NAG FL Interface
f08tcf (dspgvd)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

1: $\mathbf{itype}$ – Integer Input

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: $\mathbf{jobz}$ – 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: $\mathbf{uplo}$ – 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: $\mathbf{n}$ – Integer Input

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

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

5: $\mathbf{ap}\left(*\right)$ – Real (Kind=nag_wp) array Input/Output

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

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

On exit: the contents of ap are destroyed.

6: $\mathbf{bp}\left(*\right)$ – Real (Kind=nag_wp) array Input/Output

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

On entry: the upper or lower triangle of the $n\times n$ 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 $\mathbf{bp}\left(i+j(j-1)/2\right)$ for $i\le j$;
• if $\mathbf{uplo}=\text{'L'}$, the lower triangle of $B$ must be stored with element $B_{ij}$ in $\mathbf{bp}\left(i+(2n-j)(j-1)/2\right)$ 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: $\mathbf{w}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) array Output

On exit: the eigenvalues in ascending order.

8: $\mathbf{z}(\mathbf{ldz},*)$ – Real (Kind=nag_wp) array Output

Note: the second dimension of the array z must be at least $\max (1,\mathbf{n})$ 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: $\mathbf{ldz}$ – Integer Input

On entry: the first dimension of the array z as declared in the (sub)program from which f08tcf is called.

Constraints:

• if $\mathbf{jobz}=\text{'V'}$, $\mathbf{ldz}\ge \max (1,\mathbf{n})$;
• otherwise $\mathbf{ldz}\ge 1$.

10: $\mathbf{work}\left(\max (1,\mathbf{lwork})\right)$ – Real (Kind=nag_wp) array Workspace

On exit: if $\mathbf{info}=\mathbf{0}$, $\mathbf{work}\left(1\right)$ contains the minimum value of lwork required for optimal performance.

11: $\mathbf{lwork}$ – Integer Input

On entry: the dimension of the array work as declared in the (sub)program from which f08tcf is called.

If $\mathbf{lwork}=\mathrm{-1}$, 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: $\mathbf{iwork}\left(\max (1,\mathbf{liwork})\right)$ – Integer array Workspace

On exit: if $\mathbf{info}=\mathbf{0}$, $\mathbf{iwork}\left(1\right)$ returns the minimum liwork.

13: $\mathbf{liwork}$ – Integer Input

On entry: the dimension of the array iwork as declared in the (sub)program from which f08tcf is called.

If $\mathbf{liwork}=\mathrm{-1}$, 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: $\mathbf{info}$ – Integer Output

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.

$\mathbf{info}=1,\dots ,\mathbf{n}$

The algorithm failed to converge; $⟨\mathit{\text{value}}⟩$ off-diagonal elements of an intermediate tridiagonal form did not converge to zero.

$\mathbf{info}>\mathbf{n}$

If $\mathbf{info}=\mathbf{n}+⟨\mathit{\text{value}}⟩$, for $1\le ⟨\mathit{\text{value}}⟩\le \mathbf{n}$, then the leading minor of order $⟨\mathit{\text{value}}⟩$ of $B$ is not positive definite. The factorization of $B$ could not be completed and no eigenvalues or eigenvectors were computed.

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results