivatives E04JAF Minimum, function of several variables, quasi-Newton algorithm, simple bounds, using function values only E04MBF Linear programming problem E04NAF Quadratic programming problem E04UCF Minimum, function of several variables, sequential QP method, nonlinear constraints, using function values and optionally 1st derivatives E04UDF Read optional parameter values for E04UCF from external file E04UEF Supply optional parameter values to E04UCF E04YCF Covariance matrix for nonlinear least-squares problem F01 -- Matrix Factorizations F01BRF LU factorization of real sparse matrix F01BSF LU factorization of real sparse matrix with known sparsity pattern T F01MAF LL factorization of real sparse symmetric positive- definite matrix T F01MCF LDL factorization of real symmetric positive-definite variable-bandwidth matrix F01QCF QR factorization of real m by n matrix (m>=n) T F01QDF Operations with orthogonal matrices, compute QB or Q B after factorization by F01QCF F01QEF Operations with orthogonal matrices, form columns of Q after factorization by F01QCF F01RCF QR factorization of complex m by n matrix (m>=n) H F01RDF Operations with unitary matrices, compute QB or Q B after factorization by F01RCF F01REF Operations with unitary matrices, form columns of Q after factorization by F01RCF F02 -- Eigenvalues and Eigenvectors F02AAF All eigenvalues of real symmetric matrix F02ABF All eigenvalues and eigenvectors of real symmetric matrix F02ADF All eigenvalues of generalized real symmetric-definite eigenproblem F02AEF All eigenvalues and eigenvectors of generalized real symmetric-definite eigenproblem F02AFF All eigenvalues of real matrix F02AGF All eigenvalues and eigenvectors of real matrix F02AJF All eigenvalues of complex matrix F02AKF All eigenvalues and eigenvectors of complex matrix F02AWF All eigenvalues of complex Hermitian matrix F02AXF All eigenvalues and eigenvectors of complex Hermitian matrix F02BBF Selected eigenvalues and eigenvectors of real symmetric matrix F02BJF All eigenvalues and optionally eigenvectors of generalized eigenproblem by QZ algorithm, real matrices F02FJF Selected eigenvalues and eigenvectors of sparse symmetric eigenproblem F02WEF SVD of real matrix F02XEF SVD of complex matrix F04 -- Simultaneous Linear Equations F04ADF Approximate solution of complex simultaneous linear equations with multiple right-hand sides F04ARF Approximate solution of real simultaneous linear equations, one right-hand side F04ASF Accurate solution of real symmetric positive-definite simultaneous linear equations, one right-hand side F04ATF Accurate solution of real simultaneous linear equations, one right-hand side F04AXF Approximate solution of real sparse simultaneous linear equations (coefficient matrix already factorized by F01BRF or F01BSF) F04FAF Approximate solution of real symmetric positive-definite tridiagonal simultaneous linear equations, one right-hand side F04JGF Least-squares (if rank = n) or minimal least-squares (if rank =n F04MAF Real sparse symmetric positive-definite simultaneous linear equations (coefficient matrix already factorized) F04MBF Real sparse symmetric simultaneous linear equations F04MCF Approximate solution of real symmetric positive-definite variable-bandwidth simultaneous linear equations (coefficient matrix already factorized) F04QAF Sparse linear least-squares problem, m real equations in n unknowns F07 -- Linear Equations (LAPACK) F07ADF (DGETRF) LU factorization of real m by n matrix F07AEF (DGETRS) Solution of real system of linear equations, multiple right-hand sides, matrix already factorized by F07ADF F07FDF (DPOTRF) Cholesky factorization of real symmetric positive-definite matrix F07FEF (DPOTRS) Solution of real symmetric positive-definite system of linear equations, multiple right-hand sides, matrix already factorized by F07FDF G01 -- Simple Calculations on Statistical Data G01AAF Mean, variance, skewness, kurtosis etc, one variable, from raw data G01ADF Mean, variance, skewness, kurtosis etc, one variable, from frequency table G01AEF Frequency table from raw data G01AFF Two-way contingency table analysis, with (chi) /Fisher's exact test G01ALF Computes a five-point summary (median, hinges and extremes) G01ARF Constructs a stem and leaf plot G01EAF Computes probabilities for the standard Normal distribution G01EBF Computes probabilities for Student's t-distribution 2 G01ECF Computes probabilities for (chi) distribution G01EDF Computes probabilities for F-distribution G01EEF Computes upper and lower tail probabilities and probability density function for the beta distribution G01EFF Computes probabilities for the gamma distribution G01FAF Computes deviates for the standard Normal distribution G01FBF Computes deviates for Student's t-distribution 2 G01FCF Computes deviates for the (chi) distribution G01FDF Computes deviates for the F-distribution G01FEF Computes deviates for the beta distribution G01FFF Computes deviates for the gamma distribution G01HAF Computes probabilities for the bivariate Normal distribution G02 -- Correlation and Regression Analysis G02BNF Kendall/Spearman non-parametric rank correlation coefficients, no missing values, overwriting input data G02BQF Kendall/Spearman non-parametric rank correlation coefficients, no missing values, preserving input data G02BXF Computes (optionally weighted) correlation and covariance matrices G02CAF Simple linear regression with constant term, no missing values G02DAF Fits a general (multiple) linear regression model G02DGF Fits a general linear regression model for new dependent variable G02DNF Computes estimable function of a general linear regression model and its standard error G02FAF Calculates standardized residuals and influence statistics G02GBF Fits a generalized linear model with binomial errors G02GCF Fits a generalized linear model with Poisson errors G03 -- Multivariate Methods G03AAF Performs principal component analysis G03ADF Performs canonical correlation analysis G03BAF Computes orthogonal rotations for loading matrix, generalized orthomax criterion G05 -- Random Number Generators G05CAF Pseudo-random double precision numbers, uniform distribution over (0,1) G05CBF Initialise random number generating routines to give repeatable sequence G05CCF Initialise random number generating routines to give non- repeatable sequence G05CFF Save state of random number generating routines G05CGF Restore state of random number generating routines G05DDF Pseudo-random double precision numbers, Normal distribution G05DFF Pseudo-random double precision numbers, Cauchy distribution G05DPF Pseudo-random double precision numbers, Weibull distribution G05DYF Pseudo-random integer from uniform distribution G05DZF Pseudo-random logical (boolean) value G05EAF Set up reference vector for multivariate Normal distribution G05ECF Set up reference vector for generating pseudo-random integers, Poisson distribution G05EDF Set up reference vector for generating pseudo-random integers, binomial distribution G05EHF Pseudo-random permutation of an integer vector G05EJF Pseudo-random sample from an integer vector G05EXF Set up reference vector from supplied cumulative distribution function or probability distribution function G05EYF Pseudo-random integer from reference vector G05EZF Pseudo-random multivariate Normal vector from reference vector G05FAF Generates a vector of pseudo-random numbers from a uniform distribution G05FBF Generates a vector of pseudo-random numbers from a (negative) exponential distribution G05FDF Generates a vector of pseudo-random numbers from a Normal distribution G05FEF Generates a vector of pseudo-random numbers from a beta distribution G05FFF Generates a vector of pseudo-random numbers from a gamma distribution G05HDF Generates a realisation of a multivariate time series from a VARMA model G08 -- Nonparameteric Statistics G08AAF Sign test on two paired samples G08ACF Median test on two samples of unequal size G08AEF Friedman two-way analysis of variance on k matched samples G08AFF Kruskal-Wallis one-way analysis of variance on k samples of unequal size G08AGF Performs the Wilcoxon one sample (matched pairs) signed rank test G08AHF Performs the Mann-Whitney U test on two independent samples G08AJF Computes the exact probabilities for the Mann-Whitney U statistic, no ties in pooled sample G08AKF Computes the exact probabilities for the Mann-Whitney U statistic, ties in pooled sample 2 G08CGF Performs the (chi) goodness of fit test, for standard continuous distributions G13 -- Time Series Analysis G13AAF Univariate time series, seasonal and non-seasonal differencing G13ABF Univariate time series, sample autocorrelation function G13ACF Univariate time series, partial autocorrelations from autocorrelations G13ADF Univariate time series, preliminary estimation, seasonal ARIMA model G13AFF Univariate time series, estimation, seasonal ARIMA model G13AGF Univariate time series, update state set for forecasting G13AHF Univariate time series, forecasting from state set G13AJF Univariate time series, state set and forecasts, from fully specified seasonal ARIMA model G13ASF Univariate time series, diagnostic checking of residuals, following G13AFF G13BAF Multivariate time series, filtering (pre-whitening) by an ARIMA model G13BCF Multivariate time series, cross correlations 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 G13CBF Univariate time series, smoothed sample spectrum using spectral smoothing by the trapezium frequency (Daniell) window G13CDF Multivariate time series, smoothed sample cross spectrum using spectral smoothing by the trapezium frequency (Daniell) window M01 -- Sorting M01CAF Sort a vector, double precision numbers M01DAF Rank a vector, double precision numbers M01DEF Rank rows of a matrix, double precision numbers M01DJF Rank columns of a matrix, double precision numbers M01EAF Rearrange a vector according to given ranks, double precision numbers M01ZAF Invert a permutation S -- Approximations of Special Functions z S01EAF Complex exponential, e S13AAF Exponential integral E (x) 1 S13ACF Cosine integral Ci(x) S13ADF Sine integral Si(x) S14AAF Gamma function S14ABF Log Gamma function S14BAF Incomplete gamma functions P(a,x) and Q(a,x) S15ADF Complement of error function erfc x S15AEF Error function erf x S17ACF Bessel function Y (x) 0 S17ADF Bessel function Y (x) 1 S17AEF Bessel function J (x) 0 S17AFF Bessel function J (x) 1 S17AGF Airy function Ai(x) S17AHF Airy function Bi(x) S17AJF Airy function Ai'(x) S17AKF Airy function Bi'(x) S17DCF Bessel functions Y (z), real a>=0, complex z, (nu)+a (nu)=0,1,2,... S17DEF Bessel functions J (z), real a>=0, complex z, (nu)+a (nu)=0,1,2,... S17DGF Airy functions Ai(z) and Ai'(z), complex z S17DHF Airy functions Bi(z) and Bi'(z), complex z (j) S17DLF Hankel functions H (z), j=1,2, real a>=0, complex z, (nu)+a (nu)=0,1,2,... S18ACF Modified Bessel function K (x) 0 S18ADF Modified Bessel function K (x) 1 S18AEF Modified Bessel function I (x) 0 S18AFF Modified Bessel function I (x) 1 S18DCF Modified Bessel functions K (z), real a>=0, complex (nu)+a z, (nu)=0,1,2,... S18DEF Modified Bessel functions I (z), real a>=0, complex (nu)+a z, (nu)=0,1,2,... S19AAF Kelvin function ber x S19ABF Kelvin function bei x S19ACF Kelvin function ker x S19ADF Kelvin function kei x S20ACF Fresnel integral S(x) S20ADF Fresnel integral C(x) S21BAF Degenerate symmetrised elliptic integral of 1st kind R (x,y) C S21BBF Symmetrised elliptic integral of 1st kind R (x,y,z) F S21BCF Symmetrised elliptic integral of 2nd kind R (x,y,z) D S21BDF Symmetrised elliptic integral of 3rd kind R (x,y,z,r) J X04 -- Input/Output Utilities X04AAF Return or set unit number for error messages X04ABF Return or set unit number for advisory messages X04CAF Print a real general matrix X04DAF Print a complex general matrix X05 -- Date and Time Utilities X05AAF Return date and time as an array of integers X05ABF Convert array of integers representing date and time to character string X05ACF Compare two character strings representing date and time X05BAF Return the CPU time Fundamental Support Routines The following fundamental support routines are provided and are documented in compact form in the relevant chapter introductory material: F06 -- Linear Algebra Support Routines F06AAF (DROTG) Generate real plane rotation F06EAF (DDOT) Dot product of two real vectors F06ECF (DAXPY) Add scalar times real vector to real vector F06EDF (DSCAL) Multiply real vector by scalar F06EFF (DCOPY) Copy real vector F06EGF (DSWAP) Swap two real vectors F06EJF (DNRM2) Compute Euclidean norm of real vector F06EKF (DASUM) Sum the absolute values of real vector elements F06EPF (DROT) Apply real plane rotation F06GAF (ZDOTU) Dot product of two complex vectors, unconjugated F06GBF (ZDOTC) Dot product of two complex vectors, conjugated F06GCF (ZAXPY) Add scalar times complex vector to complex vector F06GDF (ZSCAL) Multiply complex vector by complex scalar F06GFF (ZCOPY) Copy complex vector F06GGF (ZSWAP) Swap two complex vectors F06JDF (ZDSCAL) Multiply complex vector by real scalar F06JJF (DZNRM2) Compute Euclidean norm of complex vector F06JKF (DZASUM) Sum the absolute values of complex vector elements F06JLF (IDAMAX) Index, real vector element with largest absolute value F06JMF (IZAMAX) Index, complex vector element with largest absolute value F06PAF (DGEMV) Matrix-vector product, real rectangular matrix F06PBF (DGBMV) Matrix-vector product, real rectangular band matrix F06PCF (DSYMV) Matrix-vector product, real symmetric matrix F06PDF (DSBMV) Matrix-vector product, real symmetric band matrix F06PEF (DSPMV) Matrix-vector product, real symmetric packed matrix F06PFF (DTRMV) Matrix-vector product, real triangular matrix F06PGF (DTBMV) Matrix-vector product, real triangular band matrix F06PHF (DTPMV) Matrix-vector product, real triangular packed matrix F06PJF (DTRSV) System of equations, real triangular matrix F06PKF (DTBSV) System of equations, real triangular band matrix F06PLF (DTPSV) System of equations, real triangular packed matrix F06PMF (DGER) Rank-1 update, real rectangular matrix F06PPF (DSYR) Rank-1 update, real symmetric matrix F06PQF (DSPR) Rank-1 update, real symmetric packed matrix F06PRF (DSYR2) Rank-2 update, real symmetric matrix F06PSF (DSPR2) Rank-2 update, real symmetric packed matrix F06SAF (ZGEMV) Matrix-vector product, complex rectangular matrix F06SBF (ZGBMV) Matrix-vector product, complex rectangular band matrix F06SCF (ZHEMV) Matrix-vector product, complex Hermitian matrix F06SDF (ZHBMV) Matrix-vector product, complex Hermitian band matrix F06SEF (ZHPMV) Matrix-vector product, complex Hermitian packed matrix F06SFF (ZTRMV) Matrix-vector product, complex triangular matrix F06SGF (ZTBMV) Matrix-vector product, complex triangular band matrix F06SHF (ZTPMV) Matrix-vector product, complex triangular packed matrix F06SJF (ZTRSV) System of equations, complex triangular matrix F06SKF (ZTBSV) System of equations, complex triangular band matrix F06SLF (ZTPSV) System of equations, complex triangular packed matrix F06SMF (ZGERU) Rank-1 update, complex rectangular matrix, unconjugated vector F06SNF (ZGERC) Rank-1 update, complex rectangular matrix, conjugated vector F06SPF (ZHER) Rank-1 update, complex Hermitian matrix F06SQF (ZHPR) Rank-1 update, complex Hermitian packed matrix F06SRF (ZHER2) Rank-2 update, complex Hermitian matrix F06SSF (ZHPR2) Rank-2 update, complex Hermitian packed matrix F06YAF (DGEMM) Matrix-matrix product, two real rectangular matrices F06YCF (DSYMM) Matrix-matrix product, one real symmetric matrix, one real rectangular matrix F06YFF (DTRMM) Matrix-matrix product, one real triangular matrix, one real rectangular matrix F06YJF (DTRSM) Solves a system of equations with multiple right- hand sides, real triangular coefficient matrix F06YPF (DSYRK) Rank-k update of a real symmetric matrix F06YRF (DSYR2K) Rank-2k update of a real symmetric matrix F06ZAF (ZGEMM) Matrix-matrix product, two complex rectangular matrices F06ZCF (ZHEMM) Matrix-matrix product, one complex Hermitian matrix, one complex rectangular matrix F06ZFF (ZTRMM) Matrix-matrix product, one complex triangular matrix, one complex rectangular matrix F06ZJF (ZTRSM) Solves system of equations with multiple right- hand sides, complex triangular coefficient matrix F06ZPF (ZHERK) Rank-k update of a complex Hermitian matrix F06ZRF (ZHER2K) Rank-2k update of a complex Hermitian matrix F06ZTF (ZSYMM) Matrix-matrix product, one complex symmetric matrix, one complex rectangular matrix F06ZUF (ZSYRK) Rank-k update of a complex symmetric matrix F06ZWF (ZSYR2K) Rank-2k update of a complex symmetric matrix X01 -- Mathematical Constants X01AAF (pi) X01ABF Euler's constant, (gamma) X02 -- Machine Constants X02AHF Largest permissible argument for SIN and COS X02AJF Machine precision X02AKF Smallest positive model number X02ALF Largest positive model number X02AMF Safe range of floating-point arithmetic X02ANF Safe range of complex floating-point arithmetic X02BBF Largest representable integer X02BEF Maximum number of decimal digits that can be represented X02BHF Parameter of floating-point arithmetic model, b X02BJF Parameter of floating-point arithmetic model, p X02BKF Parameter of floating-point arithmetic model, e min X02BLF Parameter of floating-point arithmetic model, e max X02DJF Parameter of floating-point arithmetic model, ROUNDS Routines from the NAG Fortran Library A number of routines from the NAG Fortran Library are used in the Foundation Library as auxiliaries and are not documented here: A00AAF A02AAF A02ABF A02ACF C02AJF C05AZF C05NCF C05PCF C06FAF C06FBF C06FCF C06FFF C06FJF C06FKF C06HAF C06HBF C06HCF C06HDF D02CBF D02CHF D02NMF D02NSF D02NVF D02PAF D02XAF D02XKF D02YAF D02ZAF E02AFF E04GBF E04GEF E04YAF F01ADF F01AEF F01AFF F01AGF F01AHF F01AJF F01AKF F01AMF F01APF F01ATF F01AUF F01AVF F01AWF F01AXF F01BCF F01BTF F01CRF F01LBF F01LZF F01QAF F01QFF F01QGF F01QJF F01QKF F01RFF F01RGF F01RJF F01RKF F02AMF F02ANF F02APF F02AQF F02AVF F02AYF F02BEF F02SWF F02SXF F02SYF F02SZF F02UWF F02UXF F02UYF F02WDF F02WUF F02XUF F03AAF F03ABF F03AEF F03AFF F04AAF F04AEF F04AFF F04AGF F04AHF F04AJF F04AMF F04ANF F04AYF F04LDF F04YAF F04YCF F06BAF F06BCF F06BLF F06BMF F06BNF F06CAF F06CCF F06CLF F06DBF F06DFF F06FBF F06FCF F06FDF F06FGF F06FJF F06FLF F06FPF F06FQF F06FRF F06FSF F06HBF F06HGF F06HQF F06HRF F06KFF F06KJF F06KLF F06QFF F06QHF F06QKF F06QRF F06QSF F06QTF F06QVF F06QWF F06QXF F06RAF F06RJF F06TFF F06THF F06TTF F06TXF F06VJF F06VKF F06VXF F07AGF F07AHF F07AJF F07FGF F07FHF F07FJF F07TJF G01CEF G02BAF G02BUF G02BWF G02DDF G13AEF M01CBF M01CCF M01DBF M01DCF M01DFF M01ZBF P01ABF P01ACF S01BAF S07AAF S15ABF X03AAF X04BAF X04BBF X04CBF X04DBF \end{verbatim} \endscroll \end{page} \begin{page}{manpageXXintro}{NAG On-line Documentation: intro} \beginscroll \begin{verbatim} INTRO(3NAG) Foundation Library (12/10/92) INTRO(3NAG) Introduction Essential Introduction Essential Introduction to the NAG Foundation Library This document is essential reading for any prospective user of the Library. This document appears in both the Handbook and the Reference Manual for the NAG Foundation Library, but with a different Section 3 to describe the different forms of routine documentation in the two publications. 1. The Library and its Documentation 1.1. Structure of the Library 1.2. Structure of the Documentation 1.3. On-line Documentation 1.4. Implementations of the Library 1.5. Library Identification 1.6. Fortran Language Standards 2. Using the Library 2.1. General Advice 2.2. Programming Advice 2.3. Error handling and the Parameter IFAIL 2.4. Input/output in the Library 2.5. Auxiliary Routines 3. Using the Reference Manual 3.1. General Guidance 3.2. Structure of Routine Documents 3.3. Specifications of Parameters 3.3.1. Classification of Parameters 3.3.2. Constraints and Suggested Values 3.3.3. Array Parameters 3.4. Implementation-dependent Information 3.5. Example Programs and Results 3.6. Summary for New Users 4. Relationship between the Foundation Library and other NAG Libraries 4.1. NAG Fortran Library 4.2. NAG Workstation Library 4.3. NAG C Library 5. Contact between Users and NAG 6. General Information about NAG 7. References 1. The Library and its Documentation 1.1. Structure of the Library The NAG Foundation Library is a comprehensive collection of Fortran 77 routines for the solution of numerical and statistical problems. The word 'routine' is used to denote 'subroutine' or ' function'. The Library is divided into chapters, each devoted to a branch of numerical analysis or statistics. Each chapter has a three- character name and a title, e.g. D01 -- Quadrature Exceptionally one chapter (S) has a one-character name. (The chapters and their names are based on the ACM modified SHARE classification index [1].) All documented routines in the Library have six-character names, beginning with the characters of the chapter name, e.g. D01AJF Note that the second and third characters are digits, not letters; e.g. 0 is the digit zero, not the letter O. The last letter of each routine name is always 'F'. 1.2. Structure of the Documentation There are two types of manual for the NAG Foundation Library: a Handbook and a Reference Manual. The Handbook has the same chapter structure as the Library: each chapter of routines in the Library has a corresponding chapter (of the same name) in the Handbook. The chapters occur in alphanumeric order. General introductory documents and indexes are placed at the beginning of the Handbook. Each chapter in the Handbook contains a Chapter Introduction, followed by concise summaries of the functionality and parameter specifications of each routine in the chapter. Exceptionally, in some chapters (F06, X01, X02) which contain simple support routines, there are no concise summaries: all the routines are described together in the Chapter Introduction. The Reference Manual provides complete reference documentation for the NAG Foundation Library. In the Reference Manual, each chapter consists of the following documents: Chapter Introduction, e.g. Introduction -- D01; Chapter Contents, e.g. Contents -- D01; routine documents, one for each documented routine in the chapter. A routine document has the same name as the routine which it describes. Within each chapter, routine documents occur in alphanumeric order. As in the Handbook, chapters F06, X01 and X02 do not contain separate documentation for individual routines. The general introductory documents, indexes and chapter introductions are the same in the Reference Manual as in the Handbook. The only exception is that the Essential Introduction contains a different Section 3 in the two publications, to describe the different forms of routine documentation. 1.3. On-line Documentation Extensive on-line documentation is included as an integral part of the Foundation Library product. This consists of a number of components: -- general introductory material, including the Essential Introduction -- a summary list of all documented routines -- a KWIC Index -- Chapter Introductions -- routine documents -- example programs, data and results. The material has been derived in a number of forms to cater for different user requirements, e.g. UNIX man pages, plain text, RICH TEXT format etc, and the appropriate version is included on the distribution media. For each implementation of the Foundation Library the specific documentation (Installers' Note, Users' Note etc) gives details of what is provided. 1.4. Implementations of the Library The NAG Foundation Library is available on many different computer systems. For each distinct system, an implementation of the Library is prepared by NAG, e.g. the IBM RISC System/6000 implementation. The implementation is distributed as a tested compiled library. An implementation is usually specific to a range of machines; it may also be specific to a particular operating system or compilation system. Essentially the same facilities are provided in all implementations of the Library, but, because of differences in arithmetic behaviour and in the compilation system, routines cannot be expected to give identical results on different systems, especially for sensitive numerical problems. The documentation supports all implementations of the Library, with the help of a few simple conventions, and a small amount of implementation-dependent information, which is published in a separate Users' Note for each implementation (see Section 3.4). 1.5. Library Identification You must know which implementation of the Library you are using or intend to use. To find out which implementation of the Library is available on your machine, you can run a program which calls the NAG Foundation Library routine A00AAF. This routine has no parameters; it simply outputs text to the advisory message unit (see Section 2.4). An example of the output is: *** Start of NAG Foundation Library implementation details *** Implementation title: IBM RISC System/6000 Precision: FORTRAN double precision Product Code: FFIB601D Release: 1 *** End of NAG Foundation Library implementation details *** (The product code can be ignored, except possibly when communicating with NAG; see Section 4.) 1.6. Fortran Language Standards All routines in the Library conform to ANSI Standard Fortran 90 [8]. Most of the routines in the Library were originally written to conform to the earlier Fortran 66 [6] and Fortran 77 [7] standards, and their calling sequences contain some parameters which are not strictly necessary in Fortran 90. 2. Using the Library 2.1. General Advice A NAG Foundation Library routine cannot be guaranteed to return meaningful results, irrespective of the data supplied to it. Care and thought must be exercised in: (a) formulating the problem; (b) programming the use of library routines; (c) assessing the significance of the results. 2.2. Programming Advice The NAG Foundation Library and its documentation are designed on the assumption that users know how to write a calling program in Fortran. When programming a call to a routine, read the routine document carefully, especially the description of the Parameters. This states clearly which parameters must have values assigned to them on entry to the routine, and which return useful values on exit. See Section 3.3 for further guidance. If a call to a Library routine results in an unexpected error message from the system (or possibly from within the Library), check the following: Has the NAG routine been called with the correct number of parameters? Do the parameters all have the correct type? Have all array parameters been dimensioned correctly? Remember that all floating-point parameters must be declared to be double precision, either with an explicit DOUBLE PRECISION declaration (or COMPLEX(KIND(1.0D0)) if they are complex), or by using a suitable IMPLICIT statement. Avoid the use of NAG-type names for your own program units or COMMON blocks: in general, do not use names which contain a three-character NAG chapter name embedded in them; they may clash with the names of an auxiliary routine or COMMON block used by the NAG Library. 2.3. Error handling and the Parameter IFAIL NAG Foundation Library routines may detect various kinds of error, failure or warning conditions. Such conditions are handled in a systematic way by the Library. They fall roughly into three classes: (i) an invalid value of a parameter on entry to a routine; (ii) a numerical failure during computation (e.g. approximate singularity of a matrix, failure of an iteration to converge); (iii) a warning that although the computation has been completed, the results cannot be guaranteed to be completely reliable. All three classes are handled in the same way by the Library, and are all referred to here simply as 'errors'. The error-handling mechanism uses the parameter IFAIL, which is the last parameter in the calling sequence of most NAG Foundation Library routines. IFAIL serves two purposes: (i) it allows users to specify what action a Library routine should take if it detects an error; (ii) it reports the outcome of a call to a Library routine, either success (IFAIL = 0) or failure (IFAIL /= 0, with different values indicating different reasons for the failure, as explained in Section 6 of the routine document) . For the first purpose IFAIL must be assigned a value before calling the routine; since IFAIL is reset by the routine, it must be passed as a variable, not as an integer constant. Allowed values on entry are: IFAIL=0: an error message is output, and execution is terminated ('hard failure'); IFAIL=+1: execution continues without any error message; IFAIL=-1: an error message is output, and execution continues. The settings IFAIL =+-1 are referred to as 'soft failure'. The safest choice is to set IFAIL to 0, but this is not always convenient: some routines return useful results even though a failure (in some cases merely a warning) is indicated. However, if IFAIL is set to +- 1 on entry, it is essential for the program to test its value on exit from the routine, and to take appropriate action. The specification of IFAIL in Section 5 of a routine document suggests a suitable setting of IFAIL for that routine. 2.4. Input/output in the Library Most NAG Foundation Library routines perform no output to an external file, except possibly to output an error message. All error messages are written to a logical error message unit. This unit number (which is set by default to 6 in most implementations) can be changed by calling the Library routine X04AAF. Some NAG Foundation Library routines may optionally output their final results, or intermediate results to monitor the course of computation. All output other than error messages is written to a logical advisory message unit. This unit number (which is also set by default to 6 in most implementations) can be changed by calling the Library routine X04ABF. Although it is logically distinct from the error message unit, in practice the two unit numbers may be the same. All output from the Library is formatted. The only Library routines which perform input from an external file are a few 'option-setting' routines in Chapter E04: the unit number is a parameter to the routine, and all input is formatted. You must ensure that the relevant Fortran unit numbers are associated with the desired external files, either by an OPEN statement in your calling program, or by operating system commands. 2.5. Auxiliary Routines In addition to those Library routines which are documented and are intended to be called by users, the Library also contains many auxiliary routines. In general, you need not be concerned with them at all, although you may be made aware of their existence if, for example, you examine a memory map of an executable program which calls NAG routines. The only exception is that when calling some NAG Foundation Library routines, you may be required or allowed to supply the name of an auxiliary routine from the Library as an external procedure parameter. The routine documents give the necessary details. In such cases, you only need to supply the name of the routine; you never need to know details of its parameter-list. NAG auxiliary routines have names which are similar to the name of the documented routine(s) to which they are related, but with last letter 'Z', 'Y', and so on, e.g. D01AJZ is an auxiliary routine called by D01AJF. 3. Using the Reference Manual 3.1. General Guidance The Reference Manual is designed to serve the following functions: -- to give background information about different areas of numerical and statistical computation; -- to advise on the choice of the most suitable NAG Foundation Library routine or routines to solve a particular problem; -- to give all the information needed to call a NAG Foundation Library routine correctly from a Fortran program, and to assess the results. At the beginning of the Manual are some general introductory documents. The following may help you to find the chapter, and possibly the routine, which you need to solve your problem: Contents -- a list of routines in the Library, by Summary chapter; KWIC Index -- a keyword index to chapters and routines. Having found a likely chapter or routine, you should read the corresponding Chapter Introduction, which gives background information about that area of numerical computation, and recommendations on the choice of a routine, including indexes, tables or decision trees. When you have chosen a routine, you must consult the routine document. Each routine document is essentially self-contained (it may contain references to related documents). It includes a description of the method, detailed specifications of each parameter, explanations of each error exit, and remarks on accuracy. Example programs which illustrate the use of each routine are distributed with the Library in machine-readable form. 3.2. Structure of Routine Documents All routine documents have the same structure, consisting of nine numbered sections: 1. Purpose 2. Specification 3. Description 4. References 5. Parameters (see Section 3.3 below) 6. Error Indicators 7. Accuracy 8. Further Comments 9. Example (see Section 3.5 below) In a few documents, Section 5 also includes a description of printed output which may optionally be produced by the routine. 3.3. Specifications of Parameters Section 5 of each routine document contains the specification of the parameters, in the order of their appearance in the parameter list. 3.3.1. Classification of Parameters Parameters are classified as follows: Input : you must assign values to these parameters on or before entry to the routine, and these values are unchanged on exit from the routine. Output : you need not assign values to these parameters on or before entry to the routine; the routine may assign values to them. Input/Output : you must assign values to these parameters on or before entry to the routine, and the routine may then change these values. Workspace: array parameters which are used as workspace by the routine. You must supply arrays of the correct type and dimension, but you need not be concerned with their contents. External Procedure: a subroutine or function which must be supplied (e.g. to evaluate an integrand or to print intermediate output). Usually it must be supplied as part of your calling program, in which case its specification includes full details of its parameter-list and specifications of its parameters (all enclosed in a box). Its parameters are classified in the same way as those of the Library routine, but because you must write the procedure rather than call it, the significance of the classification is different: Input : values may be supplied on entry, which your procedure must not change. Output : you may or must assign values to these parameters before exit from your procedure. Input/Output : values may be supplied on entry, and you may or must assign values to them before exit from your procedure. Occasionally, as mentioned in Section 2.5, the procedure can be supplied from the NAG Library, and then you only need to know its name. User Workspace: array parameters which are passed by the Library routine to an external procedure parameter. They are not used by the routine, but you may use them to pass information between your calling program and the external procedure. 3.3.2. Constraints and Suggested Values The word 'Constraint:' or 'Constraints:' in the specification of an Input parameter introduces a statement of the range of valid values for that parameter, e.g. Constraint: N > 0. If the routine is called with an invalid value for the parameter (e.g. N = 0), the routine will usually take an error exit, returning a non-zero value of IFAIL (see Section 2.3). In newer documents constraints on parameters of type CHARACTER only list uppercase alphabetic characters, e.g. Constraint: STRING = 'A' or 'B'. In practice all routines with CHARACTER parameters will permit the use of lower case characters. The phrase 'Suggested Value:' introduces a suggestion for a reasonable initial setting for an Input parameter (e.g. accuracy or maximum number of iterations) in case you are unsure what value to use; you should be prepared to use a different setting if the suggested value turns out to be unsuitable for your problem. 3.3.3. Array Parameters Most array parameters have dimensions which depend on the size of the problem. In Fortran terminology they have 'adjustable dimensions': the dimensions occurring in their declarations are integer variables which are also parameters of the Library routine. For example, a Library routine might have the specification: SUBROUTINE (M, N, A, B, LDB) INTEGER M, N, A(N), B(LDB,N), LDB For a one-dimensional array parameter, such as A in this example, the specification would begin: 3: A(N) -- DOUBLE PRECISION array Input You must ensure that the dimension of the array, as declared in your calling (sub)program, is at least as large as the value you supply for N. It may be larger; but the routine uses only the first N elements. For a two-dimensional array parameter, such as B in the example, the specification might be: 4: B(LDB,N) -- DOUBLE PRECISION array Input/Output On entry: the m by n matrix B. and the parameter LDB might be described as follows: 5: LDB -- INTEGER Input On entry: the first dimension of the array B as declared in the (sub)program from which is called. Constraint: LDB >= M. You must supply the first dimension of the array B, as declared in your calling (sub)program, through the parameter LDB, even though the number of rows actually used by the routine is determined by the parameter M. You must ensure that the first dimension of the array is at least as large as the value you supply for M. The extra parameter LDB is needed because Fortran does not allow information about the dimensions of array parameters to be passed automatically to a routine. You must also ensure that the second dimension of the array, as declared in your calling (sub)program, is at least as large as the value you supply for N. It may be larger, but the routine only uses the first N columns. A program to call the hypothetical routine used as an example in this section might include the statements: INTEGER AA(100), BB(100,50) LDB = 100 . . . M = 80 N = 20 CALL (M,N,AA,BB,LDB) Fortran requires that the dimensions which occur in array declarations, must be greater than zero. Many NAG routines are designed so that they can be called with a parameter like N in the above example set to 0 (in which case they would usually exit immediately without doing anything). If so, the declarations in the Library routine would use the 'assumed size' array dimension, and would be given as: INTEGER M, N, A(*), B(LDB,*), LDB However, the original declaration of an array in your calling program must always have constant dimensions, greater than or equal to 1. Consult an expert or a textbook on Fortran, if you have difficulty in calling NAG routines with array parameters. 3.4. Implementation-dependent Information In order to support all implementations of the Foundation Library, the Manual has adopted a convention of using bold italics to distinguish terms which have different interpretations in different implementations. For example, machine precision denotes the relative precision to which double precision floating-point numbers are stored in the computer, e.g. in an implementation with approximately 16 decimal digits of precision, machine precision has a value of - 16 approximately 10 . The precise value of machine precision is given by the function X02AJF. Other functions in Chapter X02 return the values of other implementation-dependent constants, such as the overflow threshold, or the largest representable integer. Refer to the X02 Chapter Introduction for more details. For each implementation of the Library, a separate Users' Note is provided. This is a short document, revised at each Mark. At most installations it is available in machine-readable form. It gives any necessary additional information which applies specifically to that implementation, in particular: -- the interpretation of bold italicised terms; -- the values returned by X02 routines; -- the default unit numbers for output (see Section 2.4). 3.5. Example Programs and Results The last section of each routine document describes an example problem which can be solved by simple use of the routine. The example programs themselves, together with data and results, are not printed in the routine document, but are distributed in machine-readable form with the Library. The programs are designed so that they can fairly easily be modified, and so serve as the basis for a simple program to solve a user's own problem. The results distributed with each implementation were obtained using that implementation of the Library; they may not be identical to the results obtained with other implementations. 3.6. Summary for New Users If you are unfamiliar with the NAG Foundation Library and are thinking of using a routine from it, please follow these instructions: (a) read the whole of the Essential Introduction; (b) consult the Contents Summary or KWIC Index to choose an appropriate chapter or routine; (c) read the relevant Chapter Introduction; (d) choose a routine, and read the routine document. If the routine does not after all meet your needs, return to steps (b) or (c); (e) read the Users' Note for your implementation; (f) consult local documentation, which should be provided by your local support staff, about access to the NAG Library on your computing system. You should now be in a position to include a call to the routine in a program, and to attempt to run it. You may of course need to refer back to the relevant documentation in the case of difficulties, for advice on assessment of results, and so on. As you become familiar with the Library, some of steps (a) to (f) can be omitted, but it is always essential to: -- be familiar with the Chapter Introduction; -- read the routine document; -- be aware of the Users' Note for your implementation. 4. Relationship between the Foundation Library and other NAG Libraries 4.1. NAG Fortran Library The Foundation Library is a strict subset of the full NAG Fortran Library (Mark 15 or later). Routines in both libraries have identical source code (apart from any modifications necessary for implementation on a specific system) and hence can be called in exactly the same way, though you should consult the relevant implementation-specific documentation for details such as values of machine constants. By its very nature, the Foundation Library cannot contain the same extensive range of routines as the full Fortran Library. If your application requires a routine which is not in the Foundation Library, then please consult NAG for information on relevant material available in the Fortran Library. Some routines which occur as user-callable routines in the full Fortran Library are included as auxiliary routines in the Foundation Library but they are not documented in this publication and direct calls to them should only be made if you are already familiar with their use in the Fortran Library. A list of all such auxiliary routines is given at the end of the Foundation Library Contents Summary. Whereas the full Fortran Library may be provided in either a single precision or a double precision version, the Foundation Library is always provided in double precision. 4.2. NAG Workstation Library The Foundation Library is a successor product to an earlier, smaller subset of the full NAG Fortran Library which was called the NAG Workstation Library. The Foundation Library has greater functionality than the Workstation Library but is not strictly upwards compatible, i.e., a number of routines in the earlier product have been replaced by new material to reflect recent algorithmic developments. If you have used the Workstation Library and wish to convert your programs to call routines from the Foundation Library, please consult the document 'Converting from the Workstation Library' in this Manual. 4.3. NAG C Library NAG has also developed a library of numerical and statistical software for use by C programmers. This now contains over 200 user-callable functions and provides similar (but not identical) coverage to that of the Foundation Library. Please contact NAG for further details if you have a requirement for similar quality library code in C. 5. Contact between Users and NAG If you are using the NAG Foundation Library in a multi-user environment and require further advice please consult your local support staff who will be receiving regular information from NAG. This covers such matters as: -- obtaining a copy of the Users' Note for your implementation; -- obtaining information about local access to the Library; -- seeking advice about using the Library; -- reporting suspected errors in routines or documents; -- making suggestions for new routines or features; -- purchasing NAG documentation. If you are unable to make contact with a local source of support or are in a single-user environment then please contact NAG directly at any one of the addresses given at the beginning of this publication. 6. General Information about NAG NAG produces and distributes numerical, symbolic, statistical and graphical software for the solution of problems in a wide range of applications in such areas as science, engineering, financial analysis and research. For users who write programs and build packages NAG produces sub- program libraries in a range of computer languages (Ada, C, Fortran, Pascal, Turbo Pascal). NAG also provides a number of Fortran programming support products in the NAGWare range -- Fortran 77 programming tools, Fortran 90 compilers for a number of machine platforms (including PC-DOS) and VecPar 77 for restructuring and tuning programs for execution on vector or parallel computers. For users who do not wish to program in the traditional sense but want the same reliability and qualities offered by our libraries, NAG provides several powerful mathematical and statistical packages for interactive use. A major addition to this range of packages is AXIOM -- the powerful symbolic solver which includes a Hypertext system and graphical capabilities. For further details of any of these products, please contact NAG at one of the addresses given at the beginning of this publication. References [2], [3], [4], and [5] discuss various aspects of the design and development of the NAG Library, and NAG's technical policies and organisation. 7. References [1] (1960--1976) Collected Algorithms from ACM Index by subject to algorithms. [2] Ford B (1982) Transportable Numerical Software. Lecture Notes in Computer Science. 142 128--140. [3] Ford B, Bentley J, Du Croz J J and Hague S J (1979) The NAG Library 'machine'. Software Practice and Experience. 9(1) 65--72. [4] Ford B and Pool J C T (1984) The Evolving NAG Library Service. Sources and Development of Mathematical Software. (ed W Cowell) Prentice-Hall. 375--397. [5] Hague S J, Nugent S M and Ford B (1982) Computer-based Documentation for the NAG Library. Lecture Notes in Computer Science. 142 91--127. [6] (1966) USA Standard Fortran. Publication X3.9. American National Standards Institute. [7] (1978) American National Standard Fortran. Publication X3.9. American National Standards Institute. [8] (1991) American National Standard Programming Language Fortran 90. Publication X3.198. American National Standards Institute. \end{verbatim} \endscroll \end{page} \begin{page}{manpageXXkwic}{NAG On-line Documentation: kwic} \beginscroll \begin{verbatim} KWIC(3NAG) Foundation Library (12/10/92) KWIC(3NAG) Introduction Keywords in Context Keywords in Context Pre-computed weights and D01BBF abscissae for Gaussian quadrature rules, restricted choice of ... Sum the F06EKF absolute values of real vector elements (DASUM) Sum the F06JKF absolute values of complex vector elements (DZASUM) Index, real vector element with largest F06JLF absolute value (IDAMAX) Index, complex vector element with largest F06JMF absolute value (IZAMAX) ODEs, IVP, D02CJF Adams method, until function of solution is zero, ... 1-D quadrature, D01AJF adaptive , finite interval, strategy due to Piessens and de ... 1-D quadrature, D01AKF adaptive , finite interval, method suitable for oscillating ... 1-D quadrature, D01ALF adaptive , finite interval, allowing for singularities at ... 1-D quadrature, D01AMF adaptive , infinite or semi-infinite interval 1-D quadrature, D01ANF adaptive , finite interval, weight function cos((omega)x) ... 1-D quadrature, D01APF adaptive , finite interval, weight function with end-point ... 1-D quadrature, D01AQF adaptive , finite interval, weight function 1/(x-c), ... 1-D quadrature, D01ASF adaptive , semi-infinite interval, weight function cos((omega)x) Multi-dimensional D01FCF adaptive quadrature over hyper-rectangle Add F06ECF scalar times real vector to real vector (DAXPY) Add F06GCF scalar times complex vector to complex vector (ZAXPY) Return or set unit number for X04ABF advisory messages Airy S17AGF function Ai(x) Airy S17AHF function Bi(x) Airy S17AJF function Ai'(x) Airy S17AKF function Bi'(x) Airy S17DGF functions Ai(z) and Ai('z), complex z Airy S17DHF functions Bi(z) and Bi'(z), complex z Airy function S17AGF Ai(x) Airy function S17AJF Ai'(x) Airy functions S17DGF Ai(z) and Ai'(z), complex z Airy functions Ai(z) and S17DGF Ai'(z) , complex z algebraico-logarithmic type Two-way contingency table G01AFF analysis 2 , with (chi) /Fisher's exact test Performs principal component G03AAF analysis Performs canonical correlation G03ADF analysis Friedman two-way G08AEF analysis of variance on k matched samples Kruskal-Wallis one-way G08AFF analysis of variance on k samples of unequal size L - E02GAF 1 approximation by general linear function Approximation E02 Approximation S of special functions ARIMA model Univariate time series, estimation, seasonal G13AFF ARIMA model ARIMA model ARIMA model Safe range of floating-point X02AMF arithmetic Safe range of complex floating-point X02ANF arithmetic Parameter of floating-point X02BHF arithmetic model, b Parameter of floating-point X02BJF arithmetic model, p Parameter of floating-point X02BKF arithmetic model, e min Parameter of floating-point X02BLF arithmetic model, e max Parameter of floating-point X02DJF arithmetic model, ROUNDS Univariate time series, sample G13ABF autocorrelation function Univariate time series, partial G13ACF autocorrelations from autocorrelations Univariate time series, partial autocorrelations from G13ACF autocorrelations Least-squares cubic spline curve fit, E02BEF automatic knot placement Least-squares surface fit by bicubic splines with E02DCF automatic knot placement, data on rectangular grid Least-squares surface fit by bicubic splines with E02DDF automatic knot placement, scattered data B-splines E02 Matrix-vector product, real rectangular F06PBF band matrix (DGBMV) Matrix-vector product, real symmetric F06PDF band matrix (DSBMV) Matrix-vector product, real triangular F06PGF band matrix (DTBMV) System of equations, real triangular F06PKF band matrix (DTBSV) Matrix-vector product, complex rectangular F06SBF band matrix (ZGBMV) Matrix-vector product, complex Hermitian F06SDF band matrix (ZHBMV) Matrix-vector product, complex triangular F06SGF band matrix (ZTBMV) System of equations, complex triangular F06SKF band matrix (ZTBSV) bandwidth matrix Solution of real symmetric positive-definite variable- F04MCF bandwidth simultaneous linear equations (coefficient matrix ... Basic F06 Linear Algebra Subprograms ODEs, stiff IVP, D02EJF BDF method, until function of solution is zero, intermediate ... Kelvin function S19ABF bei x Kelvin function S19AAF ber x Bessel S17ACF function Y (x) 0 Bessel S17ADF function Y (x) 1 Bessel S17AEF function J (x) 0 Bessel S17AFF function J (x) 1 Bessel S17DCF functions Y (z), complex z, real (nu)>=0, ... (nu)+n Bessel S17DEF functions J (z), complex z, real (nu)>=0, ... (nu)+n Modified S18ACF Bessel function K (x) 0 Modified S18ADF Bessel function K (x) 1 Modified S18AEF Bessel function I (x) 0 Modified S18AFF Bessel function I (x) 1 Modified S18DCF Bessel functions K (z), complex z, real (nu)>=0, ... (nu)+n Modified S18DEF Bessel functions I (z), complex z, real (nu)>=0, ... (nu)+n beta distribution Computes deviates for the G01FEF beta distribution Generates a vector of pseudo-random numbers from a G05FEF beta distribution Interpolating functions, fitting E01DAF bicubic spline, data on rectangular grid Least-squares surface fit, E02DAF bicubic splines Least-squares surface fit by E02DCF bicubic splines with automatic knot placement, data on ... Least-squares surface fit by E02DDF bicubic splines with automatic knot placement, scattered data Evaluation of a fitted E02DEF bicubic spline at a vector of points Evaluation of a fitted E02DFF bicubic spline at a mesh of points Sort 2-D data into panels for fitting E02ZAF bicubic splines Fits a generalized linear model with G02GBF binomial errors binomial distribution Computes probability for the G01HAF bivariate Normal distribution Airy function S17AHF Bi(x) Airy function S17AKF Bi'(x) Airy functions S17DHF Bi(z) and Bi'(z), complex z Airy functions Bi(z) and S17DHF Bi'(z) , complex z BLAS F06 Pseudo-random logical G05DZF (boolean) value ODEs, D02GAF boundary value problem, finite difference technique with ... ODEs, D02GBF boundary value problem, finite difference technique with ... ODEs, general nonlinear D02RAF boundary value problem, finite difference technique with ... bounds , using function values only break-points break-points Zero of continuous function in given interval, C05ADF Bus and Dekker algorithm Performs G03ADF canonical correlation analysis Carlo method Elliptic PDE, Helmholtz equation, 3-D D03FAF Cartesian co-ordinates Cauchy principal value (Hilbert transform) Pseudo-random real numbers, G05DFF Cauchy distribution character string Compare two X05ACF character strings representing date and time Evaluation of fitted polynomial in one variable from E02AEF Chebyshev series form (simplified parameter list) Derivative of fitted polynomial in E02AHF Chebyshev series form Integral of fitted polynomial in E02AJF Chebyshev series form Evaluation of fitted polynomial in one variable, from E02AKF Chebyshev series form Check C05ZAF user's routine for calculating 1st derivatives Univariate time series, diagnostic G13ASF checking of residuals, following G13AFF Cholesky F07FDF factorization of real symmetric positive-definite ... Circular C06EKF convolution or correlation of two real vectors, no ... Cosine integral S13ACF Ci(x) Interpolating functions, method of Renka and E01SAF Cline , two variables Elliptic PDE, Helmholtz equation, 3-D Cartesian D03FAF co-ordinates coefficient matrix already factorized by F01MCF) coefficient matrix (DTRSM) coefficient matrix (ZTRSM) Kendall/Spearman non-parametric rank correlation G02BNF coefficients , no missing values, overwriting input data Kendall/Spearman non-parametric rank correlation G02BQF coefficients , no missing values, preserving input data Operations with orthogonal matrices, form F01QEF columns of Q after factorization by F01QCF Operations with unitary matrices, form F01REF columns of Q after factorization by F01RCF Rank M01DJF columns of a matrix, real numbers Compare X05ACF two character strings representing date and time Complement S15ADF of error function erfcx Unconstrained minimum, pre- E04DGF conditioned conjugate gradient algorithm, function of several ... Complex C06GBF conjugate of Hermitian sequence Complex C06GCF conjugate of complex sequence Complex C06GQF conjugate of multiple Hermitian sequences Unconstrained minimum, pre-conditioned E04DGF conjugate gradient algorithm, function of several variables ... Dot product of two complex vectors, F06GBF conjugated (ZDOTC) Rank-1 update, complex rectangular matrix, F06SNF conjugated vector (ZGERC) Mathematical X01 Constants Machine X02 Constants constrained , arbitrary data points constraints , using function values and optionally 1st ... Two-way G01AFF contingency 2 table analysis, with (chi) /Fisher's ... continuation facility Zero of C05ADF continuous function in given interval, Bus and Dekker algorithm 2 continuous distributions Convert C06GSF Hermitian sequences to general complex sequences Convert X05ABF array of integers representing date and time to ... Circular C06EKF convolution or correlation of two real vectors, no extra ... Copy F06EFF real vector (DCOPY) Copy F06GFF complex vector (ZCOPY) correction , simple nonlinear problem correction , general linear problem correction , continuation facility Circular convolution or C06EKF correlation of two real vectors, no extra workspace Kendall/Spearman non-parametric rank G02BNF correlation coefficients, no missing values, overwriting ... Kendall/Spearman non-parametric rank G02BQF correlation coefficients, no missing values, preserving input ... Computes (optionally weighted) G02BXF correlation and covariance matrices Performs canonical G03ADF correlation analysis Multivariate time series, cross- G13BCF correlations cos ((omega)x) or sin((omega)x) cos ((omega)x) or sin((omega)x) Cosine S13ACF integral Ci(x) Covariance E04YCF matrix for nonlinear least-squares problem Computes (optionally weighted) correlation and G02BXF covariance matrices Return the X05BAF CPU time Multivariate time series, G13BCF cross-correlations Multivariate time series, smoothed sample G13CDF cross spectrum using spectral smoothing by the trapezium ... Interpolating functions, E01BAF cubic spline interpolant, one variable cubic Hermite, one variable Least-squares curve E02BAF cubic spline fit (including interpolation) Evaluation of fitted E02BBF cubic spline, function only Evaluation of fitted E02BCF cubic spline, function and derivatives Evaluation of fitted E02BDF cubic spline, definite integral Least-squares E02BEF cubic spline curve fit, automatic knot placement Set up reference vector from supplied G05EXF cumulative distribution function or probability distribution ... Least-squares E02ADF curve fit, by polynomials, arbitrary data points Least-squares E02BAF curve cubic spline fit (including interpolation) Least-squares cubic spline E02BEF curve fit, automatic knot placement Fresnel integral S20ADF C(x) Daniell) window Daniell) window Return X05AAF date and time as an array of integers Convert array of integers representing X05ABF date and time to character string Compare two character strings representing X05ACF date and time deferred correction, simple nonlinear problem deferred correction, general linear problem deferred correction, continuation facility Interpolated values, interpolant computed by E01BEF, E01BHF definite integral, one variable Evaluation of fitted cubic spline, E02BDF definite integral definite matrix T LDL factorization of real symmetric positive- F01MCF definite variable-bandwidth matrix definite definite Solution of real symmetric positive- F04ASF definite simultaneous linear equations, one right-hand side ... Solution of real symmetric positive- F04FAF definite tridiagonal simultaneous linear equations, one ... Real sparse symmetric positive- F04MAF definite simultaneous linear equations (coefficient matrix ... Solution of real symmetric positive- F04MCF definite variable-bandwidth simultaneous linear equations ... Cholesky factorization of real symmetric positive- F07FDF definite matrix (DPOTRF) Solution of real symmetric positive- F07FEF definite system of linear equations, multiple right-hand ... Degenerate S21BAF symmetrised elliptic integral of 1st kind R ... C Dekker algorithm Computes upper and lower tail and probability G01EEF density function probabilities for the beta distribution Solution of system of nonlinear equations using 1st C05PBF derivatives Check user's routine for calculating 1st C05ZAF derivatives derivative , one variable Least-squares polynomial fit, values and E02AGF derivatives may be constrained, arbitrary data points Derivative E02AHF of fitted polynomial in Chebyshev series form Evaluation of fitted cubic spline, function and E02BCF derivatives derivatives derivatives derivatives Computes G01FAF deviates for the standard Normal distribution Computes G01FBF deviates for Student's t-distribution Computes G01FCF deviates 2 for the (chi) distribution Computes G01FDF deviates for the F-distribution Computes G01FEF deviates for the beta distribution Computes G01FFF deviates for the gamma distribution Univariate time series, G13ASF diagnostic checking of residuals, following G13AFF ODEs, boundary value problem, finite D02GAF difference technique with deferred correction, simple ... ODEs, boundary value problem, finite D02GBF difference technique with deferred correction, general linear ... difference technique with deferred correction, continuation ... Elliptic PDE, solution of finite D03EDF difference equations by a multigrid technique Univariate time series, seasonal and non-seasonal G13AAF differencing Single 1-D real C06EAF discrete Fourier transform, no extra workspace Single 1-D Hermitian C06EBF discrete Fourier transform, no extra workspace Single 1-D complex C06ECF discrete Fourier transform, no extra workspace Multiple 1-D real C06FPF discrete Fourier transforms Multiple 1-D Hermitian C06FQF discrete Fourier transforms Multiple 1-D complex C06FRF discrete Fourier transforms 2-D complex C06FUF discrete Fourier transform Discretize D03EEF a 2nd order elliptic PDE on a rectangle Computes probabilities for the standard Normal G01EAF distribution Computes probabilities for Student's t- G01EBF distribution 2 Computes probabilities for (chi) G01ECF distribution Computes probabilities for F- G01EDF distribution distribution Computes probabilities for the gamma G01EFF distribution Computes deviates for the standard Normal G01FAF distribution Computes deviates for Student's t- G01FBF distribution 2 Computes deviates for the (chi) G01FCF distribution Computes deviates for the F- G01FDF distribution Computes deviates for the beta G01FEF distribution Computes deviates for the gamma G01FFF distribution Computes probability for the bivariate Normal G01HAF distribution Pseudo-random real numbers, uniform G05CAF distribution over (0,1) Pseudo-random real numbers, Normal G05DDF distribution Pseudo-random real numbers, Cauchy G05DFF distribution Pseudo-random real numbers, Weibull G05DPF distribution Pseudo-random integer from uniform G05DYF distribution Set up reference vector for multivariate Normal G05EAF distribution distribution distribution Set up reference vector from supplied cumulative G05EXF distribution function or probability distribution function distribution function Generates a vector of random numbers from a uniform G05FAF distribution distribution Generates a vector of random numbers from a Normal G05FDF distribution distribution distribution 2 (chi) goodness of fit test, for standard continuous G08CGF distributions Inverse G01F distributions Doncker , allowing for badly-behaved integrands Dot F06EAF product of two real vectors (DDOT) Dot F06GAF product of two complex vectors, unconjugated (ZDOTU) Dot F06GBF product of two complex vectors, conjugated (ZDOTC) eigenfunction , user-specified break-points All eigenvalues of generalized real F02ADF eigenproblem of the form Ax=(lambda)Bx where A and B are ... All eigenvalues and eigenvectors of generalized real F02AEF eigenproblem of the form Ax=(lambda)Bx where A and B are ... eigenproblem by QZ algorithm, real matrices eigenproblem eigenvalue and eigenfunction, user-specified break-points All F02AAF eigenvalues of real symmetric matrix All F02ABF eigenvalues and eigenvectors of real symmetric matrix All F02ADF eigenvalues of generalized real eigenproblem of the form Ax=(lambda)Bx All F02AEF eigenvalues and eigenvectors of generalized real ... All F02AFF eigenvalues of real matrix All F02AGF eigenvalues and eigenvectors of real matrix All F02AJF eigenvalues of complex matrix All F02AKF eigenvalues and eigenvectors of complex matrix All F02AWF eigenvalues of complex Hermitian matrix All F02AXF eigenvalues and eigenvectors of complex Hermitian matrix Selected F02BBF eigenvalues and eigenvectors of real symmetric matrix All F02BJF eigenvalues and optionally eigenvectors of generalized ... Selected F02FJF eigenvalues and eigenvectors of sparse symmetric eigenproblem All eigenvalues and F02ABF eigenvectors of real symmetric matrix All eigenvalues and F02AEF eigenvectors of generalized real eigenproblem of the form ... All eigenvalues and F02AGF eigenvectors of real matrix All eigenvalues and F02AKF eigenvectors of complex matrix All eigenvalues and F02AXF eigenvectors of complex Hermitian matrix Selected eigenvalues and F02BBF eigenvectors of real symmetric matrix All eigenvalues and optionally F02BJF eigenvectors of generalized eigenproblem by QZ algorithm, ... Selected eigenvalues and F02FJF eigenvectors of sparse symmetric eigenproblem Elliptic D03EDF PDE, solution of finite difference equations by a ... Discretize a 2nd order D03EEF elliptic PDE on a rectangle Elliptic D03FAF PDE, Helmholtz equation, 3-D Cartesian co-ordinates Degenerate symmetrised S21BAF elliptic integral of 1st kind R (x,y) C Symmetrised S21BBF elliptic integral of 1st kind R (x,y,z) F Symmetrised S21BCF elliptic integral of 2nd kind R (x,y,z) D Symmetrised S21BDF elliptic integral of 3rd kind R (x,y,z,r) J end-point singularities of algebraico-logarithmic type error Fits a generalized linear model with binomial G02GBF errors Fits a generalized linear model with Poisson G02GCF errors Complement of S15ADF error function erfc x Error S15AEF function erf x Return or set unit number for X04AAF error messages Computes G02DNF estimable function of a general linear regression model and ... Univariate time series, preliminary G13ADF estimation , seasonal ARIMA model Univariate time series, G13AFF estimation , seasonal ARIMA model Multivariate time series, preliminary G13BDF estimation of transfer function model Multivariate time series, G13BEF estimation of multi-input model Compute F06EJF Euclidean norm of real vector (DNRM2) Compute F06JJF Euclidean norm of complex vector (DZNRM2) Evaluation E02AEF of fitted polynomial in one variable from ... Evaluation E02AKF of fitted polynomial in one variable, from ... Evaluation E02BBF of fitted cubic spline, function only Evaluation E02BCF of fitted cubic spline, function and derivatives Evaluation E02BDF of fitted cubic spline, definite integral Evaluation E02DEF of a fitted bicubic spline at a vector of points Evaluation E02DFF of a fitted bicubic spline at a mesh of points 2 exact test Computes the G08AJF exact probabilities for the Mann-Whitney U statistic, no ... Computes the G08AKF exact probabilities for the Mann-Whitney U statistic, ties .. exponential distribution Complex S01EAF exponential z , e Exponential S13AAF integral E (x) 1 Computes a five-point summary (median, hinges and G01ALF extremes) Computes probabilities for G01EDF F -distribution Computes deviates for the G01FDF F -distribution LU F01BRF factorization of real sparse matrix LU F01BSF factorization of real sparse matrix with known sparsity pattern T LL F01MAF factorization of real sparse symmetric positive-definite matrix T LDL F01MCF factorization of real symmetric positive-definite ... QR F01QCF factorization of real m by n matrix (m>=n) T factorization by F01QCF factorization by F01QCF QR F01RCF factorization of complex m by n matrix (m>=n) H factorization by F01RCF factorization by F01RCF LU F07ADF factorization of real m by n matrix (DGETRF) Cholesky F07FDF factorization of real symmetric positive-definite matrix ... Multivariate time series, G13BAF filtering (pre-whitening) by an ARIMA model 1-D quadrature, adaptive, D01AJF finite interval, strategy due to Piessens and de Doncker, ... 1-D quadrature, adaptive, D01AKF finite interval, method suitable for oscillating functions 1-D quadrature, adaptive, D01ALF finite interval, allowing for singularities at user-specified 1-D quadrature, adaptive, D01ANF finite interval, weight function cos((omega)x) or sin... 1-D quadrature, adaptive, D01APF finite interval, weight function with end-point ... 1-D quadrature, adaptive, D01AQF finite interval, weight function 1/(x-c), Cauchy ... ODEs, boundary value problem, D02GAF finite difference technique with deferred correction, simple . ODEs, boundary value problem, D02GBF finite difference technique with deferred correction, general finite/infinite range, eigenvalue and eigenfunction, ... ODEs, general nonlinear boundary value problem, D02RAF finite difference technique with deferred correction, ... Elliptic PDE, solution of D03EDF finite difference equations by a multigrid technique 2 Fisher's exact test Interpolating functions, E01DAF fitting bicubic spline, data on rectangular grid Least-squares curve E02ADF fit , by polynomials, arbitrary data points Evaluation of E02AEF fitted polynomial in one variable from Chebyshev series form . Least-squares polynomial E02AGF fit , values and derivatives may be constrained, arbitrary Derivative of E02AHF fitted polynomial in Chebyshev series form Integral of E02AJF fitted polynomial in Chebyshev series form Evaluation of E02AKF fitted polynomial in one variable, from Chebyshev series form Least-squares curve cubic spline E02BAF fit (including interpolation) Evaluation of E02BBF fitted cubic spline, function only Evaluation of E02BCF fitted cubic spline, function and derivatives Evaluation of E02BDF fitted cubic spline, definite integral Least-squares cubic spline curve E02BEF fit , automatic knot placement Least-squares surface E02DAF fit , bicubic splines Least-squares surface E02DCF fit by bicubic splines with automatic knot placement, data on ... Least-squares surface E02DDF fit by bicubic splines with automatic knot placement, ... Evaluation of a E02DEF fitted bicubic spline at a vector of points Evaluation of a E02DFF fitted bicubic spline at a mesh of points Sort 2-D data into panels for E02ZAF fitting bicubic splines Fits G02DAF a general (multiple) linear regression model Fits G02DGF a general linear regression model for new dependent ... Fits G02GBF a generalized linear model with binomial errors Fits G02GCF a generalized linear model with Poisson errors 2 Performs the (chi) goodness of G08CGF fit test, for standard continuous distributions Goodness of G08 fit tests Computes a G01ALF five-point summary (median, hinges and extremes) Safe range of X02AMF floating-point arithmetic Safe range of complex X02ANF floating-point arithmetic Parameter of X02BHF floating-point arithmetic model, b Parameter of X02BJF floating-point arithmetic model, p Parameter of X02BKF floating-point arithmetic model, e min Parameter of X02BLF floating-point arithmetic model, e max Parameter of X02DJF floating-point arithmetic model, ROUNDS Univariate time series, update state set for G13AGF forecasting Univariate time series, G13AHF forecasting from state set Univariate time series, state set and G13AJF forecasts , from fully specified seasonal ARIMA model Multivariate time series, state set and G13BJF forecasts from fully specified multi-input model Single 1-D real discrete C06EAF Fourier transform, no extra workspace Single 1-D Hermitian discrete C06EBF Fourier transform, no extra workspace Single 1-D complex discrete C06ECF Fourier transform, no extra workspace Multiple 1-D real discrete C06FPF Fourier transforms Multiple 1-D Hermitian discrete C06FQF Fourier transforms Multiple 1-D complex discrete C06FRF Fourier transforms 2-D complex discrete C06FUF Fourier transform frequency table Frequency G01AEF table from raw data frequency (Daniell) window frequency (Daniell) window Fresnel S20ACF integral S(x) Fresnel S20ADF integral C(x) Friedman G08AEF two-way analysis of variance on k matched samples Computes probabilities for the G01EFF gamma distribution Computes deviates for the G01FFF gamma distribution Generates a vector of pseudo-random numbers from a G05FFF gamma distribution Gamma S14AAF function Log S14ABF Gamma function Incomplete S14BAF gamma functions P(a,x) and Q(a,x) Unconstrained minimum of a sum of squares, combined E04FDF Gauss-Newton and modified Newton algorithm using function ... Unconstrained minimum of a sum of squares, combined E04GCF Gauss-Newton and quasi-Newton algorithm, using 1st derivatives Pre-computed weights and abscissae for D01BBF Gaussian quadrature rules, restricted choice of rule All eigenvalues of F02ADF generalized real eigenproblem of the form Ax=(lambda)Bx where ... All eigenvalues and eigenvectors of F02AEF generalized real eigenproblem of the form Ax=(lambda)Bx where ... All eigenvalues and optionally eigenvectors of F02BJF generalized eigenproblem by QZ algorithm, real matrices Fits a G02GBF generalized linear model with binomial errors Fits a G02GCF generalized linear model with Poisson errors Computes orthogonal rotations for loading matrix, G03BAF generalized orthomax criterion Generate F06AAF real plane rotation (DROTG) Initialise random number G05CBF generating routines to give repeatable sequence Initialise random number G05CCF generating routines to give non-repeatable sequence Save state of random number G05CFF generating routines Restore state of random number G05CGF generating routines Set up reference vector for G05ECF generating pseudo-random integers, Poisson distribution Set up reference vector for G05EDF generating pseudo-random integers, binomial distribution Generates G05FAF a vector of random numbers from a uniform ... Generates G05FBF a vector of random numbers from an (negative) ... Generates G05FDF a vector of random numbers from a Normal distribution Generates G05FEF a vector of pseudo-random numbers from a beta ... Generates G05FFF a vector of pseudo-random numbers from a gamma ... Generates G05HDF a realisation of a multivariate time series from a ... Gill-Miller method 2 Performs the (chi) G08CGF goodness of fit test, for standard continuous distributions Goodness G08 of fit tests Unconstrained minimum, pre-conditioned conjugate E04DGF gradient algorithm, function of several variables using 1st ... Hankel S17DLF (j) functions H (z), j=1,2, ... (nu)+n Elliptic PDE, D03FAF Helmholtz equation, 3-D Cartesian co-ordinates Hermite , one variable Single 1-D C06EBF Hermitian discrete Fourier transform, no extra workspace Multiple 1-D C06FQF Hermitian discrete Fourier transforms Complex conjugate of C06GBF Hermitian sequence Complex conjugate of multiple C06GQF Hermitian sequences Convert C06GSF Hermitian sequences to general complex sequences All eigenvalues of complex F02AWF Hermitian matrix All eigenvalues and eigenvectors of complex F02AXF Hermitian matrix Matrix-vector product, complex F06SCF Hermitian matrix (ZHEMV) Matrix-vector product, complex F06SDF Hermitian band matrix (ZHBMV) Matrix-vector product, complex F06SEF Hermitian packed matrix (ZHPMV) Rank-1 update, complex F06SPF Hermitian matrix (ZHER) Rank-1 update, complex F06SQF Hermitian packed matrix (ZHPR) Rank-2 update, complex F06SRF Hermitian matrix (ZHER2) Rank-2 update, complex F06SSF Hermitian packed matrix (ZHPR2) Matrix-matrix product, one complex F06ZCF Hermitian matrix, one complex rectangular matrix (ZHEMM) Rank-k update of a complex F06ZPF Hermitian matrix (ZHERK) Rank-2k update of a complex F06ZRF Hermitian matrix (ZHER2K) Hilbert transform) Computes a five-point summary (median, G01ALF hinges and extremes) Multi-dimensional adaptive quadrature over D01FCF hyper-rectangle Multi-dimensional quadrature over D01GBF hyper-rectangle , Monte Carlo method Incomplete S14BAF gamma functions P(a,x) and Q(a,x) Index F06JLF , real vector element with largest absolute value (IDAMAX) Index F06JMF , complex vector element with largest absolute value .. 1-D quadrature, adaptive, D01AMF infinite or semi-infinite interval 1-D quadrature, adaptive, infinite or semi- D01AMF infinite interval 1-D quadrature, adaptive, semi- D01ASF infinite interval, weight function cos((omega)x) or ... infinite range, eigenvalue and eigenfunction, user-specified ... Calculates standardized residuals and G02FAF influence statistics Initialise G05CBF random number generating routines to give ... Initialise G05CCF random number generating routines to give ... input data input data Multivariate time series, estimation of multi- G13BEF input model input model Input/output X04 utilities Pseudo-random G05DYF integer from uniform distribution Set up reference vector for generating pseudo-random G05ECF integers , Poisson distribution Set up reference vector for generating pseudo-random G05EDF integers , binomial distribution Pseudo-random permutation of an G05EHF integer vector Pseudo-random sample from an G05EJF integer vector Pseudo-random G05EYF integer from reference vector Largest representable X02BBF integer Return date and time as an array of X05AAF integers Convert array of X05ABF integers representing date and time to character string integral , one variable Integral E02AJF of fitted polynomial in Chebyshev series form Evaluation of fitted cubic spline, definite E02BDF integral Exponential S13AAF integral E (x) 1 Cosine S13ACF integral Ci(x) Sine S13ADF integral Si(x) Fresnel S20ACF integral S(x) Fresnel S20ADF integral C(x) Degenerate symmetrised elliptic S21BAF integral of 1st kind R (x,y) C Symmetrised elliptic S21BBF integral of 1st kind R (x,y,z) F Symmetrised elliptic S21BCF integral of 2nd kind R (x,y,z) D Symmetrised elliptic S21BDF integral of 3rd kind R (x,y,z,r) J 1-D quadrature, D01GAF integration of function defined by data values, Gill-Miller ... Numerical D01 integration Interpolating E01BAF functions, cubic spline interpolant, one variable Interpolating functions, cubic spline E01BAF interpolant , one variable Interpolating E01BEF functions, monotonicity-preserving, piecewise ... Interpolated E01BFF values, interpolant computed by E01BEF, function ... Interpolated values, E01BFF interpolant computed by E01BEF, function only, one variable Interpolated E01BGF values, interpolant computed by E01BEF, function ... Interpolated values, E01BGF interpolant computed by E01BEF, function and 1st derivative, ... Interpolated E01BHF values, interpolant computed by E01BEF, definite ... Interpolated values, E01BHF interpolant computed by E01BEF, definite integral, one variable Interpolating E01DAF functions, fitting bicubic spline, data on ... Interpolating E01SAF functions, method of Renka and Cline, two ... Interpolating E01SEF functions, modified Shepard's method, two ... Least-squares curve cubic spline fit (including E02BAF interpolation) Inverse G01F distributions Invert M01ZAF a permutation iterative refinement iterative refinement ODEs, D02BBF IVP , Runge-Kutta-Merson method, over a range, intermediate ODEs, D02BHF IVP , Runge-Kutta-Merson method, until function of solution is ... ODEs, D02CJF IVP , Adams method, until function of solution is zero, ... ODEs, stiff D02EJF IVP , BDF method, until function of solution is zero, ... Kelvin function S19ADF kei x Kelvin S19AAF function ber x Kelvin S19ABF function bei x Kelvin S19ACF function ker x Kelvin S19ADF function kei x Kendall/Spearman G02BNF non-parametric rank correlation ... Kendall/Spearman G02BQF non-parametric rank correlation ... Kelvin function S19ACF ker x Least-squares cubic spline curve fit, automatic E02BEF knot placement knot placement, data on rectangular grid knot placement, scattered data Kruskal-Wallis G08AFF one-way analysis of variance on k samples of ... Mean, variance, skewness, G01AAF kurtosis etc, one variable, from raw data Mean, variance, skewness, G01ADF kurtosis etc, one variable, from frequency table All zeros of complex polynomial, modified C02AFF Laguerre method All zeros of real polynomial, modified C02AGF Laguerre method Index, real vector element with F06JLF largest absolute value (IDAMAX) Index, complex vector element with F06JMF largest absolute value (IZAMAX) Largest X02ALF positive model number Largest X02BBF representable integer LDL F01MCF T factorization of real symmetric positive-definite ... Constructs a stem and G01ARF leaf plot Least-squares E02ADF curve fit, by polynomials, arbitrary data points Least-squares E02AGF polynomial fit, values and derivatives may be ... Least-squares E02BAF curve cubic spline fit (including interpolation) Least-squares E02BEF cubic spline curve fit, automatic knot placement Least-squares E02DAF surface fit, bicubic splines Least-squares E02DCF surface fit by bicubic splines with automatic ... Least-squares E02DDF surface fit by bicubic splines with automatic ... Covariance matrix for nonlinear E04YCF least-squares problem Least-squares F04JGF (if rank=n) or minimal least-squares (if ... Least-squares (if rank=n) or minimal F04JGF least-squares (if rank=n Rank-1 F06PMF update, real rectangular matrix (DGER) Rank-1 F06PPF update, real symmetric matrix (DSYR) Rank-1 F06PQF update, real symmetric packed matrix (DSPR) Rank-2 F06PRF update, real symmetric matrix (DSYR2) Rank-2 F06PSF update, real symmetric packed matrix (DSPR2) Rank-1 F06SMF update, complex rectangular matrix, unconjugated ... Rank-1 F06SNF update, complex rectangular matrix, conjugated vector . Rank-1 F06SPF update, complex Hermitian matrix (ZHER) Rank-1 F06SQF update, complex Hermitian packed matrix (ZHPR) Rank-2 F06SRF update, complex Hermitian matrix (ZHER2) Rank-2 F06SSF update, complex Hermitian packed matrix (ZHPR2) Rank-k F06YPF update of a real symmetric matrix (DSYRK) Rank-2k F06YRF update of a real symmetric matrix (DSYR2K) Rank-k F06ZPF update of a complex Hermitian matrix (ZHERK) Rank-2k F06ZRF update of a complex Hermitian matrix (ZHER2K) Rank-k F06ZUF update of a complex symmetric matrix (ZSYRK) Rank-2k F06ZWF update of a complex symmetric matrix (ZHER2K) Kendall/Spearman non-parametric G02BNF rank correlation coefficients, no missing values, overwriting ... Kendall/Spearman non-parametric G02BQF rank correlation coefficients, no missing values, preserving rank test Rank M01DAF a vector, real numbers Rank M01DEF rows of a matrix, real numbers Rank M01DJF columns of a matrix, real numbers Rearrange a vector according to given M01EAF ranks , real numbers Generates a G05HDF realisation of a multivariate time series from a VARMA model Rearrange M01EAF a vector according to given ranks, real numbers Multi-dimensional adaptive quadrature over hyper- D01FCF rectangle Multi-dimensional quadrature over hyper- D01GBF rectangle , Monte Carlo method Discretize a 2nd order elliptic PDE on a D03EEF rectangle rectangular grid rectangular grid Matrix-vector product, real F06PAF rectangular matrix (DGEMV) Matrix-vector product, real F06PBF rectangular band matrix (DGBMV) Rank-1 update, real F06PMF rectangular matrix (DGER) Matrix-vector product, complex F06SAF rectangular matrix (ZGEMV) Matrix-vector product, complex F06SBF rectangular band matrix (ZGBMV) Rank-1 update, complex F06SMF rectangular matrix, unconjugated vector (ZGERU) Rank-1 update, complex F06SNF rectangular matrix, conjugated vector (ZGERC) Matrix-matrix product, two real F06YAF rectangular matrices (DGEMM) rectangular matrix (DSYMM) rectangular matrix (DTRMM) Matrix-matrix product, two complex F06ZAF rectangular matrices (ZGEMM) rectangular matrix (ZHEMM) rectangular matrix (ZTRMM) rectangular matrix (ZSYMM) Set up G05EAF reference vector for multivariate Normal distribution Set up G05ECF reference vector for generating pseudo-random integers, ... Set up G05EDF reference vector for generating pseudo-random integers, ... Set up G05EXF reference vector from supplied cumulative distribution ... Pseudo-random integer from G05EYF reference vector Pseudo-random multivariate Normal vector from G05EZF reference vector refinement refinement Simple linear G02CAF regression with constant term, no missing values Fits a general (multiple) linear G02DAF regression model Fits a general linear G02DGF regression model for new dependent variable Computes estimable function of a general linear G02DNF regression model and its standard error Nonlinear E04 regression 2nd order Sturm-Liouville problem, D02KEF regular/singular system, finite/infinite range, eigenvalue ... Interpolating functions, method of E01SAF Renka and Cline, two variables Calculates standardized G02FAF residuals and influence statistics Univariate time series, diagnostic checking of G13ASF residuals , following G13AFF right-hand sides Solution of real simultaneous linear equations, one F04ARF right-hand side right-hand side using iterative refinement Solution of real simultaneous linear equations, one F04ATF right-hand side using iterative refinement right-hand side Solves a system of equations with multiple F06YJF right-hand sides, real triangular coefficient matrix (DTRSM) Solves system of equations with multiple F06ZJF right-hand sides, complex triangular coefficient matrix (ZTRSM) Solution of real system of linear equations, multiple F07AEF right-hand sides, matrix already factorized by F07ADF (DGETRS) right-hand sides, matrix already factorized by F07FDF (DPOTRS) Generate real plane F06AAF rotation (DROTG) Apply real plane F06EPF rotation (DROT) Computes orthogonal G03BAF rotations for loading matrix, generalized orthomax criterion rules , restricted choice of rule rule ODEs, IVP, D02BBF Runge-Kutta-Merson method, over a range, intermediate output ODEs, IVP, D02BHF Runge-Kutta-Merson method, until function of solution is zero ... Safe X02AMF range of floating-point arithmetic Safe X02ANF range of complex floating-point arithmetic Pseudo-random G05EJF sample from an integer vector Sign test on two paired G08AAF samples Median test on two G08ACF samples of unequal size Friedman two-way analysis of variance on k matched G08AEF samples Kruskal-Wallis one-way analysis of variance on k G08AFF samples of unequal size Performs the Wilcoxon one- G08AGF sample (matched pairs) signed rank test Performs the Mann-Whitney U test on two independent G08AHF samples sample sample Univariate time series, G13ABF sample autocorrelation function Univariate time series, smoothed G13CBF sample spectrum using spectral smoothing by the trapezium ... Multivariate time series, smoothed G13CDF sample cross spectrum using spectral smoothing by the ... Add F06ECF scalar times real vector to real vector (DAXPY) Multiply real vector by F06EDF scalar (DSCAL) Add F06GCF scalar times complex vector to complex vector (ZAXPY) Multiply complex vector by complex F06GDF scalar (ZSCAL) Multiply complex vector by real F06JDF scalar (ZDSCAL) scattered data Univariate time series, G13AAF seasonal and non-seasonal differencing Univariate time series, seasonal and non- G13AAF seasonal differencing Univariate time series, preliminary estimation, G13ADF seasonal ARIMA model Univariate time series, estimation, G13AFF seasonal ARIMA model seasonal ARIMA model 1-D quadrature, adaptive, infinite or D01AMF semi-infinite interval 1-D quadrature, adaptive, D01ASF semi-infinite interval, weight function cos((omega)x) ... Complex conjugate of Hermitian C06GBF sequence Complex conjugate of complex C06GCF sequence Complex conjugate of multiple Hermitian C06GQF sequences Convert Hermitian C06GSF sequences to general complex sequences Convert Hermitian sequences to general complex C06GSF sequences sequence sequence Minimum, function of several variables, E04UCF sequential QP method, nonlinear constraints, using function ... Interpolating functions, modified E01SEF Shepard's method, two variables Sign G08AAF test on two paired samples Performs the Wilcoxon one-sample (matched pairs) G08AGF signed rank test Solution of complex F04ADF simultaneous linear equations with multiple right-hand sides Solution of real F04ARF simultaneous linear equations, one right-hand side Solution of real symmetric positive-definite F04ASF simultaneous linear equations, one right-hand side using ... Solution of real F04ATF simultaneous linear equations, one right-hand side using ... Solution of real sparse F04AXF simultaneous linear equations (coefficient matrix already ... simultaneous linear equations, one right-hand side Real sparse symmetric positive-definite F04MAF simultaneous linear equations (coefficient matrix already ... Real sparse symmetric F04MBF simultaneous linear equations simultaneous linear equations (coefficient matrix already ... sin ((omega)x) sin ((omega)x) Sine S13ADF integral Si(x) 2nd order Sturm-Liouville problem, regular/ D02KEF singular system, finite/infinite range, eigenvalue and ... singularities at user-specified break-points singularities of algebraico-logarithmic type Mean, variance, G01AAF skewness , kurtosis etc, one variable, from raw data Mean, variance, G01ADF skewness , kurtosis etc, one variable, from frequency table Smallest X02AKF positive model number Univariate time series, G13CBF smoothed sample spectrum using spectral smoothing by the ... smoothing by the trapezium frequency (Daniell) window Multivariate time series, G13CDF smoothed sample cross spectrum using spectral smoothing by ... smoothing by the trapezium frequency (Daniell) window Sort E02ZAF 2-D data into panels for fitting bicubic splines Sort M01CAF a vector, real numbers LU factorization of real F01BRF sparse matrix LU factorization of real F01BSF sparse matrix with known sparsity pattern LU factorization of real sparse matrix with known F01BSF sparsity pattern T LL factorization of real F01MAF sparse symmetric positive-definite matrix Selected eigenvalues and eigenvectors of F02FJF sparse symmetric eigenproblem Solution of real F04AXF sparse simultaneous linear equations (coefficient matrix ... Real F04MAF sparse symmetric positive-definite simultaneous linear ... Real F04MBF sparse symmetric simultaneous linear equations Sparse F04QAF linear least-squares problem, m real equations in ... Kendall/ G02BNF Spearman non-parametric rank correlation coefficients, no ... Kendall/ G02BQF Spearman non-parametric rank correlation coefficients, no ... Approximation of S special functions Univariate time series, smoothed sample G13CBF spectrum using spectral smoothing by the trapezium frequency ... spectral smoothing by the trapezium frequency (Daniell) window Multivariate time series, smoothed sample cross G13CDF spectrum using spectral smoothing by the trapezium frequency ... spectral smoothing by the trapezium frequency (Daniell) window Interpolating functions, cubic E01BAF spline interpolant, one variable Interpolating functions, fitting bicubic E01DAF spline , data on rectangular grid Least-squares curve cubic E02BAF spline fit (including interpolation) Evaluation of fitted cubic E02BBF spline , function only Evaluation of fitted cubic E02BCF spline , function and derivatives Evaluation of fitted cubic E02BDF spline , definite integral Least-squares cubic E02BEF spline curve fit, automatic knot placement Least-squares surface fit, bicubic E02DAF splines Least-squares surface fit by bicubic E02DCF splines with automatic knot placement, data on rectangular grid Least-squares surface fit by bicubic E02DDF splines with automatic knot placement, scattered data Evaluation of a fitted bicubic E02DEF spline at a vector of points Evaluation of a fitted bicubic E02DFF spline at a mesh of points Sort 2-D data into panels for fitting bicubic E02ZAF splines B- E02 splines Least- E02ADF squares curve fit, by polynomials, arbitrary data points Least- E02AGF squares polynomial fit, values and derivatives may be ... Least- E02BAF squares curve cubic spline fit (including interpolation) Least- E02BEF squares cubic spline curve fit, automatic knot placement Least- E02DAF squares surface fit, bicubic splines Least- E02DCF squares surface fit by bicubic splines with automatic knot ... Least- E02DDF squares surface fit by bicubic splines with automatic knot ... Unconstrained minimum of a sum of E04FDF squares , combined Gauss-Newton and modified Newton algorithm . Unconstrained minimum of a sum of E04GCF squares , combined Gauss-Newton and quasi-Newton algorithm, ... Covariance matrix for nonlinear least- E04YCF squares problem Least- F04JGF squares (if rank=n) or minimal least-squares (if ... Least-squares (if rank=n) or minimal least- F04JGF squares (if rank