void	nag_zhbgst (Nag_OrderType order, Nag_VectType vect, Nag_UploType uplo, Integer n, Integer ka, Integer kb, Complex ab[], Integer pdab, const Complex bb[], Integer pdbb, Complex x[], Integer pdx, NagError *fail)

3

Description

To reduce the complex Hermitian-definite generalized eigenproblem

A z = λ B z

to the standard form

C y = λ y

, where

A

B

and

C

are banded, nag_zhbgst (f08usc) must be preceded by a call to nag_zpbstf (f08utc) which computes the split Cholesky factorization of the positive definite matrix

B

B = S^{H} S

. The split Cholesky factorization, compared with the ordinary Cholesky factorization, allows the work to be approximately halved.

This function overwrites

A

with

C = X^{H} A X

, where

X = S^{- 1} Q

and

Q

is a unitary matrix chosen (implicitly) to preserve the bandwidth of

A

. The function also has an option to allow the accumulation of

X

, and then, if

z

is an eigenvector of

C

X z

is an eigenvector of the original system.

4

References

Crawford C R (1973) Reduction of a band-symmetric generalized eigenvalue problem Comm. ACM 16 41–44

Kaufman L (1984) Banded eigenvalue solvers on vector machines ACM Trans. Math. Software 10 73–86

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: $vect$ – Nag_VectTypeInput

On entry: indicates whether

X

is to be returned.

$vect = Nag_DoNotForm$: $X$ is not returned.
$vect = Nag_FormX$: $X$ is returned.

Constraint:

vect = Nag_DoNotForm

Nag_FormX

3: $uplo$ – Nag_UploTypeInput

On entry: indicates whether the upper or lower triangular part of

A

is stored.

$uplo = Nag_Upper$: 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

4: $n$ – IntegerInput

On entry:

n

, the order of the matrices

A

and

B

Constraint:

n \geq 0

5: $ka$ – IntegerInput

On entry: if

uplo = Nag_Upper

, the number of superdiagonals,

k_{a}

, of the matrix

A

uplo = Nag_Lower

, the number of subdiagonals,

k_{a}

, of the matrix

A

Constraint:

ka \geq 0

6: $kb$ – IntegerInput

On entry: if

uplo = Nag_Upper

, the number of superdiagonals,

k_{b}

, of the matrix

B

uplo = Nag_Lower

, the number of subdiagonals,

k_{b}

, of the matrix

B

Constraint:

ka \geq kb \geq 0

7: $ab [\dim]$ – ComplexInput/Output

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

\max (1, pdab \times n)

On entry: the upper or lower triangle of the

n

n

Hermitian band matrix

A

This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of

A_{i j}

, depends on the order and uplo arguments as follows:

if $order = Nag_ColMajor$ and $uplo = Nag_Upper$ ,
$A_{i j}$ is stored in $ab [k_{a} + i - j + (j - 1) \times pdab]$ , for $j = 1, \dots, n$ and $i = \max (1, j - k_{a}), \dots, j$ ;
if $order = Nag_ColMajor$ and $uplo = Nag_Lower$ ,
$A_{i j}$ is stored in $ab [i - j + (j - 1) \times pdab]$ , for $j = 1, \dots, n$ and $i = j, \dots, \min (n, j + k_{a})$ ;
if $order = Nag_RowMajor$ and $uplo = Nag_Upper$ ,
$A_{i j}$ is stored in $ab [j - i + (i - 1) \times pdab]$ , for $i = 1, \dots, n$ and $j = i, \dots, \min (n, i + k_{a})$ ;
if $order = Nag_RowMajor$ and $uplo = Nag_Lower$ ,
$A_{i j}$ is stored in $ab [k_{a} + j - i + (i - 1) \times pdab]$ , for $i = 1, \dots, n$ and $j = \max (1, i - k_{a}), \dots, i$ .

On exit: the upper or lower triangle of ab is overwritten by the corresponding upper or lower triangle of

C

as specified by uplo.

8: $pdab$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) of the matrix

A

in the array ab.

Constraint:

pdab \geq ka + 1

9: $bb [\dim]$ – const ComplexInput

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

\max (1, pdbb \times n)

On entry: the banded split Cholesky factor of

B

as specified by uplo, n and kb and returned by nag_zpbstf (f08utc).

10: $pdbb$ – IntegerInput

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

Constraint:

pdbb \geq kb + 1

11: $x [\dim]$ – ComplexOutput

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

$\max (1, pdx \times n)$ when $vect = Nag_FormX$ ;
$1$ when $vect = Nag_DoNotForm$ .

The

(i, j)

th element of the matrix

X

is stored in

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

On exit: the

n

n

matrix

X = S^{- 1} Q

, if

vect = Nag_FormX

vect = Nag_DoNotForm

, x is not referenced.

12: $pdx$ – IntegerInput

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

Constraints:

if $vect = Nag_FormX$ , $pdx \geq \max (1, n)$ ;
if $vect = Nag_DoNotForm$ , $pdx \geq 1$ .

13: $fail$ – 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_ENUM_INT_2: On entry, $vect = 〈value〉$ , $pdx = 〈value〉$ and $n = 〈value〉$ .
Constraint: if $vect = Nag_FormX$ , $pdx \geq \max (1, n)$ ;
if $vect = Nag_DoNotForm$ , $pdx \geq 1$ .
NE_INT: On entry, $ka = 〈value〉$ .
Constraint: $ka \geq 0$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .

On entry, $pdab = 〈value〉$ .
Constraint: $pdab > 0$ .

On entry, $pdbb = 〈value〉$ .
Constraint: $pdbb > 0$ .

On entry, $pdx = 〈value〉$ .
Constraint: $pdx > 0$ .
NE_INT_2: On entry, $ka = 〈value〉$ and $kb = 〈value〉$ .
Constraint: $ka \geq kb \geq 0$ .

On entry, $pdab = 〈value〉$ and $ka = 〈value〉$ .
Constraint: $pdab \geq ka + 1$ .

On entry, $pdbb = 〈value〉$ and $kb = 〈value〉$ .
Constraint: $pdbb \geq kb + 1$ .
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

Forming the reduced matrix

C

is a stable procedure. However it involves implicit multiplication by

B^{- 1}

. When nag_zhbgst (f08usc) is used as a step in the computation of eigenvalues and eigenvectors of the original problem, there may be a significant loss of accuracy if

B

is ill-conditioned with respect to inversion.

8

Parallelism and Performance

nag_zhbgst (f08usc) 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 real floating-point operations is approximately

20 n^{2} k_{B}

, when

vect = Nag_DoNotForm

, assuming

n ≫ k_{A}, k_{B}

; there are an additional

5 n^{3} (k_{B} / k_{A})

operations when

vect = Nag_FormX

The real analogue of this function is nag_dsbgst (f08uec).

10

Example

This example computes all the eigenvalues of

A z = λ B z

, where

A = (\begin{array}{r} - 1.13 + 0.00 i & 1.94 - 2.10 i & - 1.40 + 0.25 i & 0.00 + 0.00 i \\ 1.94 + 2.10 i & - 1.91 + 0.00 i & - 0.82 - 0.89 i & - 0.67 + 0.34 i \\ - 1.40 - 0.25 i & - 0.82 + 0.89 i & - 1.87 + 0.00 i & - 1.10 - 0.16 i \\ 0.00 + 0.00 i & - 0.67 - 0.34 i & - 1.10 + 0.16 i & 0.50 + 0.00 i \end{array})

and

B = (\begin{array}{r} 9.89 + 0.00 i & 1.08 - 1.73 i & 0.00 + 0.00 i & 0.00 + 0.00 i \\ 1.08 + 1.73 i & 1.69 + 0.00 i & - 0.04 + 0.29 i & 0.00 + 0.00 i \\ 0.00 + 0.00 i & - 0.04 - 0.29 i & 2.65 + 0.00 i & - 0.33 + 2.24 i \\ 0.00 + 0.00 i & 0.00 + 0.00 i & - 0.33 - 2.24 i & 2.17 + 0.00 i \end{array}) .

Here

A

is Hermitian,

B

is Hermitian positive definite, and

A

and

B

are treated as band matrices.

B

must first be factorized by nag_zpbstf (f08utc). The program calls nag_zhbgst (f08usc) to reduce the problem to the standard form

C y = λ y

, then nag_zhbtrd (f08hsc) to reduce

C

to tridiagonal form, and nag_dsterf (f08jfc) to compute the eigenvalues.

NAG Library Function Document

nag_zhbgst (f08usc)

▸▿ 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