NAG Library Routine Document

F08TPF (ZHPGVX)

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:     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 $\left(\mathbf{VL},\mathbf{VU}\right]$ 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:     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:     N – INTEGERInput

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

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

6:     AP($*$) – COMPLEX (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 $n$ Hermitian 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.

7:     BP($*$) – COMPLEX (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 $n$ Hermitian 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 $B=U^{\mathrm{H}}U$ or $B=L{L}^{\mathrm{H}}$, in the same storage format as $B$.

8:     VL – REAL (KIND=nag_wp)Input

9:     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:   IL – INTEGERInput

11:   IU – INTEGERInput

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

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:   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 $\left[a,b\right]$ of width less than or equal to

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

where $\epsilon$ is the machine precision. If ABSTOL is less than or equal to zero, then $\epsilon {‖T‖}_1$ 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 $2\times \mathbf{X02AMF}\left( \right)$, not zero. If this routine returns with $\mathbf{INFO}=\mathbf{1}\text{ to }\mathbf{N}$, indicating that some eigenvectors did not converge, try setting ABSTOL to $2\times \mathbf{X02AMF}\left( \right)$. See Demmel and Kahan (1990).

13:   M – INTEGEROutput

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

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

15:   Z(LDZ,$*$) – COMPLEX (KIND=nag_wp) arrayOutput

Note: the second dimension of the array Z must be at least $\max \left(1,\mathbf{M}\right)$ 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 $2$, $Z^{\mathrm{H}}BZ=I$;
  • if $\mathbf{ITYPE}=3$, $Z^{\mathrm{H}}{B}^{-1}Z=I$;
• if an eigenvector fails to converge ($\mathbf{INFO}=\mathbf{1}\text{ to }\mathbf{N}$), 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 $\max \left(1,\mathbf{M}\right)$ 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:   LDZ – INTEGERInput

On entry: the first dimension of the array Z as declared in the (sub)program from which F08TPF (ZHPGVX) is called.

Constraints:

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

17:   WORK($2\times \mathbf{N}$) – COMPLEX (KIND=nag_wp) arrayWorkspace

18:   RWORK($7\times \mathbf{N}$) – REAL (KIND=nag_wp) arrayWorkspace

19:   IWORK($5\times \mathbf{N}$) – INTEGER arrayWorkspace

20:   JFAIL($*$) – INTEGER arrayOutput

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

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

• if $\mathbf{INFO}=\mathbf{0}$, the first M elements of JFAIL are zero;
• if $\mathbf{INFO}=\mathbf{1}\text{ to }\mathbf{N}$, JFAIL contains the indices of the eigenvectors that failed to converge.

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

21:   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.

$\mathbf{INFO}=1\text{ to }\mathbf{N}$

If $\mathbf{INFO}=i$, F08GPF (ZHPEVX) failed to converge; $i$ eigenvectors failed to converge. Their indices are stored in array JFAIL.

$\mathbf{INFO}>\mathbf{N}$

F07GRF (ZPPTRF) returned an error code; i.e., 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 (f08tpfe.f90)

9.2  Program Data

Program Data (f08tpfe.d)

9.3  Program Results

Program Results (f08tpfe.r)