F08GUF (ZUPMTR) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document


Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

F08GUF (ZUPMTR) multiplies an arbitrary complex matrix C by the complex unitary matrix Q which was determined by F08GSF (ZHPTRD) when reducing a complex Hermitian matrix to tridiagonal form.

2  Specification

COMPLEX (KIND=nag_wp)  AP(*), TAU(*), C(LDC,*), WORK(*)
The routine may be called by its LAPACK name zupmtr.

3  Description

F08GUF (ZUPMTR) is intended to be used after a call to F08GSF (ZHPTRD), which reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by a unitary similarity transformation: A=QTQH. F08GSF (ZHPTRD) represents the unitary matrix Q as a product of elementary reflectors.
This routine may be used to form one of the matrix products
QC , QHC , CQ ​ or ​ CQH ,
overwriting the result on C (which may be any complex rectangular matrix).
A common application of this routine is to transform a matrix Z of eigenvectors of T to the matrix QZ of eigenvectors of A.

4  References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5  Parameters

1:     SIDE – CHARACTER(1)Input
On entry: indicates how Q or QH is to be applied to C.
Q or QH is applied to C from the left.
Q or QH is applied to C from the right.
Constraint: SIDE='L' or 'R'.
2:     UPLO – CHARACTER(1)Input
On entry: this must be the same parameter UPLO as supplied to F08GSF (ZHPTRD).
Constraint: UPLO='U' or 'L'.
3:     TRANS – CHARACTER(1)Input
On entry: indicates whether Q or QH is to be applied to C.
Q is applied to C.
QH is applied to C.
Constraint: TRANS='N' or 'C'.
4:     M – INTEGERInput
On entry: m, the number of rows of the matrix C; m is also the order of Q if SIDE='L'.
Constraint: M0.
5:     N – INTEGERInput
On entry: n, the number of columns of the matrix C; n is also the order of Q if SIDE='R'.
Constraint: N0.
6:     AP(*) – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array AP must be at least max1, M × M+1 / 2  if SIDE='L' and at least max1, N × N+1 / 2  if SIDE='R'.
On entry: details of the vectors which define the elementary reflectors, as returned by F08GSF (ZHPTRD).
On exit: is used as internal workspace prior to being restored and hence is unchanged.
7:     TAU(*) – COMPLEX (KIND=nag_wp) arrayInput
Note: the dimension of the array TAU must be at least max1,M-1 if SIDE='L' and at least max1,N-1 if SIDE='R'.
On entry: further details of the elementary reflectors, as returned by F08GSF (ZHPTRD).
8:     C(LDC,*) – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array C must be at least max1,N.
On entry: the m by n matrix C.
On exit: C is overwritten by QC or QHC or CQ or CQH as specified by SIDE and TRANS.
9:     LDC – INTEGERInput
On entry: the first dimension of the array C as declared in the (sub)program from which F08GUF (ZUPMTR) is called.
Constraint: LDCmax1,M.
10:   WORK(*) – COMPLEX (KIND=nag_wp) arrayWorkspace
Note: the dimension of the array WORK must be at least max1,N if SIDE='L' and at least max1,M if SIDE='R'.
11:   INFO – INTEGEROutput
On exit: INFO=0 unless the routine detects an error (see Section 6).

6  Error Indicators and Warnings

Errors or warnings detected by the routine:
If INFO=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.

7  Accuracy

The computed result differs from the exact result by a matrix E such that
E2 = Oε C2 ,
where ε is the machine precision.

8  Further Comments

The total number of real floating point operations is approximately 8m2n if SIDE='L' and 8mn2 if SIDE='R'.
The real analogue of this routine is F08GGF (DOPMTR).

9  Example

This example computes the two smallest eigenvalues, and the associated eigenvectors, of the matrix A, where
A = -2.28+0.00i 1.78-2.03i 2.26+0.10i -0.12+2.53i 1.78+2.03i -1.12+0.00i 0.01+0.43i -1.07+0.86i 2.26-0.10i 0.01-0.43i -0.37+0.00i 2.31-0.92i -0.12-2.53i -1.07-0.86i 2.31+0.92i -0.73+0.00i ,
using packed storage. Here A is Hermitian and must first be reduced to tridiagonal form T by F08GSF (ZHPTRD). The program then calls F08JJF (DSTEBZ) to compute the requested eigenvalues and F08JXF (ZSTEIN) to compute the associated eigenvectors of T. Finally F08GUF (ZUPMTR) is called to transform the eigenvectors to those of A.

9.1  Program Text

Program Text (f08gufe.f90)

9.2  Program Data

Program Data (f08gufe.d)

9.3  Program Results

Program Results (f08gufe.r)

F08GUF (ZUPMTR) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012