NAG Library Routine Document

F08SPF (ZHEGVX)

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:     A(LDA,$*$) – COMPLEX (KIND=nag_wp) arrayInput/Output

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

On entry: the $n$ by $n$ Hermitian matrix $A$.

• If $\mathbf{UPLO}=\text{'U'}$, the upper triangular part of $A$ must be stored and the elements of the array below the diagonal are not referenced.
• If $\mathbf{UPLO}=\text{'L'}$, the lower triangular part of $A$ must be stored and the elements of the array above the diagonal are not referenced.

On exit: the lower triangle (if $\mathbf{UPLO}=\text{'L'}$) or the upper triangle (if $\mathbf{UPLO}=\text{'U'}$) of A, including the diagonal, is overwritten.

7:     LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F08SPF (ZHEGVX) is called.

Constraint: $\mathbf{LDA}\ge \max \left(1,\mathbf{N}\right)$.

8:     B(LDB,$*$) – COMPLEX (KIND=nag_wp) arrayInput/Output

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

On entry: the $n$ by $n$ Hermitian matrix $B$.

• If $\mathbf{UPLO}=\text{'U'}$, the upper triangular part of $B$ must be stored and the elements of the array below the diagonal are not referenced.
• If $\mathbf{UPLO}=\text{'L'}$, the lower triangular part of $B$ must be stored and the elements of the array above the diagonal are not referenced.

On exit: the triangular factor $U$ or $L$ from the Cholesky factorization $B=U^{\mathrm{H}}U$ or $B=L{L}^{\mathrm{H}}$.

9:     LDB – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F08SPF (ZHEGVX) is called.

Constraint: $\mathbf{LDB}\ge \max \left(1,\mathbf{N}\right)$.

10:   VL – REAL (KIND=nag_wp)Input

11:   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 $\mathbf{RANGE}=\text{'V'}$, $\mathbf{VL}<\mathbf{VU}$.

12:   IL – INTEGERInput

13:   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 $\mathbf{N}>0$, $1\le \mathbf{IL}\le \mathbf{IU}\le \mathbf{N}$.

14:   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).

15:   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 $\mathbf{RANGE}=\text{'I'}$, $\mathbf{M}=\mathbf{IU}-\mathbf{IL}+1$.

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

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

17:   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.

18:   LDZ – INTEGERInput

On entry: the first dimension of the array Z as declared in the (sub)program from which F08SPF (ZHEGVX) is called.

Constraints:

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

19:   WORK($\max \left(1,\mathbf{LWORK}\right)$) – COMPLEX (KIND=nag_wp) arrayWorkspace

On exit: if $\mathbf{INFO}=\mathbf{0}$, the real part of $\mathbf{WORK}\left(1\right)$ contains the minimum value of LWORK required for optimal performance.

20:   LWORK – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which F08SPF (ZHEGVX) is called.

If $\mathbf{LWORK}=-1$, a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued.

Suggested value: for optimal performance, $\mathbf{LWORK}\ge \left(\mathit{nb}+1\right)\times \mathbf{N}$, where $\mathit{nb}$ is the optimal block size for F08FSF (ZHETRD).

Constraint: $\mathbf{LWORK}\ge \max \left(1,2\times \mathbf{N}\right)$.

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

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

23:   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.

24:   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 $\mathbf{INFO}=i$, F08FPF (ZHEEVX) failed to converge; $i$ eigenvectors failed to converge. Their indices are stored in array JFAIL.

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

F07FRF (ZPOTRF) 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 (f08spfe.f90)

9.2  Program Data

Program Data (f08spfe.d)

9.3  Program Results

Program Results (f08spfe.r)