NAG FL Interface
f08tbf (dspgvx)

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: $\mathbf{range}$ – Character(1) Input

On entry: if $\mathbf{range}=\text{'A'}$, all eigenvalues will be found.

If $\mathbf{range}=\text{'V'}$, all eigenvalues in the half-open interval $(\mathbf{vl},\mathbf{vu}]$ will be found.

If $\mathbf{range}=\text{'I'}$, the ilth to iuth eigenvalues will be found.

Constraint: $\mathbf{range}=\text{'A'}$, $\text{'V'}$ or $\text{'I'}$.

4: $\mathbf{uplo}$ – 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 $\text{'L'}$.

5: $\mathbf{n}$ – Integer Input

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

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

6: $\mathbf{ap}\left(*\right)$ – 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.

7: $\mathbf{bp}\left(*\right)$ – 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 $B$.

8: $\mathbf{vl}$ – Real (Kind=nag_wp) Input

9: $\mathbf{vu}$ – Real (Kind=nag_wp) Input

On entry: if $\mathbf{range}=\text{'V'}$, the lower and upper bounds of the interval to be searched for eigenvalues.

If $\mathbf{range}=\text{'A'}$ or $\text{'I'}$, vl and vu are not referenced.

Constraint: if $\mathbf{range}=\text{'V'}$, $\mathbf{vl}<\mathbf{vu}$.

10: $\mathbf{il}$ – Integer Input

11: $\mathbf{iu}$ – Integer Input

On entry: if $\mathbf{range}=\text{'I'}$, il and iu specify the indices (in ascending order) of the smallest and largest eigenvalues to be returned, respectively.

If $\mathbf{range}=\text{'A'}$ or $\text{'V'}$, il and iu are not referenced.

Constraints:

• if $\mathbf{range}=\text{'I'}$ and $\mathbf{n}=0$, $\mathbf{il}=1$ and $\mathbf{iu}=0$;
• if $\mathbf{range}=\text{'I'}$ and $\mathbf{n}>0$, $1\le \mathbf{il}\le \mathbf{iu}\le \mathbf{n}$.

12: $\mathbf{abstol}$ – Real (Kind=nag_wp) Input

On entry: the absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval $[a,b]$ of width less than or equal to

$$\mathbf{abstol}+\epsilon \max (\left|a\right|,\left|b\right|)\text{,}$$

where $\epsilon$ is the machine precision. If abstol is less than or equal to zero, then $\epsilon {\Vert T\Vert }_1$ will be used in its place, where $T$ is the tridiagonal matrix obtained by reducing $C$ to tridiagonal form. Eigenvalues will be computed most accurately when abstol is set to twice the underflow threshold $2\times \mathbf{x02amf}( )$, not zero. If this routine returns with $\mathbf{info}=\mathbf{1},\dots ,\mathbf{n}$, indicating that some eigenvectors did not converge, try setting abstol to $2\times \mathbf{x02amf}( )$. See Demmel and Kahan (1990).

13: $\mathbf{m}$ – Integer Output

On exit: the total number of eigenvalues found. $0\le \mathbf{m}\le \mathbf{n}$.

If $\mathbf{range}=\text{'A'}$, $\mathbf{m}=\mathbf{n}$.

If $\mathbf{range}=\text{'I'}$, $\mathbf{m}=\mathbf{iu}-\mathbf{il}+1$.

14: $\mathbf{w}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) array Output

On exit: the first m elements contain the selected eigenvalues in ascending order.

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

Note: the second dimension of the array z must be at least $\max (1,\mathbf{m})$ if $\mathbf{jobz}=\text{'V'}$, and at least $1$ otherwise.

On exit: if $\mathbf{jobz}=\text{'V'}$, then

• if $\mathbf{info}=\mathbf{0}$, the first m columns of $Z$ contain the orthonormal eigenvectors of the matrix $A$ corresponding to the selected eigenvalues, with the $i$th column of $Z$ holding the eigenvector associated with $\mathbf{w}\left(i\right)$. 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 an eigenvector fails to converge ($\mathbf{info}=\mathbf{1},\dots ,\mathbf{n}$), then that column of $Z$ contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in jfail.

If $\mathbf{jobz}=\text{'N'}$, z is not referenced.

Note:  you must ensure that at least $\max (1,\mathbf{m})$ columns are supplied in the array z; if $\mathbf{range}=\text{'V'}$, the exact value of m is not known in advance and an upper bound of at least n must be used.

16: $\mathbf{ldz}$ – Integer Input

On entry: the first dimension of the array z as declared in the (sub)program from which f08tbf is called.

Constraints:

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

17: $\mathbf{work}\left(8\times \mathbf{n}\right)$ – Real (Kind=nag_wp) array Workspace

18: $\mathbf{iwork}\left(5\times \mathbf{n}\right)$ – Integer array Workspace

19: $\mathbf{jfail}\left(*\right)$ – Integer array Output

Note: the dimension of the array jfail must be at least $\max (1,\mathbf{n})$.

On exit: if $\mathbf{jobz}=\text{'V'}$, then

• if $\mathbf{info}=\mathbf{0}$, the first m elements of jfail are zero;
• if $\mathbf{info}=\mathbf{1},\dots ,\mathbf{n}$, the first info elements of jfail contains the indices of the eigenvectors that failed to converge.

If $\mathbf{jobz}=\text{'N'}$, jfail is not referenced.

20: $\mathbf{info}$ – 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; $⟨\mathit{\text{value}}⟩$ eigenvectors did not converge. Their indices are stored in array jfail.

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

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