NAG Library Routine Document

F08UBF (DSBGVX)

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     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 $\text{'V'}$.

2:     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 $\text{'I'}$.

3:     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:     N – INTEGERInput

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

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

5:     KA – INTEGERInput

On entry: if $\mathbf{UPLO}=\text{'U'}$, the number of superdiagonals, $k_a$, of the matrix $A$.

If $\mathbf{UPLO}=\text{'L'}$, the number of subdiagonals, $k_a$, of the matrix $A$.

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

6:     KB – INTEGERInput

On entry: if $\mathbf{UPLO}=\text{'U'}$, the number of superdiagonals, $k_b$, of the matrix $B$.

If $\mathbf{UPLO}=\text{'L'}$, the number of subdiagonals, $k_b$, of the matrix $B$.

Constraint: $\mathbf{KA}\ge \mathbf{KB}\ge 0$.

7:     AB(LDAB,$*$) – REAL (KIND=nag_wp) arrayInput/Output

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

On entry: the upper or lower triangle of the $n$ by $n$ symmetric band matrix $A$.

The matrix is stored in rows $1$ to $k_a+1$, more precisely,

• if $\mathbf{UPLO}=\text{'U'}$, the elements of the upper triangle of $A$ within the band must be stored with element $A_{ij}$ in $\mathbf{AB}\left(k_a+1+i-j,j\right)\text{ for }\max \left(1,j-k_a\right)\le i\le j$;
• if $\mathbf{UPLO}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element $A_{ij}$ in $\mathbf{AB}\left(1+i-j,j\right)\text{ for }j\le i\le \min \left(n,j+k_a\right)\text{.}$

On exit: the contents of AB are overwritten.

8:     LDAB – INTEGERInput

On entry: the first dimension of the array AB as declared in the (sub)program from which F08UBF (DSBGVX) is called.

Constraint: $\mathbf{LDAB}\ge \mathbf{KA}+1$.

9:     BB(LDBB,$*$) – REAL (KIND=nag_wp) arrayInput/Output

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

On entry: the upper or lower triangle of the $n$ by $n$ symmetric positive definite band matrix $B$.

The matrix is stored in rows $1$ to $k_b+1$, more precisely,

• if $\mathbf{UPLO}=\text{'U'}$, the elements of the upper triangle of $B$ within the band must be stored with element $B_{ij}$ in $\mathbf{BB}\left(k_b+1+i-j,j\right)\text{ for }\max \left(1,j-k_b\right)\le i\le j$;
• if $\mathbf{UPLO}=\text{'L'}$, the elements of the lower triangle of $B$ within the band must be stored with element $B_{ij}$ in $\mathbf{BB}\left(1+i-j,j\right)\text{ for }j\le i\le \min \left(n,j+k_b\right)\text{.}$

On exit: the factor $S$ from the split Cholesky factorization $B=S^{\mathrm{T}}S$, as returned by F08UFF (DPBSTF).

10:   LDBB – INTEGERInput

On entry: the first dimension of the array BB as declared in the (sub)program from which F08UBF (DSBGVX) is called.

Constraint: $\mathbf{LDBB}\ge \mathbf{KB}+1$.

11:   Q(LDQ,$*$) – REAL (KIND=nag_wp) arrayOutput

Note: the second dimension of the array Q must be at least $\max \left(1,\mathbf{N}\right)$ if $\mathbf{JOBZ}=\text{'V'}$, and at least $1$ otherwise.

On exit: if $\mathbf{JOBZ}=\text{'V'}$, the $n$ by $n$ matrix, $Q$ used in the reduction of the standard form, i.e., $Cx=\lambda x$, from symmetric banded to tridiagonal form.

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

12:   LDQ – INTEGERInput

On entry: the first dimension of the array Q as declared in the (sub)program from which F08UBF (DSBGVX) is called.

Constraints:

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

13:   VL – REAL (KIND=nag_wp)Input

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

15:   IL – INTEGERInput

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

17:   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).

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

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

On exit: the eigenvalues in ascending order.

20:   Z(LDZ,$*$) – REAL (KIND=nag_wp) arrayOutput

Note: the second dimension of the array Z must be at least $\max \left(1,\mathbf{N}\right)$ if $\mathbf{JOBZ}=\text{'V'}$, and at least $1$ otherwise.

On exit: if $\mathbf{JOBZ}=\text{'V'}$, Z contains the matrix $Z$ of eigenvectors, with the $i$th column of $Z$ holding the eigenvector associated with $\mathbf{W}\left(i\right)$. The eigenvectors are normalized so that $Z^{\mathrm{T}}BZ=I$.

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

21:   LDZ – INTEGERInput

On entry: the first dimension of the array Z as declared in the (sub)program from which F08UBF (DSBGVX) is called.

Constraints:

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

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

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

24:   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.

25:   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 $\mathbf{INFO}=i$, then $i$ eigenvectors failed to converge. Their indices are stored in array JFAIL. Please see ABSTOL.

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

F08UFF (DPBSTF) 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 (f08ubfe.f90)

9.2  Program Data

Program Data (f08ubfe.d)

9.3  Program Results

Program Results (f08ubfe.r)