f08jxc computes the eigenvectors of a real symmetric tridiagonal matrix corresponding to specified eigenvalues, by inverse iteration, storing the eigenvectors in a complex array.

2 Specification

#include <nag.h>

void	f08jxc (Nag_OrderType order, Integer n, const double d[], const double e[], Integer m, const double w[], const Integer iblock[], const Integer isplit[], Complex z[], Integer pdz, Integer ifailv[], NagError *fail)

The function may be called by the names: f08jxc, nag_lapackeig_zstein or nag_zstein.

3 Description

f08jxc computes the eigenvectors of a real symmetric tridiagonal matrix

T

corresponding to specified eigenvalues, by inverse iteration (see Jessup and Ipsen (1992)). It is designed to be used in particular after the specified eigenvalues have been computed by f08jjc with

rank = Nag_ByBlock

, but may also be used when the eigenvalues have been computed by other functions in Chapters F02 or F08.

The eigenvectors of

T

are real, but are stored by this function in a complex array. If

T

has been formed by reduction of a full complex Hermitian matrix

A

to tridiagonal form, then eigenvectors of

T

may be transformed to (complex) eigenvectors of

A

by a call to f08fuc or f08guc.

f08jjc determines whether the matrix

T

splits into block diagonal form:

T = (\begin{array}{r} T_{1} \\ T_{2} \\ . \\ . \\ . \\ T_{p} \end{array})

and passes details of the block structure to this function in the arrays iblock and isplit. This function can then take advantage of the block structure by performing inverse iteration on each block

T_{i}

separately, which is more efficient than using the whole matrix.

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Jessup E and Ipsen I C F (1992) Improving the accuracy of inverse iteration SIAM J. Sci. Statist. Comput. 13 550–572

5 Arguments

1: $order$ – Nag_OrderType Input

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $n$ – Integer Input

On entry:

n

, the order of the matrix

T

Constraint:

n \geq 0

3: $d [\dim]$ – const double Input

Note: the dimension, dim, of the array d must be at least

\max (1, n)

On entry: the diagonal elements of the tridiagonal matrix

T

4: $e [\dim]$ – const double Input

Note: the dimension, dim, of the array e must be at least

\max (1, n - 1)

On entry: the off-diagonal elements of the tridiagonal matrix

T

5: $m$ – Integer Input

On entry:

m

, the number of eigenvectors to be returned.

Constraint:

0 \leq m \leq n

6: $w [\dim]$ – const double Input

Note: the dimension, dim, of the array w must be at least

\max (1, n)

On entry: the eigenvalues of the tridiagonal matrix

T

stored in

w [0]

w [m - 1]

, as returned by f08jjc with

rank = Nag_ByBlock

. 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

iblock [i] = iblock [i + 1]

w [i] \leq w [i + 1]

, for

i = 0, 1, \dots, m - 2

7: $iblock [\dim]$ – const Integer Input

Note: the dimension, dim, of the array iblock must be at least

\max (1, n)

On entry: the first

m

elements must contain the sub-matrix indices associated with the specified eigenvalues, as returned by f08jjc with

rank = Nag_ByBlock

. If the eigenvalues were not computed by f08jjc with

rank = Nag_ByBlock

, set

iblock [i - 1]

1

, for

i = 1, 2, \dots, m

Constraint:

iblock [i] \leq iblock [i + 1]

, for

i = 0, 1, \dots, m - 2

8: $isplit [\dim]$ – const Integer Input

Note: the dimension, dim, of the array isplit must be at least

\max (1, n)

On entry: the points at which

T

breaks up into sub-matrices, as returned by f08jjc with

rank = Nag_ByBlock

. If the eigenvalues were not computed by f08jjc with

rank = Nag_ByBlock

, set

isplit [0]

to n.

9: $z [\dim]$ – Complex Output

Note: the dimension, dim, of the array z must be at least

$\max (1, pdz \times m)$ when $order = Nag_ColMajor$ ;
$\max (1, n \times pdz)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

Z

is stored in

$z [(j - 1) \times pdz + i - 1]$ when $order = Nag_ColMajor$ ;
$z [(i - 1) \times pdz + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the

m

eigenvectors, stored as columns of

Z

; the

i

th column corresponds to the

i

th specified eigenvalue, unless

fail . code =

NE_CONVERGENCE (in which case see Section 6).

10: $pdz$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array z.

Constraints:

if $order = Nag_ColMajor$ , $pdz \geq \max (1, n)$ ;
if $order = Nag_RowMajor$ , $pdz \geq \max (1, m)$ .

11: $ifailv [m]$ – Integer Output

On exit: if

fail . errnum = 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

12: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_CONSTRAINT: Constraint: $iblock [i] \leq iblock [i + 1]$ , for $i = 0, 1, \dots, m - 2$ .

Constraint: if $iblock [i] = iblock [i + 1]$ , $w [i] \leq w [i + 1]$ , for $i = 0, 1, \dots, m - 2$ .
NE_CONVERGENCE: $⟨ value ⟩$ eigenvectors (as indicated by argument ifailv) each failed to converge in five iterations. The current iterate after five iterations is stored in the corresponding column of z.
NE_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

On entry, $pdz = ⟨ value ⟩$ .
Constraint: $pdz > 0$ .
NE_INT_2: On entry, $m = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $0 \leq m \leq n$ .

On entry, $pdz = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: $pdz \geq \max (1, m)$ .

On entry, $pdz = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pdz \geq \max (1, n)$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

7 Accuracy

Each computed eigenvector

z_{i}

is the exact eigenvector of a nearby matrix

A + E_{i}

, such that

‖ E_{i} ‖ = O (ε) ‖ A ‖,

where

ε

is the machine precision. Hence the residual is small:

‖ A z_{i} - λ_{i} z_{i} ‖ = O (ε) ‖ A ‖ .

However, a set of eigenvectors computed by this function may not be orthogonal to so high a degree of accuracy as those computed by f08jsc.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

f08jxc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08jxc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.