NAG Library Function Document

nag_zstegr (f08jyc)

void	nag_zstegr (Nag_OrderType order, Nag_JobType job, Nag_RangeType range, Integer n, double d[], double e[], double vl, double vu, Integer il, Integer iu, Integer m, double w[], Complex z[], Integer pdz, Integer isuppz[], NagError fail)

3 Description

nag_zstegr (f08jyc) computes all the eigenvalues and, optionally, the eigenvectors, of a real symmetric tridiagonal matrix

T

. That is, the function computes the spectral factorization of

T

given by

T = Z Λ Z^{T},

where

Λ

is a diagonal matrix whose diagonal elements are the eigenvalues,

λ_{i}

, of

T

and

Z

is an orthogonal matrix whose columns are the eigenvectors,

z_{i}

, of

T

. Thus

T z_{i} = λ_{i} z_{i}, i = 1, 2, \dots, n .

The function stores the real orthogonal matrix

Z

in a complex array, so that it may also be used to compute all the eigenvalues and eigenvectors of a complex Hermitian matrix

A

which has been reduced to tridiagonal form

T

\begin{array}{l} A & = Q T Q^{H}, where ​ Q ​ is unitary \\ = (Q Z) Λ {(Q Z)}^{H} . \end{array}

In this case, the matrix

Q

must be explicitly applied to the output matrix

Z

. The functions which must be called to perform the reduction to tridiagonal form and apply

Q

are:

full matrix	nag_zhetrd (f08fsc) and nag_zunmtr (f08fuc)
full matrix, packed storage	nag_zhptrd (f08gsc) and nag_zupmtr (f08guc)
band matrix	nag_zhbtrd (f08hsc) with $vect = Nag_FormQ$ and nag_zgemm (f16zac).

This function uses the dqds and the Relatively Robust Representation algorithms to compute the eigenvalues and eigenvectors respectively; see for example Parlett and Dhillon (2000) and Dhillon and Parlett (2004) for further details. nag_zstegr (f08jyc) can usually compute all the eigenvalues and eigenvectors in

O (n^{2})

floating-point operations and so, for large matrices, is often considerably faster than the other symmetric tridiagonal functions in this chapter when all the eigenvectors are required, particularly so compared to those functions that are based on the

Q R

algorithm.

4 References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug

Barlow J and Demmel J W (1990) Computing accurate eigensystems of scaled diagonally dominant matrices SIAM J. Numer. Anal. 27 762–791

Dhillon I S and Parlett B N (2004) Orthogonal eigenvectors and relative gaps. SIAM J. Appl. Math. 25 858–899

Parlett B N and Dhillon I S (2000) Relatively robust representations of symmetric tridiagonals Linear Algebra Appl. 309 121–151

5 Arguments

1: order – Nag_OrderTypeInput

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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: job – Nag_JobTypeInput

On entry: indicates whether eigenvectors are computed.

$job = Nag_EigVals$: Only eigenvalues are computed.
$job = Nag_DoBoth$: Eigenvalues and eigenvectors are computed.

Constraint:

job = Nag_EigVals

Nag_DoBoth

3: range – Nag_RangeTypeInput

On entry: indicates which eigenvalues should be returned.

$range = Nag_AllValues$: All eigenvalues will be found.
$range = Nag_Interval$: All eigenvalues in the half-open interval $(vl, vu]$ will be found.
$range = Nag_Indices$: The ilth through iuth eigenvectors will be found.

Constraint:

range = Nag_AllValues

Nag_Interval

Nag_Indices

4: n – IntegerInput

On entry:

n

, the order of the matrix

T

Constraint:

n \geq 0

5: d[ $\dim$ ] – doubleInput/Output

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

\max (1, n)

On entry: the

n

diagonal elements of the tridiagonal matrix

T

On exit: d is overwritten.

6: e[ $\dim$ ] – doubleInput/Output

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

\max (1, n)

On entry:

e [0]

e [n - 2]

are the subdiagonal elements of the tridiagonal matrix

T

e [n - 1]

need not be set.

On exit: e is overwritten.

7: vl – doubleInput

8: vu – doubleInput

On entry: if

range = Nag_Interval

, vl and vu contain the lower and upper bounds respectively of the interval to be searched for eigenvalues.

range = Nag_AllValues

Nag_Indices

, vl and vu are not referenced.

Constraint: if

range = Nag_Interval

vl < vu

9: il – IntegerInput

10: iu – IntegerInput

On entry: if

range = Nag_Indices

, il and iu contains the indices (in ascending order) of the smallest and largest eigenvalues to be returned, respectively.

range = Nag_AllValues

Nag_Interval

, il and iu are not referenced.

Constraints:

if $range = Nag_Indices$ and $n > 0$ , $1 \leq il \leq iu \leq n$ ;
if $range = Nag_Indices$ and $n = 0$ , $il = 1$ and $iu = 0$ .

11: m – Integer *Output

On exit: the total number of eigenvalues found.

0 \leq m \leq n

range = Nag_AllValues

m = n

range = Nag_Indices

m = iu - il + 1

12: w[ $\dim$ ] – doubleOutput

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

\max (1, n)

On exit: the eigenvalues in ascending order.

13: z[ $\dim$ ] – ComplexOutput

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

$\max (1, pdz \times n)$ when $job = Nag_DoBoth$ ;
$1$ otherwise.

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: if

job = Nag_DoBoth

, then if

fail . code =

NE_NOERROR, the columns of z contain the orthonormal eigenvectors of the matrix

T

, with the

i

th column of

Z

holding the eigenvector associated with

w [i - 1]

job = Nag_EigVals

, z is not referenced.

14: pdz – IntegerInput

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

Constraints:

if $job = Nag_DoBoth$ , $pdz \geq \max (1, n)$ ;
otherwise $pdz \geq 1$ .

15: isuppz[ $\dim$ ] – IntegerOutput

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

\max (1, 2 \times m)

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

isuppz [2 \times i - 2]

through

isuppz [2 \times i - 1]

16: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_CONVERGENCE: Inverse iteration failed to converge.
The $dqds$ algorithm failed to converge.
NE_ENUM_INT_2: On entry, $job = ⟨value⟩$ , $pdz = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: if $job = Nag_DoBoth$ , $pdz \geq \max (1, n)$ ;
otherwise $pdz \geq 1$ .
NE_ENUM_INT_3: On entry, $range = ⟨value⟩$ , $il = ⟨value⟩$ , $iu = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: if $range = Nag_Indices$ and $n > 0$ , $1 \leq il \leq iu \leq n$ ;
if $range = Nag_Indices$ and $n = 0$ , $il = 1$ and $iu = 0$ .
NE_ENUM_REAL_2: On entry, $range = ⟨value⟩$ , $vl = ⟨value⟩$ and $vu = ⟨value⟩$ .
Constraint: if $range = Nag_Interval$ , $vl < vu$ .
NE_INT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .
On entry, $pdz = ⟨value⟩$ .
Constraint: $pdz > 0$ .
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.

7 Accuracy

See Section 4.7 of Anderson et al. (1999) and Barlow and Demmel (1990) for further details.

8 Parallelism and Performance

nag_zstegr (f08jyc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_zstegr (f08jyc) 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 Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The total number of floating-point operations required to compute all the eigenvalues and eigenvectors is approximately proportional to

n^{2}

The real analogue of this function is nag_dstegr (f08jlc).

10 Example

This example finds all the eigenvalues and eigenvectors of the symmetric tridiagonal matrix

T = (\begin{array}{r} 1.0 & 1.0 & 0 & 0 \\ 1.0 & 4.0 & 2.0 & 0 \\ 0 & 2.0 & 9.0 & 3.0 \\ 0 & 0 & 3.0 & 16.0 \end{array}) .

NAG Library Function Documentnag_zstegr (f08jyc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_zstegr (f08jyc)