NAG FL Interfacef08ggf (dopmtr)

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1Purpose

f08ggf multiplies an arbitrary real matrix $C$ by the real orthogonal matrix $Q$ which was determined by f08gef when reducing a real symmetric matrix to tridiagonal form.

2Specification

Fortran Interface
 Subroutine f08ggf ( side, uplo, m, n, ap, tau, c, ldc, work, info)
 Integer, Intent (In) :: m, n, ldc Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (In) :: tau(*) Real (Kind=nag_wp), Intent (Inout) :: ap(*), c(ldc,*), work(*) Character (1), Intent (In) :: side, uplo, trans
#include <nag.h>
 void f08ggf_ (const char *side, const char *uplo, const char *trans, const Integer *m, const Integer *n, double ap[], const double tau[], double c[], const Integer *ldc, double work[], Integer *info, const Charlen length_side, const Charlen length_uplo, const Charlen length_trans)
The routine may be called by the names f08ggf, nagf_lapackeig_dopmtr or its LAPACK name dopmtr.

3Description

f08ggf is intended to be used after a call to f08gef, which reduces a real symmetric matrix $A$ to symmetric tridiagonal form $T$ by an orthogonal similarity transformation: $A=QT{Q}^{\mathrm{T}}$. f08gef represents the orthogonal matrix $Q$ as a product of elementary reflectors.
This routine may be used to form one of the matrix products
 $QC , QTC , CQ ​ or ​ CQT ,$
overwriting the result on $C$ (which may be any real 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 $A$.

4References

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

5Arguments

1: $\mathbf{side}$Character(1) Input
On entry: indicates how $Q$ or ${Q}^{\mathrm{T}}$ is to be applied to $C$.
${\mathbf{side}}=\text{'L'}$
$Q$ or ${Q}^{\mathrm{T}}$ is applied to $C$ from the left.
${\mathbf{side}}=\text{'R'}$
$Q$ or ${Q}^{\mathrm{T}}$ is applied to $C$ from the right.
Constraint: ${\mathbf{side}}=\text{'L'}$ or $\text{'R'}$.
2: $\mathbf{uplo}$Character(1) Input
On entry: this must be the same argument uplo as supplied to f08gef.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
3: $\mathbf{trans}$Character(1) Input
On entry: indicates whether $Q$ or ${Q}^{\mathrm{T}}$ is to be applied to $C$.
${\mathbf{trans}}=\text{'N'}$
$Q$ is applied to $C$.
${\mathbf{trans}}=\text{'T'}$
${Q}^{\mathrm{T}}$ is applied to $C$.
Constraint: ${\mathbf{trans}}=\text{'N'}$ or $\text{'T'}$.
4: $\mathbf{m}$Integer Input
On entry: $m$, the number of rows of the matrix $C$; $m$ is also the order of $Q$ if ${\mathbf{side}}=\text{'L'}$.
Constraint: ${\mathbf{m}}\ge 0$.
5: $\mathbf{n}$Integer Input
On entry: $n$, the number of columns of the matrix $C$; $n$ is also the order of $Q$ if ${\mathbf{side}}=\text{'R'}$.
Constraint: ${\mathbf{n}}\ge 0$.
6: $\mathbf{ap}\left(*\right)$Real (Kind=nag_wp) array Input/Output
Note: the dimension of the array ap must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}×\left({\mathbf{m}}+1\right)/2\right)$ if ${\mathbf{side}}=\text{'L'}$ and at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$ if ${\mathbf{side}}=\text{'R'}$.
On entry: details of the vectors which define the elementary reflectors, as returned by f08gef.
On exit: is used as internal workspace prior to being restored and hence is unchanged.
7: $\mathbf{tau}\left(*\right)$Real (Kind=nag_wp) array Input
Note: the dimension of the array tau must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}-1\right)$ if ${\mathbf{side}}=\text{'L'}$ and at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$ if ${\mathbf{side}}=\text{'R'}$.
On entry: further details of the elementary reflectors, as returned by f08gef.
8: $\mathbf{c}\left({\mathbf{ldc}},*\right)$Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array c must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $m×n$ matrix $C$.
On exit: c is overwritten by $QC$ or ${Q}^{\mathrm{T}}C$ or $CQ$ or $C{Q}^{\mathrm{T}}$ as specified by side and trans.
9: $\mathbf{ldc}$Integer Input
On entry: the first dimension of the array c as declared in the (sub)program from which f08ggf is called.
Constraint: ${\mathbf{ldc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
10: $\mathbf{work}\left(*\right)$Real (Kind=nag_wp) array Workspace
Note: the dimension of the array work must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{side}}=\text{'L'}$ and at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$ if ${\mathbf{side}}=\text{'R'}$.
11: $\mathbf{info}$Integer Output
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

7Accuracy

The computed result differs from the exact result by a matrix $E$ such that
 $‖E‖2 = O(ε) ‖C‖2 ,$
where $\epsilon$ is the machine precision.

8Parallelism and Performance

f08ggf 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.

The total number of floating-point operations is approximately $2{m}^{2}n$ if ${\mathbf{side}}=\text{'L'}$ and $2m{n}^{2}$ if ${\mathbf{side}}=\text{'R'}$.
The complex analogue of this routine is f08guf.

10Example

This example computes the two smallest eigenvalues, and the associated eigenvectors, of the matrix $A$, where
 $A = ( 2.07 3.87 4.20 -1.15 3.87 -0.21 1.87 0.63 4.20 1.87 1.15 2.06 -1.15 0.63 2.06 -1.81 ) ,$
using packed storage. Here $A$ is symmetric and must first be reduced to tridiagonal form $T$ by f08gef. The program then calls f08jjf to compute the requested eigenvalues and f08jkf to compute the associated eigenvectors of $T$. Finally f08ggf is called to transform the eigenvectors to those of $A$.

10.1Program Text

Program Text (f08ggfe.f90)

10.2Program Data

Program Data (f08ggfe.d)

10.3Program Results

Program Results (f08ggfe.r)