NAG Library Routine Document

F08WSF (ZGGHRD)

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     $\mathrm{COMPQ}$ – CHARACTER(1)Input

On entry: specifies the form of the computed unitary matrix $Q$.

$\mathbf{COMPQ}=\text{'N'}$

Do not compute $Q$.

$\mathbf{COMPQ}=\text{'I'}$

The unitary matrix $Q$ is returned.

$\mathbf{COMPQ}=\text{'V'}$

Q must contain a unitary matrix $Q_1$, and the product $Q_{1}Q$ is returned.

Constraint: $\mathbf{COMPQ}=\text{'N'}$, $\text{'I'}$ or $\text{'V'}$.

2:     $\mathrm{COMPZ}$ – CHARACTER(1)Input

On entry: specifies the form of the computed unitary matrix $Z$.

$\mathbf{COMPZ}=\text{'N'}$

Do not compute $Z$.

$\mathbf{COMPZ}=\text{'V'}$

Z must contain a unitary matrix $Z_1$, and the product $Z_{1}Z$ is returned.

$\mathbf{COMPZ}=\text{'I'}$

The unitary matrix $Z$ is returned.

Constraint: $\mathbf{COMPZ}=\text{'N'}$, $\text{'V'}$ or $\text{'I'}$.

3:     $\mathrm{N}$ – INTEGERInput

On entry: $n$, the order of the matrices $A$ and $B$.

Constraint: $\mathbf{N}\ge 0$.

4:     $\mathrm{ILO}$ – INTEGERInput

5:     $\mathrm{IHI}$ – INTEGERInput

On entry: $i_{\mathrm{lo}}$ and $i_{\mathrm{hi}}$ as determined by a previous call to F08WVF (ZGGBAL). 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:     $\mathrm{A}\left(\mathbf{LDA},*\right)$ – COMPLEX (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array A must be at least $\max \left(1,\mathbf{N}\right)$.

On entry: the matrix $A$ of the matrix pair $\left(A,B\right)$. Usually, this is the matrix $A$ returned by F08AUF (ZUNMQR).

On exit: A is overwritten by the upper Hessenberg matrix $H$.

7:     $\mathrm{LDA}$ – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F08WSF (ZGGHRD) is called.

Constraint: $\mathbf{LDA}\ge \max \left(1,\mathbf{N}\right)$.

8:     $\mathrm{B}\left(\mathbf{LDB},*\right)$ – COMPLEX (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array B must be at least $\max \left(1,\mathbf{N}\right)$.

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 F08ASF (ZGEQRF).

On exit: B is overwritten by the upper triangular matrix $T$.

9:     $\mathrm{LDB}$ – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F08WSF (ZGGHRD) is called.

Constraint: $\mathbf{LDB}\ge \max \left(1,\mathbf{N}\right)$.

10:   $\mathrm{Q}\left(\mathbf{LDQ},*\right)$ – COMPLEX (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array Q must be at least $\max \left(1,\mathbf{N}\right)$ 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 a unitary matrix $Q_1$.

If $\mathbf{COMPQ}=\text{'N'}$, Q is not referenced.

On exit: if $\mathbf{COMPQ}=\text{'I'}$, Q contains the unitary matrix $Q$.

Iif $\mathbf{COMPQ}=\text{'V'}$, Q is overwritten by $Q_{1}Q$.

11:   $\mathrm{LDQ}$ – INTEGERInput

On entry: the first dimension of the array Q as declared in the (sub)program from which F08WSF (ZGGHRD) is called.

Constraints:

• if $\mathbf{COMPQ}=\text{'I'}$ or $\text{'V'}$, $\mathbf{LDQ}\ge \max \left(1,\mathbf{N}\right)$;
• if $\mathbf{COMPQ}=\text{'N'}$, $\mathbf{LDQ}\ge 1$.

12:   $\mathrm{Z}\left(\mathbf{LDZ},*\right)$ – COMPLEX (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array Z must be at least $\max \left(1,\mathbf{N}\right)$ 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 a unitary matrix $Z_1$.

If $\mathbf{COMPZ}=\text{'N'}$, Z is not referenced.

On exit: if $\mathbf{COMPZ}=\text{'I'}$, Z contains the unitary matrix $Z$.

If $\mathbf{COMPZ}=\text{'V'}$, Z is overwritten by $Z_{1}Z$.

13:   $\mathrm{LDZ}$ – INTEGERInput

On entry: the first dimension of the array Z as declared in the (sub)program from which F08WSF (ZGGHRD) is called.

Constraints:

• if $\mathbf{COMPZ}=\text{'V'}$ or $\text{'I'}$, $\mathbf{LDZ}\ge \max \left(1,\mathbf{N}\right)$;
• if $\mathbf{COMPZ}=\text{'N'}$, $\mathbf{LDZ}\ge 1$.

14:   $\mathrm{INFO}$ – INTEGEROutput

On exit: $\mathbf{INFO}=0$ unless the routine detects an error (see Section 6).

6  Error 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.

7  Accuracy

8  Parallelism and Performance

9  Further Comments

10  Example