NAG Library Routine Document

f08jxf  (zstein)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{n}$ – IntegerInput

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

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

2:     $\mathbf{d}\left(*\right)$ – Real (Kind=nag_wp) arrayInput

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 $T$.

3:     $\mathbf{e}\left(*\right)$ – Real (Kind=nag_wp) arrayInput

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

On entry: the off-diagonal elements of the tridiagonal matrix $T$.

4:     $\mathbf{m}$ – IntegerInput

On entry: $m$, the number of eigenvectors to be returned.

Constraint: $0\le \mathbf{m}\le \mathbf{n}$.

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

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

On entry: the eigenvalues of the tridiagonal matrix $T$ stored in $\mathbf{w}\left(1\right)$ to $\mathbf{w}\left(m\right)$, as returned by f08jjf (dstebz) with $\mathbf{order}=\text{'B'}$. Eigenvalues associated with the first sub-matrix must be supplied first, in nondecreasing order; then those associated with the second sub-matrix, again in nondecreasing order; and so on.

Constraint: if $\mathbf{iblock}\left(\mathit{i}\right)=\mathbf{iblock}\left(\mathit{i}+1\right)$, $\mathbf{w}\left(\mathit{i}\right)\le \mathbf{w}\left(\mathit{i}+1\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{m}-1$.

6:     $\mathbf{iblock}\left(*\right)$ – Integer arrayInput

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

On entry: the first $m$ elements must contain the sub-matrix indices associated with the specified eigenvalues, as returned by f08jjf (dstebz) with $\mathbf{order}=\text{'B'}$. If the eigenvalues were not computed by f08jjf (dstebz) with $\mathbf{order}=\text{'B'}$, set $\mathbf{iblock}\left(\mathit{i}\right)$ to $1$, for $\mathit{i}=1,2,\dots ,m$.

Constraint: $\mathbf{iblock}\left(\mathit{i}\right)\le \mathbf{iblock}\left(\mathit{i}+1\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{m}-1$.

7:     $\mathbf{isplit}\left(*\right)$ – Integer arrayInput

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

On entry: the points at which $T$ breaks up into sub-matrices, as returned by f08jjf (dstebz) with $\mathbf{order}=\text{'B'}$. If the eigenvalues were not computed by f08jjf (dstebz) with $\mathbf{order}=\text{'B'}$, set $\mathbf{isplit}\left(1\right)$ to n.

8:     $\mathbf{z}\left(\mathbf{ldz},*\right)$ – Complex (Kind=nag_wp) arrayOutput

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

On exit: the $m$ eigenvectors, stored as columns of $Z$; the $i$th column corresponds to the $i$th specified eigenvalue, unless $\mathbf{info}>\mathbf{0}$ (in which case see Section 6).

9:     $\mathbf{ldz}$ – IntegerInput

On entry: the first dimension of the array z as declared in the (sub)program from which f08jxf (zstein) is called.

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

10:   $\mathbf{work}\left(5\times \mathbf{n}\right)$ – Real (Kind=nag_wp) arrayWorkspace

11:   $\mathbf{iwork}\left(\mathbf{n}\right)$ – Integer arrayWorkspace

12:   $\mathbf{ifailv}\left(\mathbf{m}\right)$ – Integer arrayOutput

On exit: if $\mathbf{info}=i>0$, the first $i$ elements of ifailv contain the indices of any eigenvectors which have failed to converge. The rest of the first m elements of ifailv are set to $0$.

13:   $\mathbf{info}$ – 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.

$\mathbf{info}>0$

If $\mathbf{info}=i$, then $i$ eigenvectors (as indicated by the argument ifailv above) each failed to converge in five iterations. The current iterate after five iterations is stored in the corresponding column of z.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example