nag_dsbgst (f08uec) : NAG Library, Mark 24

3 Description

To reduce the real symmetric-definite generalized eigenproblem

A z = λ B z

to the standard form

C y = λ y

, where

A

B

and

C

are banded, nag_dsbgst (f08uec) must be preceded by a call to nag_dpbstf (f08ufc) which computes the split Cholesky factorization of the positive definite matrix

B

B = S^{T} 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^{T} A X

, where

X = S^{- 1} Q

and

Q

is a orthogonal 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.

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: 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

] – doubleInput/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

symmetric 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 doubleInput

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_dpbstf (f08ufc).

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

] – doubleOutput

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 3.6 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_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.

7 Accuracy

Forming the reduced matrix

C

is a stable procedure. However it involves implicit multiplication by

B^{- 1}

. When nag_dsbgst (f08uec) 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_dsbgst (f08uec) is not threaded by NAG in any implementation.

nag_dsbgst (f08uec) 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.

10 Example

This example computes all the eigenvalues of

A z = λ B z

, where

A = (\begin{array}{r} 0.24 & 0.39 & 0.42 & 0.00 \\ 0.39 & - 0.11 & 0.79 & 0.63 \\ 0.42 & 0.79 & - 0.25 & 0.48 \\ 0.00 & 0.63 & 0.48 & - 0.03 \end{array}) and B = (\begin{array}{r} 2.07 & 0.95 & 0.00 & 0.00 \\ 0.95 & 1.69 & - 0.29 & 0.00 \\ 0.00 & - 0.29 & 0.65 & - 0.33 \\ 0.00 & 0.00 & - 0.33 & 1.17 \end{array}) .

Here

A

is symmetric,

B

is symmetric positive definite, and

A

and

B

are treated as band matrices.

B

must first be factorized by nag_dpbstf (f08ufc). The program calls nag_dsbgst (f08uec) to reduce the problem to the standard form

C y = λ y

, then nag_dsbtrd (f08hec) to reduce

C

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

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_dsbgst (f08uec)

+− 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 Documentnag_dsbgst (f08uec)

+− 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_dsbgst (f08uec)