NAG Library Routine Document

F08JHF (DSTEDC)

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     COMPZ – CHARACTER(1)Input

On entry: indicates whether the eigenvectors are to be computed.

$\mathbf{COMPZ}=\text{'N'}$

Only the eigenvalues are computed (and the array Z is not referenced).

$\mathbf{COMPZ}=\text{'I'}$

The eigenvalues and eigenvectors of $T$ are computed (and the array Z is initialized by the routine).

$\mathbf{COMPZ}=\text{'V'}$

The eigenvalues and eigenvectors of $A$ are computed (and the array Z must contain the matrix $Q$ on entry).

Constraint: $\mathbf{COMPZ}=\text{'N'}$, $\text{'V'}$ or $\text{'I'}$.

2:     N – INTEGERInput

On entry: $n$, the order of the symmetric tridiagonal matrix $T$.

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

3:     D($*$) – REAL (KIND=nag_wp) arrayInput/Output

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

On entry: the diagonal elements of the tridiagonal matrix.

On exit: if $\mathbf{INFO}=\mathbf{0}$, the eigenvalues in ascending order.

4:     E($*$) – REAL (KIND=nag_wp) arrayInput/Output

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

On entry: the subdiagonal elements of the tridiagonal matrix.

On exit: E is overwritten.

5:     Z(LDZ,$*$) – REAL (KIND=nag_wp) arrayInput/Output

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

On entry: if $\mathbf{COMPZ}=\text{'V'}$, Z must contain the orthogonal matrix $Q$ used in the reduction to tridiagonal form.

On exit: if $\mathbf{COMPZ}=\text{'V'}$, Z contains the orthonormal eigenvectors of the original symmetric matrix $A$, and if $\mathbf{COMPZ}=\text{'I'}$, Z contains the orthonormal eigenvectors of the symmetric tridiagonal matrix $T$.

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

6:     LDZ – INTEGERInput

On entry: the first dimension of the array Z as declared in the (sub)program from which F08JHF (DSTEDC) is called.

Constraints:

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

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

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

8:     LWORK – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which F08JHF (DSTEDC) is called.

If $\mathbf{LWORK}=-1$, a workspace query is assumed; the routine only calculates the minimum sizes of the WORK and IWORK arrays, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued.

Constraints:

if $\mathbf{LWORK}\ne -1$,

• if $\mathbf{COMPZ}=\text{'N'}$ or $\mathbf{N}\le 1$, LWORK must be at least $1$;
• if $\mathbf{COMPZ}=\text{'V'}$ and $\mathbf{N}>1$, LWORK must be at least $\left(1+3\times \mathbf{N}+2\times \mathbf{N}\times \lg \left(\mathbf{N}\right)+4\times {\mathbf{N}}^2\right)$, where $\lg \left(\mathbf{N}\right)=$ smallest integer $k$ such that $2^k\ge \mathbf{N}$;
• if $\mathbf{COMPZ}=\text{'I'}$ and $\mathbf{N}>1$, LWORK must be at least $\left(1+4\times \mathbf{N}+{\mathbf{N}}^2\right)$.

Note: that for $\mathbf{COMPZ}=\text{'I'}$ or $\text{'V'}$ then if N is less than or equal to the minimum divide size, usually $25$, then LWORK need only be $\max \left(1,2\times \left(\mathbf{N}-1\right)\right)$.

9:     IWORK($\max \left(1,\mathbf{LIWORK}\right)$) – INTEGER arrayWorkspace

On exit: if $\mathbf{INFO}=\mathbf{0}$, $\mathbf{IWORK}\left(1\right)$ returns the minimum LIWORK.

10:   LIWORK – INTEGERInput

On entry: the dimension of the array IWORK as declared in the (sub)program from which F08JHF (DSTEDC) is called.

If $\mathbf{LIWORK}=-1$, a workspace query is assumed; the routine only calculates the minimum sizes of the WORK and IWORK arrays, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued.

Constraints:

if $\mathbf{LIWORK}\ne -1$,

• if $\mathbf{COMPZ}=\text{'N'}$ or $\mathbf{N}\le 1$, LIWORK must be at least $1$;
• if $\mathbf{COMPZ}=\text{'V'}$ and $\mathbf{N}>1$, LIWORK must be at least $\left(6+6\times \mathbf{N}+5\times \mathbf{N}\times \lg \left(\mathbf{N}\right)\right)$;
• if $\mathbf{COMPZ}=\text{'I'}$ and $\mathbf{N}>1$, LIWORK must be at least $\left(3+5\times \mathbf{N}\right)$.

Note: that for $\mathbf{COMPZ}=\text{'I'}$ or $\text{'V'}$, then if N is less than or equal to the minimum divide size, usually $25$, then LIWORK need only be $1$.

11:   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.

$\mathbf{INFO}>0$

The algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns $\mathbf{INFO}/\left(\mathbf{N}+1\right)$ through $\mathbf{INFO}\mathrm{ mod }\left(\mathbf{N}+1\right)$.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (f08jhfe.f90)

9.2  Program Data

Program Data (f08jhfe.d)

9.3  Program Results

Program Results (f08jhfe.r)