NAG FL Interface
f08nuf (zunmhr)

Settings help

FL Name Style:

FL Specification Language:

1 Purpose

f08nuf multiplies an arbitrary complex matrix C by the complex unitary matrix Q which was determined by f08nsf when reducing a complex general matrix to Hessenberg form.

2 Specification

Fortran Interface
Subroutine f08nuf ( side, trans, m, n, ilo, ihi, a, lda, tau, c, ldc, work, lwork, info)
Integer, Intent (In) :: m, n, ilo, ihi, lda, ldc, lwork
Integer, Intent (Out) :: info
Complex (Kind=nag_wp), Intent (In) :: tau(*)
Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*), c(ldc,*)
Complex (Kind=nag_wp), Intent (Out) :: work(max(1,lwork))
Character (1), Intent (In) :: side, trans
C Header Interface
#include <nag.h>
void  f08nuf_ (const char *side, const char *trans, const Integer *m, const Integer *n, const Integer *ilo, const Integer *ihi, Complex a[], const Integer *lda, const Complex tau[], Complex c[], const Integer *ldc, Complex work[], const Integer *lwork, Integer *info, const Charlen length_side, const Charlen length_trans)
The routine may be called by the names f08nuf, nagf_lapackeig_zunmhr or its LAPACK name zunmhr.

3 Description

f08nuf is intended to be used following a call to f08nsf, which reduces a complex general matrix A to upper Hessenberg form H by a unitary similarity transformation: A=QHQH. f08nsf represents the matrix Q as a product of ihi-ilo elementary reflectors. Here ilo and ihi are values determined by f08nvf when balancing the matrix; if the matrix has not been balanced, ilo=1 and ihi=n.
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 V of eigenvectors of H to the matrix QV 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 Arguments

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: 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'.
3: m Integer Input
On entry: m, the number of rows of the matrix C; m is also the order of Q if side='L'.
Constraint: m0.
4: n Integer Input
On entry: n, the number of columns of the matrix C; n is also the order of Q if side='R'.
Constraint: n0.
5: ilo Integer Input
6: ihi Integer Input
On entry: these must be the same arguments ilo and ihi, respectively, as supplied to f08nsf.
  • if side='L' and m>0, 1 ilo ihi m ;
  • if side='L' and m=0, ilo=1 and ihi=0;
  • if side='R' and n>0, 1 ilo ihi n ;
  • if side='R' and n=0, ilo=1 and ihi=0.
7: a(lda,*) Complex (Kind=nag_wp) array Input
Note: the second dimension of the array a must be at least max(1,m) if side='L' and at least max(1,n) if side='R'.
On entry: details of the vectors which define the elementary reflectors, as returned by f08nsf.
8: lda Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08nuf is called.
  • if side='L', lda max(1,m) ;
  • if side='R', lda max(1,n) .
9: tau(*) Complex (Kind=nag_wp) array Input
Note: the dimension of the array tau must be at least max(1,m-1) if side='L' and at least max(1,n-1) if side='R'.
On entry: further details of the elementary reflectors, as returned by f08nsf.
10: c(ldc,*) Complex (Kind=nag_wp) array Input/Output
Note: the second dimension of the array c must be at least max(1,n).
On entry: the m×n matrix C.
On exit: c is overwritten by QC or QHC or CQ or CQH as specified by side and trans.
11: ldc Integer Input
On entry: the first dimension of the array c as declared in the (sub)program from which f08nuf is called.
Constraint: ldcmax(1,m).
12: work(max(1,lwork)) Complex (Kind=nag_wp) array Workspace
On exit: if info=0, the real part of work(1) contains the minimum value of lwork required for optimal performance.
13: lwork Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f08nuf is called.
If lwork=−1, a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.
Suggested value: for optimal performance, lworkn×nb if side='L' and at least m×nb if side='R', where nb is the optimal block size.
  • if side='L', lworkmax(1,n) or lwork=−1;
  • if side='R', lworkmax(1,m) or lwork=−1.
14: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

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 = O(ε) C2 ,  
where ε is the machine precision.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08nuf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08nuf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The total number of real floating-point operations is approximately 8nq2 if side='L' and 8mq2 if side='R', where q=ihi-ilo.
The real analogue of this routine is f08ngf.

10 Example

This example computes all the eigenvalues of the matrix A, where
A = ( -3.97-5.04i -4.11+3.70i -0.34+1.01i 1.29-0.86i 0.34-1.50i 1.52-0.43i 1.88-5.38i 3.36+0.65i 3.31-3.85i 2.50+3.45i 0.88-1.08i 0.64-1.48i -1.10+0.82i 1.81-1.59i 3.25+1.33i 1.57-3.44i ) ,  
and those eigenvectors which correspond to eigenvalues λ such that Re(λ)<0. Here A is general and must first be reduced to upper Hessenberg form H by f08nsf. The program then calls f08psf to compute the eigenvalues, and f08pxf to compute the required eigenvectors of H by inverse iteration. Finally f08nuf is called to transform the eigenvectors of H back to eigenvectors of the original matrix A.

10.1 Program Text

Program Text (f08nufe.f90)

10.2 Program Data

Program Data (f08nufe.d)

10.3 Program Results

Program Results (f08nufe.r)