NAG Library Function Document

nag_dspgv (f08tac)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:    $\mathbf{order}$ – Nag_OrderTypeInput

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by $\mathbf{order}=\mathrm{Nag\_RowMajor}$. See Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint: $\mathbf{order}=\mathrm{Nag\_RowMajor}$ or $\mathrm{Nag\_ColMajor}$.

2:    $\mathbf{itype}$ – 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 $3$.

3:    $\mathbf{job}$ – Nag_JobTypeInput

On entry: indicates whether eigenvectors are computed.

$\mathbf{job}=\mathrm{Nag\_EigVals}$

Only eigenvalues are computed.

$\mathbf{job}=\mathrm{Nag\_DoBoth}$

Eigenvalues and eigenvectors are computed.

Constraint: $\mathbf{job}=\mathrm{Nag\_EigVals}$ or $\mathrm{Nag\_DoBoth}$.

4:    $\mathbf{uplo}$ – Nag_UploTypeInput

On entry: if $\mathbf{uplo}=\mathrm{Nag\_Upper}$, the upper triangles of $A$ and $B$ are stored.

If $\mathbf{uplo}=\mathrm{Nag\_Lower}$, the lower triangles of $A$ and $B$ are stored.

Constraint: $\mathbf{uplo}=\mathrm{Nag\_Upper}$ or $\mathrm{Nag\_Lower}$.

5:    $\mathbf{n}$ – IntegerInput

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

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

6:    $\mathbf{ap}\left[\mathit{dim}\right]$ – doubleInput/Output

Note: the dimension, dim, 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 rows or columns.

The storage of elements $A_{ij}$ depends on the order and uplo arguments as follows:

• if $\mathbf{order}=\mathrm{Nag\_ColMajor}$ and $\mathbf{uplo}=\mathrm{Nag\_Upper}$,
            $A_{ij}$ is stored in $\mathbf{ap}\left[\left(j-1\right)\times j/2+i-1\right]$, for $i\le j$;
• if $\mathbf{order}=\mathrm{Nag\_ColMajor}$ and $\mathbf{uplo}=\mathrm{Nag\_Lower}$,
            $A_{ij}$ is stored in $\mathbf{ap}\left[\left(2n-j\right)\times \left(j-1\right)/2+i-1\right]$, for $i\ge j$;
• if $\mathbf{order}=\mathrm{Nag\_RowMajor}$ and $\mathbf{uplo}=\mathrm{Nag\_Upper}$,
            $A_{ij}$ is stored in $\mathbf{ap}\left[\left(2n-i\right)\times \left(i-1\right)/2+j-1\right]$, for $i\le j$;
• if $\mathbf{order}=\mathrm{Nag\_RowMajor}$ and $\mathbf{uplo}=\mathrm{Nag\_Lower}$,
            $A_{ij}$ is stored in $\mathbf{ap}\left[\left(i-1\right)\times i/2+j-1\right]$, for $i\ge j$.

On exit: the contents of ap are destroyed.

7:    $\mathbf{bp}\left[\mathit{dim}\right]$ – doubleInput/Output

Note: the dimension, dim, 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 rows or columns.

The storage of elements $B_{ij}$ depends on the order and uplo arguments as follows:

• if $\mathbf{order}=\mathrm{Nag\_ColMajor}$ and $\mathbf{uplo}=\mathrm{Nag\_Upper}$,
            $B_{ij}$ is stored in $\mathbf{bp}\left[\left(j-1\right)\times j/2+i-1\right]$, for $i\le j$;
• if $\mathbf{order}=\mathrm{Nag\_ColMajor}$ and $\mathbf{uplo}=\mathrm{Nag\_Lower}$,
            $B_{ij}$ is stored in $\mathbf{bp}\left[\left(2n-j\right)\times \left(j-1\right)/2+i-1\right]$, for $i\ge j$;
• if $\mathbf{order}=\mathrm{Nag\_RowMajor}$ and $\mathbf{uplo}=\mathrm{Nag\_Upper}$,
            $B_{ij}$ is stored in $\mathbf{bp}\left[\left(2n-i\right)\times \left(i-1\right)/2+j-1\right]$, for $i\le j$;
• if $\mathbf{order}=\mathrm{Nag\_RowMajor}$ and $\mathbf{uplo}=\mathrm{Nag\_Lower}$,
            $B_{ij}$ is stored in $\mathbf{bp}\left[\left(i-1\right)\times i/2+j-1\right]$, 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 $B$.

8:    $\mathbf{w}\left[\mathbf{n}\right]$ – doubleOutput

On exit: the eigenvalues in ascending order.

9:    $\mathbf{z}\left[\mathit{dim}\right]$ – doubleOutput

Note: the dimension, dim, of the array z must be at least

• $\max \left(1,\mathbf{pdz}\times \mathbf{n}\right)$ when $\mathbf{job}=\mathrm{Nag\_DoBoth}$;
• $1$ otherwise.

The $\left(i,j\right)$th element of the matrix $Z$ is stored in

• $\mathbf{z}\left[\left(j-1\right)\times \mathbf{pdz}+i-1\right]$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\mathbf{z}\left[\left(i-1\right)\times \mathbf{pdz}+j-1\right]$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$.

On exit: if $\mathbf{job}=\mathrm{Nag\_DoBoth}$, 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 $\mathbf{job}=\mathrm{Nag\_EigVals}$, z is not referenced.

10:  $\mathbf{pdz}$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array z.

Constraints:

• if $\mathbf{job}=\mathrm{Nag\_DoBoth}$, $\mathbf{pdz}\ge \max \left(1,\mathbf{n}\right)$;
• otherwise $\mathbf{pdz}\ge 1$.

11:  $\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.

NE_BAD_PARAM

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.

NE_CONVERGENCE

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

NE_ENUM_INT_2

On entry, $\mathbf{job}=〈\mathit{\text{value}}〉$, $\mathbf{pdz}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: if $\mathbf{job}=\mathrm{Nag\_DoBoth}$, $\mathbf{pdz}\ge \max \left(1,\mathbf{n}\right)$;
otherwise $\mathbf{pdz}\ge 1$.

NE_INT

On entry, $\mathbf{itype}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{itype}=1$, $2$ or $3$.

On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{n}\ge 0$.

On entry, $\mathbf{pdz}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pdz}>0$.

NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.

NE_MAT_NOT_POS_DEF

If $\mathbf{fail}\mathbf{.}\mathbf{errnum}=\mathbf{n}+〈\mathit{\text{value}}〉$, 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.

NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f08tace.c)

10.2

Program Data

Program Data (f08tace.d)

10.3

Program Results

Program Results (f08tace.r)