NAG Library Routine Document

1Purpose

f08wff (dgghd3) reduces a pair of real matrices $\left(A,B\right)$, where $B$ is upper triangular, to the generalized upper Hessenberg form using orthogonal transformations.

2Specification

Fortran Interface
 Subroutine f08wff ( n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, work, info)
 Integer, Intent (In) :: n, ilo, ihi, lda, ldb, ldq, ldz, lwork Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (Inout) :: a(lda,*), b(ldb,*), q(ldq,*), z(ldz,*) Real (Kind=nag_wp), Intent (Out) :: work(max(1,lwork)) Character (1), Intent (In) :: compq, compz
C Header Interface
#include <nagmk26.h>
 void f08wff_ (const char *compq, const char *compz, const Integer *n, const Integer *ilo, const Integer *ihi, double a[], const Integer *lda, double b[], const Integer *ldb, double q[], const Integer *ldq, double z[], const Integer *ldz, double work[], const Integer *lwork, Integer *info, const Charlen length_compq, const Charlen length_compz)
The routine may be called by its LAPACK name dgghd3.

3Description

f08wff (dgghd3) is the third step in the solution of the real generalized eigenvalue problem
 $Ax=λBx.$
The (optional) first step balances the two matrices using f08whf (dggbal). In the second step, matrix $B$ is reduced to upper triangular form using the $QR$ factorization routine f08aef (dgeqrf) and this orthogonal transformation $Q$ is applied to matrix $A$ by calling f08agf (dormqr). The driver, f08wcf (dggev3), solves the real generalized eigenvalue problem by combining all the required steps including those just listed.
f08wff (dgghd3) reduces a pair of real matrices $\left(A,B\right)$, where $B$ is upper triangular, to the generalized upper Hessenberg form using orthogonal transformations. This two-sided transformation is of the form
 $QTAZ=H, QTBZ=T$
where $H$ is an upper Hessenberg matrix, $T$ is an upper triangular matrix and $Q$ and $Z$ are orthogonal matrices determined as products of Givens rotations. They may either be formed explicitly, or they may be postmultiplied into input matrices ${Q}_{1}$ and ${Z}_{1}$, so that
 $Q1AZ1T=Q1QHZ1ZT, Q1BZ1T=Q1QTZ1ZT.$

4References

Golub G H and Van Loan C F (2012) Matrix Computations (4th Edition) Johns Hopkins University Press, Baltimore
Moler C B and Stewart G W (1973) An algorithm for generalized matrix eigenproblems SIAM J. Numer. Anal. 10 241–256

5Arguments

1:     $\mathbf{compq}$ – Character(1)Input
On entry: specifies the form of the computed orthogonal matrix $Q$.
${\mathbf{compq}}=\text{'N'}$
Do not compute $Q$.
${\mathbf{compq}}=\text{'I'}$
The orthogonal matrix $Q$ is returned.
${\mathbf{compq}}=\text{'V'}$
q must contain an orthogonal matrix ${Q}_{1}$, and the product ${Q}_{1}Q$ is returned.
Constraint: ${\mathbf{compq}}=\text{'N'}$, $\text{'I'}$ or $\text{'V'}$.
2:     $\mathbf{compz}$ – Character(1)Input
On entry: specifies the form of the computed orthogonal matrix $Z$.
${\mathbf{compz}}=\text{'N'}$
Do not compute $Z$.
${\mathbf{compz}}=\text{'I'}$
The orthogonal matrix $Z$ is returned.
${\mathbf{compz}}=\text{'V'}$
z must contain an orthogonal matrix ${Z}_{1}$, and the product ${Z}_{1}Z$ is returned.
Constraint: ${\mathbf{compz}}=\text{'N'}$, $\text{'V'}$ or $\text{'I'}$.
3:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrices $A$ and $B$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     $\mathbf{ilo}$ – IntegerInput
5:     $\mathbf{ihi}$ – IntegerInput
On entry: ${i}_{\mathrm{lo}}$ and ${i}_{\mathrm{hi}}$ as determined by a previous call to f08whf (dggbal). Otherwise, they should be set to $1$ and $n$, respectively.
Constraints:
• if ${\mathbf{n}}>0$, $1\le {\mathbf{ilo}}\le {\mathbf{ihi}}\le {\mathbf{n}}$;
• if ${\mathbf{n}}=0$, ${\mathbf{ilo}}=1$ and ${\mathbf{ihi}}=0$.
6:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the matrix $A$ of the matrix pair $\left(A,B\right)$. Usually, this is the matrix $A$ returned by f08agf (dormqr).
On exit: a is overwritten by the upper Hessenberg matrix $H$.
7:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08wff (dgghd3) is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
8:     $\mathbf{b}\left({\mathbf{ldb}},*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the upper triangular matrix $B$ of the matrix pair $\left(A,B\right)$. Usually, this is the matrix $B$ returned by the $QR$ factorization routine f08aef (dgeqrf).
On exit: b is overwritten by the upper triangular matrix $T$.
9:     $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f08wff (dgghd3) is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
10:   $\mathbf{q}\left({\mathbf{ldq}},*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array q must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{compq}}=\text{'I'}$ or $\text{'V'}$ and at least $1$ if ${\mathbf{compq}}=\text{'N'}$.
On entry: if ${\mathbf{compq}}=\text{'V'}$, q must contain an orthogonal matrix ${Q}_{1}$.
If ${\mathbf{compq}}=\text{'N'}$, q is not referenced.
On exit: if ${\mathbf{compq}}=\text{'I'}$, q contains the orthogonal matrix $Q$.
If ${\mathbf{compq}}=\text{'V'}$, q is overwritten by ${Q}_{1}Q$.
11:   $\mathbf{ldq}$ – IntegerInput
On entry: the first dimension of the array q as declared in the (sub)program from which f08wff (dgghd3) is called.
Constraints:
• if ${\mathbf{compq}}=\text{'I'}$ or $\text{'V'}$, ${\mathbf{ldq}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{compq}}=\text{'N'}$, ${\mathbf{ldq}}\ge 1$.
12:   $\mathbf{z}\left({\mathbf{ldz}},*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array z must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{compz}}=\text{'V'}$ or $\text{'I'}$ and at least $1$ if ${\mathbf{compz}}=\text{'N'}$.
On entry: if ${\mathbf{compz}}=\text{'V'}$, z must contain an orthogonal matrix ${Z}_{1}$.
If ${\mathbf{compz}}=\text{'N'}$, z is not referenced.
On exit: if ${\mathbf{compz}}=\text{'I'}$, z contains the orthogonal matrix $Z$.
If ${\mathbf{compz}}=\text{'V'}$, z is overwritten by ${Z}_{1}Z$.
13:   $\mathbf{ldz}$ – IntegerInput
On entry: the first dimension of the array z as declared in the (sub)program from which f08wff (dgghd3) is called.
Constraints:
• if ${\mathbf{compz}}=\text{'V'}$ or $\text{'I'}$, ${\mathbf{ldz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{compz}}=\text{'N'}$, ${\mathbf{ldz}}\ge 1$.
14:   $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$ – Real (Kind=nag_wp) arrayWorkspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{work}}\left(1\right)$ contains the minimum value of lwork required for optimal performance.
15:   $\mathbf{lwork}$ – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08wff (dgghd3) is called.
If ${\mathbf{lwork}}=-1$, 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, lwork must generally be larger than the minimum; increase workspace by, say, $6×\mathit{nb}×{\mathbf{n}}$, where $\mathit{nb}$ is the optimal block size.
16:   $\mathbf{info}$ – IntegerOutput
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 reduction to the generalized Hessenberg form is implemented using orthogonal transformations which are backward stable.

8Parallelism and Performance

f08wff (dgghd3) 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.

9Further Comments

This routine is usually followed by f08xef (dhgeqz) which implements the $QZ$ algorithm for computing generalized eigenvalues of a reduced pair of matrices.
The complex analogue of this routine is f08wtf (zgghd3).

10Example

See Section 10 in f08xef (dhgeqz) and f08ykf (dtgevc).