Bunch–Kaufman factorization of complex symmetric matrix, packed storage

はじめる前に
新機能一覧
Maple ワークシートの作成
Mapleワークシートを共有
Maple ウィンドウのカスタマイズ
Maple システムのカスタマイズ
コネクティビティ
Mathematics
数学
物理
プログラミング
グラフィックス
Student パッケージ
Science and Engineering
科学・エンジニアリング
アプリケーションと例題
リファレンス
システム
MapleSim
MapleSim ツールボックス
MapleSim ヘルプ
Tasks
Toolboxes
- BlockImporter
- Global Optimization
- Grid Computing
- Maple-NAG Connector
  - General information
  - Computational Routines
    - a02 (Complex Arithmetic)
    - c02 (Zeros of Polynomials)
    - c05 (Roots of One or More Transcendental Equations)
    - c06 (Fourier Transforms)
    - d01 (Quadrature)
    - d02 (Ordinary Differential Equations)
    - d03 (Partial Differential Equations)
    - d06 (Mesh Generation)
    - e01 (Interpolation)
    - e02 (Curve and Surface Fitting)
    - e04 (Minimizing or Maximizing a Function)
    - f01 (Matrix Factorizations)
    - f02 (Eigenvalues and Eigenvectors)
    - f03 (Determinants)
    - f04 (Simultaneous Linear Equations)
    - f06 (Linear Algebra Support Functions)
    - f07 (Linear Equations (LAPACK))
      - f07 Introduction
      - f07adc (nag_dgetrf)
      - f07aec (nag_dgetrs)
      - f07agc (nag_dgecon)
      - f07ahc (nag_dgerfs)
      - f07ajc (nag_dgetri)
      - f07arc (nag_zgetrf)
      - f07asc (nag_zgetrs)
      - f07auc (nag_zgecon)
      - f07avc (nag_zgerfs)
      - f07awc (nag_zgetri)
      - f07bdc (nag_dgbtrf)
      - f07bec (nag_dgbtrs)
      - f07bgc (nag_dgbcon)
      - f07bhc (nag_dgbrfs)
      - f07brc (nag_zgbtrf)
      - f07bsc (nag_zgbtrs)
      - f07buc (nag_zgbcon)
      - f07bvc (nag_zgbrfs)
      - f07fdc (nag_dpotrf)
      - f07fec (nag_dpotrs)
      - f07fgc (nag_dpocon)
      - f07fhc (nag_dporfs)
      - f07fjc (nag_dpotri)
      - f07frc (nag_zpotrf)
      - f07fsc (nag_zpotrs)
      - f07fuc (nag_zpocon)
      - f07fvc (nag_zporfs)
      - f07fwc (nag_zpotri)
      - f07gdc (nag_dpptrf)
      - f07gec (nag_dpptrs)
      - f07ggc (nag_dppcon)
      - f07ghc (nag_dpprfs)
      - f07gjc (nag_dpptri)
      - f07grc (nag_zpptrf)
      - f07gsc (nag_zpptrs)
      - f07guc (nag_zppcon)
      - f07gvc (nag_zpprfs)
      - f07gwc (nag_zpptri)
      - f07hdc (nag_dpbtrf)
      - f07hec (nag_dpbtrs)
      - f07hgc (nag_dpbcon)
      - f07hhc (nag_dpbrfs)
      - f07hrc (nag_zpbtrf)
      - f07hsc (nag_zpbtrs)
      - f07huc (nag_zpbcon)
      - f07hvc (nag_zpbrfs)
      - f07mdc (nag_dsytrf)
      - f07mec (nag_dsytrs)
      - f07mgc (nag_dsycon)
      - f07mhc (nag_dsyrfs)
      - f07mjc (nag_dsytri)
      - f07mrc (nag_zhetrf)
      - f07msc (nag_zhetrs)
      - f07muc (nag_zhecon)
      - f07mvc (nag_zherfs)
      - f07mwc (nag_zhetri)
      - f07nrc (nag_zsytrf)
      - f07nsc (nag_zsytrs)
      - f07nuc (nag_zsycon)
      - f07nvc (nag_zsyrfs)
      - f07nwc (nag_zsytri)
      - f07pdc (nag_dsptrf)
      - f07pec (nag_dsptrs)
      - f07pgc (nag_dspcon)
      - f07phc (nag_dsprfs)
      - f07pjc (nag_dsptri)
      - f07prc (nag_zhptrf)
      - f07psc (nag_zhptrs)
      - f07puc (nag_zhpcon)
      - f07pvc (nag_zhprfs)
      - f07pwc (nag_zhptri)
      - f07qrc (nag_zsptrf)
      - f07qsc (nag_zsptrs)
      - f07quc (nag_zspcon)
      - f07qvc (nag_zsprfs)
      - f07qwc (nag_zsptri)
      - f07tec (nag_dtrtrs)
      - f07tgc (nag_dtrcon)
      - f07thc (nag_dtrrfs)
      - f07tjc (nag_dtrtri)
      - f07tsc (nag_ztrtrs)
      - f07tuc (nag_ztrcon)
      - f07tvc (nag_ztrrfs)
      - f07twc (nag_ztrtri)
      - f07uec (nag_dtptrs)
      - f07ugc (nag_dtpcon)
      - f07uhc (nag_dtprfs)
      - f07ujc (nag_dtptri)
      - f07usc (nag_ztptrs)
      - f07uuc (nag_ztpcon)
      - f07uvc (nag_ztprfs)
      - f07uwc (nag_ztptri)
      - f07vec (nag_dtbtrs)
      - f07vgc (nag_dtbcon)
      - f07vhc (nag_dtbrfs)
      - f07vsc (nag_ztbtrs)
      - f07vuc (nag_ztbcon)
      - f07vvc (nag_ztbrfs)
    - f08 (Least-squares and Eigenvalue Problems (LAPACK))
    - f11 (Large Scale Linear Systems)
    - f16 (NAG Interface to BLAS)
    - g01 (Simple Calculations on Statistical Data)
    - g02 (Correlation and Regression Analysis)
    - g03 (Multivariate Methods)
    - g04 (Analysis of Variance)
    - g05 (Random Number Generators)
    - g07 (Univariate Estimation)
    - g08 (Nonparametric Statistics)
    - g10 (Smoothing in Statistics)
    - g11 (Contingency Table Analysis)
    - g12 (Survival Analysis)
    - g13 (Time Series Analysis)
    - h (Operations Research)
    - m01 (Sorting)
    - s (Approximations of Special Functions)
    - x01 (Mathematical Constants)
    - x02 (Machine Constants)
    - x04 (Input-Output Utilities)
  - Getting Started
  - Example Worksheet
エラーメッセージガイド
マニュアル
数学アプリ

NAG[f07qrc] NAG[nag_zsptrf] - Bunch–Kaufman factorization of complex symmetric matrix, packed storage

Calling Sequence

f07qrc(uplo, n, ap, ipiv, 'fail'=fail)

nag_zsptrf(. . .)

Parameters

uplo - String;

On entry: indicates whether the upper or lower triangular part of is stored and how is to be factorized.

The upper triangular part of is stored and is factorized as , where is upper triangular.

The lower triangular part of is stored and is factorized as , where is lower triangular.

Constraint: "Nag_Upper" or "Nag_Lower". .

n - integer;

On entry: , the order of the matrix .

Constraint: . .

ap - Vector(1..dim, datatype=complex[8]);

Note: the dimension, dim, of the array ap must be at least .

On entry: the by symmetric matrix , packed by rows or columns. The storage of elements depends on the storage order and uplo arguments as follows:

if and , is stored in , for ;

if and , is stored in , for .

On exit: is overwritten by details of the block diagonal matrix and the multipliers used to obtain the factor or as specified by uplo.

ipiv - Vector(1..dim, datatype=integer[kernelopts('wordsize')/8]);

Note: the dimension, dim, of the array ipiv must be at least .

On exit: details of the interchanges and the block structure of . More precisely,

if , is a by pivot block and the th row and column of were interchanged with the th row and column;

if and , is a by pivot block and the th row and column of were interchanged with the th row and column;

if and , $Matrix(2, 2, {(1, 1) = d[i^2], (1, 2) = d[i+1, i], (2, 1) = d[i+1, i], (2, 2) = d[i+1, i+1]})$ is a by pivot block and the th row and column of were interchanged with the th row and column.

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_zsptrf (f07qrc) computes the Bunch–Kaufman factorization of a complex symmetric matrix, using packed storage.

Description

nag_zsptrf (f07qrc) factorizes a complex symmetric matrix , using the Bunch–Kaufman diagonal pivoting method and packed storage. is factorized as either if or if , where is a permutation matrix, (or ) is a unit upper (or lower) triangular matrix and is a symmetric block diagonal matrix with 1 by 1 and 2 by 2 diagonal blocks; (or ) has 2 by 2 unit diagonal blocks corresponding to the 2 by 2 blocks of . Row and column interchanges are performed to ensure numerical stability while preserving symmetry.

	Error Indicators and Warnings
	"NE_BAD_PARAM" On entry, argument had an illegal value. "NE_INT" On entry, . Constraint: . "NE_INTERNAL_ERROR" An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please consult NAG for assistance. "NE_SINGULAR" The block diagonal matrix is exactly singular.

	Accuracy
	If , the computed factors and are the exact factors of a perturbed matrix , where is a modest linear function of , and is the machine precision. If , a similar statement holds for the computed factors and .

Further Comments

The elements of overwrite the corresponding elements of ; if has 2 by 2 blocks, only the upper or lower triangle is stored, as specified by uplo.

The unit diagonal elements of or and the 2 by 2 unit diagonal blocks are not stored. The remaining elements of or overwrite elements in the corresponding columns of , but additional row interchanges must be applied to recover or explicitly (this is seldom necessary). If , for , then or are stored explicitly in packed form (except for their unit diagonal elements which are equal to 1).

The total number of real floating-point operations is approximately .

A call to nag_zsptrf (f07qrc) may be followed by calls to the functions:

f07qsc (nag_zsptrs) to solve ;

f07quc (nag_zspcon) to estimate the condition number of ;

f07qwc (nag_zsptri) to compute the inverse of .

The real analogue of this function is f07pdc (nag_dsptrf).

Examples

uplo := "Nag_Upper":
n := 4:
ap := Vector([-0.39 -0.71*I, 5.14 -0.64*I, -7.86 -2.96*I, 3.8 +0.92*I, 8.859999999999999 +1.81*I, -3.52 +0.58*I, 5.32 -1.59*I, -2.83 -0.03*I, -1.54 -2.86*I, -0.5600000000000001 +0.12*I], datatype=complex[8], order='C_order'):
ipiv := Vector(4, datatype=integer[kernelopts('wordsize')/8]):
NAG:-f07qrc(uplo, n, ap, ipiv):

ap;

ipiv;

	See Also
	Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore f07 Chapter Introduction. NAG Toolbox Overview. NAG Web Site.