Left and right eigenvectors of real upper quasi-triangular matrix

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[f08qkc] NAG[nag_dtrevc] - Left and right eigenvectors of real upper quasi-triangular matrix

Calling Sequence

f08qkc(side, how_many, select, t, vl, vr, m, 'n'=n, 'mm'=mm, 'fail'=fail)

nag_dtrevc(. . .)

Parameters

side - String;

On entry: indicates whether left and/or right eigenvectors are to be computed.

Only right eigenvectors are computed.

Only left eigenvectors are computed.

Both left and right eigenvectors are computed.

Constraint: "Nag_RightSide", "Nag_LeftSide" or "Nag_BothSides". .

how_many - String;

On entry: indicates how many eigenvectors are to be computed.

All eigenvectors (as specified by side) are computed.

or "Nag_BackTransform" All eigenvectors (as specified by side) are computed and then pre-multiplied by the matrix (which is overwritten).

Selected eigenvectors (as specified by side and select) are computed.

Constraint: "Nag_ComputeAll", , "Nag_BackTransform" or "Nag_ComputeSelected". .

select - Vector(1..dim, datatype=boolean);

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

when ;

otherwise.

On entry: specifies which eigenvectors are to be computed if . To obtain the real eigenvector corresponding to the real eigenvalue , must be set true. To select the complex eigenvector corresponding to a complex conjugate pair of eigenvalues and , and/or must be set true; the eigenvector corresponding to the first eigenvalue in the pair is computed.

On exit: if a complex eigenvector was selected as specified above, then is set to true and to false.

If "Nag_ComputeAll", or "Nag_BackTransform", select is not referenced.

t - Matrix(1..dim1, 1..dim2, datatype=float[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 upper quasi-triangular matrix in canonical Schur form, as returned by f08pec (nag_dhseqr).

vl - Matrix(1..dim1, 1..dim2, datatype=float[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: if or "Nag_BackTransform" and "Nag_LeftSide" or "Nag_BothSides", vl must contain an by matrix (usually the matrix of Schur vectors returned by f08pec (nag_dhseqr)).

If "Nag_ComputeAll" or "Nag_ComputeSelected", vl need not be set.

On exit: if "Nag_LeftSide" or "Nag_BothSides", vl contains the computed left eigenvectors (as specified by how_many and select). The eigenvectors are stored consecutively in the rows or columns of the array, in the same order as their eigenvalues. Corresponding to each real eigenvalue is a real eigenvector, occupying one row or column. Corresponding to each complex conjugate pair of eigenvalues, is a complex eigenvector occupying two rows or columns; the first row or column holds the real part and the second row or column holds the imaginary part.

If , vl is not referenced.

vr - Matrix(1..dim1, 1..dim2, datatype=float[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: if or "Nag_BackTransform" and "Nag_RightSide" or "Nag_BothSides", vr must contain an by matrix (usually the matrix of Schur vectors returned by f08pec (nag_dhseqr)).

If "Nag_ComputeAll" or "Nag_ComputeSelected", vr need not be set.

On exit: if "Nag_RightSide" or "Nag_BothSides", vr contains the computed right eigenvectors (as specified by how_many and select). The eigenvectors are stored consecutively in the rows or columns of the array, in the same order as their eigenvalues. Corresponding to each real eigenvalue is a real eigenvector, occupying one row or column. Corresponding to each complex conjugate pair of eigenvalues, is a complex eigenvector occupying two rows or columns; the first row or column holds the real part and the second row or column holds the imaginary part.

If , vr is not referenced.

m - assignable;

Note: On exit the variable m will have a value of type integer.

On exit: , the number of rows or columns of vl and/or vr actually used to store the computed eigenvectors. If "Nag_ComputeAll", or "Nag_BackTransform", m is set to .

'n'=n - integer; (optional)

Default value: the dimension of the array t.

On entry: , the order of the matrix .

Constraint: . .

'mm'=mm - integer; (optional)

Default value: the second dimension of the arrays vl, vr.

On entry: the number of rows or columns in the arrays vl and/or vr. The precise number of rows or columns required (depending on the value of storage order), , is if "Nag_ComputeAll", or "Nag_BackTransform"; if , is obtained by counting 1 for each selected real eigenvector and 2 for each selected complex eigenvector (see select), in which case .

Constraint: . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_dtrevc (f08qkc) computes selected left and/or right eigenvectors of a real upper quasi-triangular matrix.

Description

nag_dtrevc (f08qkc) computes left and/or right eigenvectors of a real upper quasi-triangular matrix in canonical Schur form. Such a matrix arises from the Schur factorization of a real general matrix, as computed by f08pec (nag_dhseqr), for example.

The right eigenvector , and the left eigenvector , corresponding to an eigenvalue , are defined by:

Note that even though is real, , and may be complex. If is an eigenvector corresponding to a complex eigenvalue , then the complex conjugate vector is the eigenvector corresponding to the complex conjugate eigenvalue .

The function can compute the eigenvectors corresponding to selected eigenvalues, or it can compute all the eigenvectors. In the latter case the eigenvectors may optionally be pre-multiplied by an input matrix . Normally is an orthogonal matrix from the Schur factorization of a matrix as ; if is a (left or right) eigenvector of , then is an eigenvector of .

The eigenvectors are computed by forward or backward substitution. They are scaled so that, for a real eigenvector , , and for a complex eigenvector, .

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: , where is obtained by counting 1 for each selected real eigenvector and 2 for each selected complex eigenvector.

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.

	Accuracy
	If is an exact right eigenvector, and is the corresponding computed eigenvector, then the angle between them is bounded as follows: where is the reciprocal condition number of . The condition number may be computed by calling f08qlc (nag_dtrsna).

	Further Comments
	For a description of canonical Schur form, see the document for f08pec (nag_dhseqr). The complex analogue of this function is f08qxc (nag_ztrevc).

Examples

side := "Nag_BothSides":
how_many := "Nag_ComputeAll":
n := 4:
mm := 4:
selector := Vector([false], datatype=boolean):
t := Matrix([[0.7995, -0.1144, 0.006, 0.0336], [0, -0.0994, 0.2478, 0.3474], [0, -0.6483, -0.0994, 0.2026], [0, 0, 0, -0.1007]], datatype=float[8]):
vl := Matrix([[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]], datatype=float[8]):
vr := Matrix([[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]], datatype=float[8]):
NAG:-f08qkc(side, how_many, selector, t, vl, vr, m, 'n' = n, 'mm' = mm):

selector;

vl;

vr;

	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.