Unitary reduction of complex general matrix to upper Hessenberg form

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))
    - f08 (Least-squares and Eigenvalue Problems (LAPACK))
      - f08 Introduction
      - f08aec (nag_dgeqrf)
      - f08afc (nag_dorgqr)
      - f08agc (nag_dormqr)
      - f08ahc (nag_dgelqf)
      - f08ajc (nag_dorglq)
      - f08akc (nag_dormlq)
      - f08asc (nag_zgeqrf)
      - f08atc (nag_zungqr)
      - f08auc (nag_zunmqr)
      - f08avc (nag_zgelqf)
      - f08awc (nag_zunglq)
      - f08axc (nag_zunmlq)
      - f08bec (nag_dgeqpf)
      - f08bsc (nag_zgeqpf)
      - f08fcc (nag_dsyevd)
      - f08fec (nag_dsytrd)
      - f08ffc (nag_dorgtr)
      - f08fgc (nag_dormtr)
      - f08fqc (nag_zheevd)
      - f08fsc (nag_zhetrd)
      - f08ftc (nag_zungtr)
      - f08fuc (nag_zunmtr)
      - f08gcc (nag_dspevd)
      - f08gec (nag_dsptrd)
      - f08gfc (nag_dopgtr)
      - f08ggc (nag_dopmtr)
      - f08gqc (nag_zhpevd)
      - f08gsc (nag_zhptrd)
      - f08gtc (nag_zupgtr)
      - f08guc (nag_zupmtr)
      - f08hcc (nag_dsbevd)
      - f08hec (nag_dsbtrd)
      - f08hqc (nag_zhbevd)
      - f08hsc (nag_zhbtrd)
      - f08jcc (nag_dstevd)
      - f08jec (nag_dsteqr)
      - f08jfc (nag_dsterf)
      - f08jgc (nag_dpteqr)
      - f08jjc (nag_dstebz)
      - f08jkc (nag_dstein)
      - f08jsc (nag_zsteqr)
      - f08juc (nag_zpteqr)
      - f08jxc (nag_zstein)
      - f08kec (nag_dgebrd)
      - f08kfc (nag_dorgbr)
      - f08kgc (nag_dormbr)
      - f08ksc (nag_zgebrd)
      - f08ktc (nag_zungbr)
      - f08kuc (nag_zunmbr)
      - f08lec (nag_dgbbrd)
      - f08lsc (nag_zgbbrd)
      - f08mec (nag_dbdsqr)
      - f08msc (nag_zbdsqr)
      - f08nec (nag_dgehrd)
      - f08nfc (nag_dorghr)
      - f08ngc (nag_dormhr)
      - f08nhc (nag_dgebal)
      - f08njc (nag_dgebak)
      - f08nsc (nag_zgehrd)
      - f08ntc (nag_zunghr)
      - f08nuc (nag_zunmhr)
      - f08nvc (nag_zgebal)
      - f08nwc (nag_zgebak)
      - f08pec (nag_dhseqr)
      - f08pkc (nag_dhsein)
      - f08psc (nag_zhseqr)
      - f08pxc (nag_zhsein)
      - f08qfc (nag_dtrexc)
      - f08qgc (nag_dtrsen)
      - f08qhc (nag_dtrsyl)
      - f08qkc (nag_dtrevc)
      - f08qlc (nag_dtrsna)
      - f08qtc (nag_ztrexc)
      - f08quc (nag_ztrsen)
      - f08qvc (nag_ztrsyl)
      - f08qxc (nag_ztrevc)
      - f08qyc (nag_ztrsna)
      - f08sec (nag_dsygst)
      - f08ssc (nag_zhegst)
      - f08tec (nag_dspgst)
      - f08tsc (nag_zhpgst)
      - f08uec (nag_dsbgst)
      - f08ufc (nag_dpbstf)
      - f08usc (nag_zhbgst)
      - f08utc (nag_zpbstf)
      - f08wec (nag_dgghrd)
      - f08whc (nag_dggbal)
      - f08wjc (nag_dggbak)
      - f08wsc (nag_zgghrd)
      - f08wvc (nag_zggbal)
      - f08wwc (nag_zggbak)
      - f08xec (nag_dhgeqz)
      - f08xsc (nag_zhgeqz)
      - f08ykc (nag_dtgevc)
      - f08yxc (nag_ztgevc)
    - 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[f08nsc] NAG[nag_zgehrd] - Unitary reduction of complex general matrix to upper Hessenberg form

Calling Sequence

f08nsc(ilo, ihi, a, tau, 'n'=n, 'fail'=fail)

nag_zgehrd(. . .)

Parameters

ilo - integer;
ihi - integer;

On entry: if has been output by f08nvc (nag_zgebal), then ilo and ihi must contain the values returned by that function. Otherwise, ilo must be set to 1 and ihi to n.

Constraints:

if , ;

if , and .

a - Matrix(1..dim1, 1..dim2, datatype=complex[8], order=order);

Note: this array may be supplied in Fortran_order or C_order , as specified by order. All array parameters must use a consistent order.

On entry: the by general matrix .

On exit: is overwritten by the upper Hessenberg matrix and details of the unitary matrix . The subdiagonal elements of are real.

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

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

On exit: further details of the unitary matrix .

'n'=n - integer; (optional)

Default value: the dimension of the array a.

On entry: , the order of the matrix .

Constraint: . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_zgehrd (f08nsc) reduces a complex general matrix to Hessenberg form.

Description

nag_zgehrd (f08nsc) reduces a complex general matrix to upper Hessenberg form by a unitary similarity transformation: . has real subdiagonal elements.

The matrix is not formed explicitly, but is represented as a product of elementary reflectors (see the f08 Chapter Introduction for details). Functions are provided to work with in this representation (see Section [Further Comments]).

The function can take advantage of a previous call to f08nvc (nag_zgebal), which may produce a matrix with the structure:

([[A[11],A[12],A[13]],[,A[22],A[23]],[,,A[33]]])

where and are upper triangular. If so, only the central diagonal block , in rows and columns to , needs to be reduced to Hessenberg form (the blocks and will also be affected by the reduction). Therefore the values of and determined by f08nvc (nag_zgebal) can be supplied to the function directly. If f08nvc (nag_zgebal) has not previously been called however, then must be set to 1 and to .

Error Indicators and Warnings

"NE_ALLOC_FAIL"

Dynamic memory allocation failed.

"NE_BAD_PARAM"

On entry, argument had an illegal value.

"NE_INT"

On entry, . Constraint: .

"NE_INT_3"

On entry, , , . Constraint: if , and .

On entry, , , . Constraint: if , .

"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.

	Accuracy
	The computed Hessenberg matrix is exactly similar to a nearby matrix , where is a modestly increasing function of , and is the machine precision. The elements of themselves may be sensitive to small perturbations in or to rounding errors in the computation, but this does not affect the stability of the eigenvalues, eigenvectors or Schur factorization.

Further Comments

The total number of real floating-point operations is approximately , where ; if and , the number is approximately .

To form the unitary matrix nag_zgehrd (f08nsc) may be followed by a call to f08ntc (nag_zunghr):

nag_zunghr (ilo,ihi,a,tau,'n'=n)

To apply to an by complex matrix nag_zgehrd (f08nsc) may be followed by a call to f08nuc (nag_zunmhr). For example,

nag_zunmhr ("Nag_LeftSide","Nag_NoTrans",ilo,ihi,a,tau,c,'m'=m,'n'=n)

forms the matrix product .

The real analogue of this function is f08nec (nag_dgehrd).

Examples

n := 4:
ilo := 1:
ihi := 4:
a := Matrix([[-3.97 -5.04*I , -4.11 +3.7*I , -0.34 +1.01*I , 1.29 -0.86*I ], [0.34 -1.5*I , 1.52 -0.43*I , 1.88 -5.38*I , 3.36 +0.65*I ], [3.31 -3.85*I , 2.5 +3.45*I , 0.88 -1.08*I , 0.64 -1.48*I ], [-1.1 +0.82*I , 1.81 -1.59*I , 3.25 +1.33*I , 1.57 -3.44*I ]], datatype=complex[8]):
tau := Vector(3, datatype=complex[8]):
NAG:-f08nsc(ilo, ihi, a, tau, 'n' = n):

tau;

	See Also
	Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore f08 Chapter Introduction. NAG Toolbox Overview. NAG Web Site.