NAG Library Routine Document

F08TCF (DSPGVD)

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     ITYPE – INTEGERInput

On entry: specifies the problem type to be solved.

$\mathbf{ITYPE}=1$

$Az=\lambda Bz$.

$\mathbf{ITYPE}=2$

$ABz=\lambda z$.

$\mathbf{ITYPE}=3$

$BAz=\lambda z$.

Constraint: $\mathbf{ITYPE}=1$, $2$ or $3$.

2:     JOBZ – CHARACTER(1)Input

On entry: indicates whether eigenvectors are computed.

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

Only eigenvalues are computed.

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

Eigenvalues and eigenvectors are computed.

Constraint: $\mathbf{JOBZ}=\text{'N'}$ or $\text{'V'}$.

3:     UPLO – CHARACTER(1)Input

On entry: if $\mathbf{UPLO}=\text{'U'}$, the upper triangles of $A$ and $B$ are stored.

If $\mathbf{UPLO}=\text{'L'}$, the lower triangles of $A$ and $B$ are stored.

Constraint: $\mathbf{UPLO}=\text{'U'}$ or $\text{'L'}$.

4:     N – INTEGERInput

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

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

5:     AP($*$) – REAL (KIND=nag_wp) arrayInput/Output

Note: the dimension of the array AP must be at least $\max \left(1,\mathbf{N}\times \left(\mathbf{N}+1\right)/2\right)$.

On entry: the upper or lower triangle of the $n$ by $n$ symmetric matrix $A$, packed by columns.

More precisely,

• if $\mathbf{UPLO}=\text{'U'}$, the upper triangle of $A$ must be stored with element $A_{ij}$ in $\mathbf{AP}\left(i+j\left(j-1\right)/2\right)$ for $i\le j$;
• if $\mathbf{UPLO}=\text{'L'}$, the lower triangle of $A$ must be stored with element $A_{ij}$ in $\mathbf{AP}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for $i\ge j$.

On exit: the contents of AP are destroyed.

6:     BP($*$) – REAL (KIND=nag_wp) arrayInput/Output

Note: the dimension of the array BP must be at least $\max \left(1,\mathbf{N}\times \left(\mathbf{N}+1\right)/2\right)$.

On entry: the upper or lower triangle of the $n$ by $n$ symmetric matrix $B$, packed by columns.

More precisely,

• if $\mathbf{UPLO}=\text{'U'}$, the upper triangle of $B$ must be stored with element $B_{ij}$ in $\mathbf{BP}\left(i+j\left(j-1\right)/2\right)$ for $i\le j$;
• if $\mathbf{UPLO}=\text{'L'}$, the lower triangle of $B$ must be stored with element $B_{ij}$ in $\mathbf{BP}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for $i\ge j$.

On exit: the triangular factor $U$ or $L$ from the Cholesky factorization $B=U^{\mathrm{T}}U$ or $B=L{L}^{\mathrm{T}}$, in the same storage format as $B$.

7:     W(N) – REAL (KIND=nag_wp) arrayOutput

On exit: the eigenvalues in ascending order.

8:     Z(LDZ,$*$) – REAL (KIND=nag_wp) arrayOutput

Note: the second dimension of the array Z must be at least $\max \left(1,\mathbf{N}\right)$ if $\mathbf{JOBZ}=\text{'V'}$, and at least $1$ otherwise.

On exit: if $\mathbf{JOBZ}=\text{'V'}$, Z contains the matrix $Z$ of eigenvectors. The eigenvectors are normalized as follows:

• if $\mathbf{ITYPE}=1$ or $2$, $Z^{\mathrm{T}}BZ=I$;
• if $\mathbf{ITYPE}=3$, $Z^{\mathrm{T}}{B}^{-1}Z=I$.

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

9:     LDZ – INTEGERInput

On entry: the first dimension of the array Z as declared in the (sub)program from which F08TCF (DSPGVD) is called.

Constraints:

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

10:   WORK($\max \left(1,\mathbf{LWORK}\right)$) – 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.

11:   LWORK – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which F08TCF (DSPGVD) is called.

If $\mathbf{LWORK}=-1$, a workspace query is assumed; the routine only calculates the minimum sizes of the WORK and IWORK arrays, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued.

Constraints:

• if $\mathbf{N}\le 1$, $\mathbf{LWORK}\ge 1$;
• if $\mathbf{JOBZ}=\text{'N'}$ and $\mathbf{N}>1$, $\mathbf{LWORK}\ge 2\times \mathbf{N}$;
• if $\mathbf{JOBZ}=\text{'V'}$ and $\mathbf{N}>1$, $\mathbf{LWORK}\ge 1+6\times \mathbf{N}+2\times {\mathbf{N}}^2$.

12:   IWORK($\max \left(1,\mathbf{LIWORK}\right)$) – INTEGER arrayWorkspace

On exit: if $\mathbf{INFO}=\mathbf{0}$, $\mathbf{IWORK}\left(1\right)$ returns the minimum LIWORK.

13:   LIWORK – INTEGERInput

On entry: the dimension of the array IWORK as declared in the (sub)program from which F08TCF (DSPGVD) is called.

If $\mathbf{LIWORK}=-1$, a workspace query is assumed; the routine only calculates the minimum sizes of the WORK and IWORK arrays, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued.

Constraints:

• if $\mathbf{JOBZ}=\text{'N'}$ or $\mathbf{N}\le 1$, $\mathbf{LIWORK}\ge 1$;
• if $\mathbf{JOBZ}=\text{'V'}$ and $\mathbf{N}>1$, $\mathbf{LIWORK}\ge 3+5\times \mathbf{N}$.

14:   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.

$\mathbf{INFO}>0$

F07GDF (DPPTRF) or F08GCF (DSPEVD) returned an error code:

$\le \mathbf{N}$	if $\mathbf{INFO}=i$, F08GCF (DSPEVD) failed to converge; $i$ off-diagonal elements of an intermediate tridiagonal form did not converge to zero;
$>\mathbf{N}$	if $\mathbf{INFO}=\mathbf{N}+i$, for $1\le i\le \mathbf{N}$, then the leading minor of order $i$ of $B$ is not positive definite. The factorization of $B$ could not be completed and no eigenvalues or eigenvectors were computed.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (f08tcfe.f90)

9.2  Program Data

Program Data (f08tcfe.d)

9.3  Program Results

Program Results (f08tcfe.r)