NAG Library Routine Document

f08jdf  (dstevr)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{jobz}$ – 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 $\text{'V'}$.

2:     $\mathbf{range}$ – 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 $\left(\mathbf{vl},\mathbf{vu}\right]$ 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:     $\mathbf{n}$ – IntegerInput

On entry: $n$, the order of the matrix.

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

4:     $\mathbf{d}\left(*\right)$ – 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 $n$ diagonal elements of the tridiagonal matrix $T$.

On exit: may be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues.

5:     $\mathbf{e}\left(*\right)$ – 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 $\left(n-1\right)$ subdiagonal elements of the tridiagonal matrix $T$.

On exit: may be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues.

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

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

8:     $\mathbf{il}$ – IntegerInput

9:     $\mathbf{iu}$ – 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 $\mathbf{n}>0$, $1\le \mathbf{il}\le \mathbf{iu}\le \mathbf{n}$.

10:   $\mathbf{abstol}$ – 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 $\left[a,b\right]$ 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. See Demmel and Kahan (1990).

If high relative accuracy is important, set abstol to $\mathbf{x02amf}\left( \right)$, although doing so does not currently guarantee that eigenvalues are computed to high relative accuracy. See Barlow and Demmel (1990) for a discussion of which matrices can define their eigenvalues to high relative accuracy.

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

12:   $\mathbf{w}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput

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

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

13:   $\mathbf{z}\left(\mathbf{ldz},*\right)$ – Real (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 $\mathbf{jobz}=\text{'V'}$, 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 $\mathbf{w}\left(i\right)$.

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.

14:   $\mathbf{ldz}$ – IntegerInput

On entry: the first dimension of the array z as declared in the (sub)program from which f08jdf (dstevr) is called.

Constraints:

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

15:   $\mathbf{isuppz}\left(*\right)$ – Integer arrayOutput

Note: the dimension of the array isuppz must be at least $\max \left(1,2\times \mathbf{m}\right)$.

On exit: the support of the eigenvectors in z, i.e., the indices indicating the nonzero elements in z. The $i$th eigenvector is nonzero only in elements $\mathbf{isuppz}\left(2\times i-1\right)$ through $\mathbf{isuppz}\left(2\times i\right)$. Implemented only for $\mathbf{range}=\text{'A'}$ or $\text{'I'}$ and $\mathbf{iu}-\mathbf{il}=\mathbf{n}-1$.

16:   $\mathbf{work}\left(\max \left(1,\mathbf{lwork}\right)\right)$ – 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.

17:   $\mathbf{lwork}$ – IntegerInput

On entry: the dimension of the array work as declared in the (sub)program from which f08jdf (dstevr) 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.

Constraint: $\mathbf{lwork}\ge \max \left(1,20\times \mathbf{n}\right)$.

18:   $\mathbf{iwork}\left(\max \left(1,\mathbf{liwork}\right)\right)$ – Integer arrayWorkspace

On exit: if $\mathbf{info}=\mathbf{0}$, $\mathbf{iwork}\left(1\right)$ returns the minimum liwork.

19:   $\mathbf{liwork}$ – IntegerInput

On entry: the dimension of the array iwork as declared in the (sub)program from which f08jdf (dstevr) 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.

Constraint: $\mathbf{liwork}\ge \max \left(1,120\times \mathbf{n}\right)$.

20:   $\mathbf{info}$ – 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.

$\mathbf{info}>0$

An internal error has occurred in this routine. Please refer to info in f08jjf (dstebz).

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f08jdfe.f90)

10.2

Program Data

Program Data (f08jdfe.d)

10.3

Program Results

Program Results (f08jdfe.r)