void	nag_zheevr (Nag_OrderType order, Nag_JobType job, Nag_RangeType range, Nag_UploType uplo, Integer n, Complex a[], Integer pda, double vl, double vu, Integer il, Integer iu, double abstol, Integer m, double w[], Complex z[], Integer pdz, Integer isuppz[], NagError fail)

3

Description

The Hermitian matrix is first reduced to a real tridiagonal matrix

T

, using unitary similarity transformations. Then whenever possible, nag_zheevr (f08frc) computes the eigenspectrum using Relatively Robust Representations. nag_zheevr (f08frc) computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various ‘good’

L D L^{T}

representations (also known as Relatively Robust Representations). Gram–Schmidt orthogonalization is avoided as far as possible. More specifically, the various steps of the algorithm are as follows. For the

i

th unreduced block of

T

(a)	compute $T - σ_{i} I = L_{i} D_{i} L_{i}^{T}$ , such that $L_{i} D_{i} L_{i}^{T}$ is a relatively robust representation,
(b)	compute the eigenvalues, $λ_{j}$ , of $L_{i} D_{i} L_{i}^{T}$ to high relative accuracy by the dqds algorithm,
(c)	if there is a cluster of close eigenvalues, ‘choose’ $σ_{i}$ close to the cluster, and go to (a),
(d)	given the approximate eigenvalue $λ_{j}$ of $L_{i} D_{i} L_{i}^{T}$ , compute the corresponding eigenvector by forming a rank-revealing twisted factorization.

The desired accuracy of the output can be specified by the argument abstol. For more details, see Dhillon (1997) and Parlett and Dhillon (2000).

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

Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput. 11 873–912

Dhillon I (1997) A new

O (n^{2})

algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem Computer Science Division Technical Report No. UCB//CSD-97-971 UC Berkeley

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

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 3.3.1.3 in How to Use the NAG Library and its Documentation 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: if

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 to iuth eigenvalues will be found.

For

range = Nag_Interval

Nag_Indices

and

iu - il < n - 1

, nag_dstebz (f08jjc) and nag_zstein (f08jxc) are called.

Constraint:

range = Nag_AllValues

Nag_Interval

Nag_Indices

4: $uplo$ – Nag_UploTypeInput

On entry: if

uplo = Nag_Upper

, the upper triangular part of

A

is stored.

uplo = Nag_Lower

, the lower triangular part of

A

is stored.

Constraint:

uplo = Nag_Upper

Nag_Lower

5: $n$ – IntegerInput

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

6: $a [\dim]$ – ComplexInput/Output

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

\max (1, pda \times n)

On entry: the

n

n

Hermitian matrix

A

order = Nag_ColMajor

A_{i j}

is stored in

a [(j - 1) \times pda + i - 1]

order = Nag_RowMajor

A_{i j}

is stored in

a [(i - 1) \times pda + j - 1]

uplo = Nag_Upper

, the upper triangular part of

A

must be stored and the elements of the array below the diagonal are not referenced.

uplo = Nag_Lower

, the lower triangular part of

A

must be stored and the elements of the array above the diagonal are not referenced.

On exit: the lower triangle (if

uplo = Nag_Lower

) or the upper triangle (if

uplo = Nag_Upper

) of a, including the diagonal, is overwritten.

7: $pda$ – IntegerInput

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

Constraint:

pda \geq \max (1, n)

8: $vl$ – doubleInput

9: $vu$ – doubleInput

On entry: if

range = Nag_Interval

, the lower and upper bounds 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

10: $il$ – IntegerInput

11: $iu$ – IntegerInput

On entry: if

range = Nag_Indices

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

range = Nag_AllValues

Nag_Interval

, il and iu are not referenced.

Constraints:

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

12: $abstol$ – doubleInput

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

[a, b]

of width less than or equal to

abstol + ε \max (|a|, |b|),

where

ε

is the machine precision. If abstol is less than or equal to zero, then

ε {‖T‖}_{1}

will be used in its place, where

T

is the real tridiagonal matrix obtained by reducing

A

to tridiagonal form. See Demmel and Kahan (1990).

If high relative accuracy is important, set abstol to

nag_real_safe_small_number ()

, 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.

13: $m$ – 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

14: $w [\dim]$ – doubleOutput

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

\max (1, n)

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

15: $z [\dim]$ – 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

, 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

w [i - 1]

job = Nag_EigVals

, z is not referenced.

16: $pdz$ – 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 $pdz \geq 1$ .

17: $isuppz [\dim]$ – 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]

. Implemented only for

range = Nag_AllValues

Nag_Indices

and

iu - il = n - 1

18: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_CONVERGENCE: nag_zheevr (f08frc) 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 $n = 0$ , $il = 1$ and $iu = 0$ ;
if $range = Nag_Indices$ and $n > 0$ , $1 \leq il \leq iu \leq n$ .
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, $pda = 〈value〉$ .
Constraint: $pda > 0$ .

On entry, $pdz = 〈value〉$ .
Constraint: $pdz > 0$ .
NE_INT_2: On entry, $pda = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pda \geq \max (1, n)$ .
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix

(A + E)

, where

{‖E‖}_{2} = O (ε) {‖A‖}_{2},

and

ε

is the machine precision. See Section 4.7 of Anderson et al. (1999) for further details.

8

Parallelism and Performance

nag_zheevr (f08frc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_zheevr (f08frc) 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.

9

Further Comments

The total number of floating-point operations is proportional to

n^{3}

The real analogue of this function is nag_dsyevr (f08fdc).

10

Example

This example finds the eigenvalues with indices in the range

[2, 3]

, and the corresponding eigenvectors, of the Hermitian matrix

A = (\begin{array}{l} 1 & 2 - i & 3 - i & 4 - i \\ 2 + i & 2 & 3 - 2 i & 4 - 2 i \\ 3 + i & 3 + 2 i & 3 & 4 - 3 i \\ 4 + i & 4 + 2 i & 4 + 3 i & 4 \end{array}) .

NAG Library Function Document

nag_zheevr (f08frc)

▸▿ 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

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