void	f08uec (Nag_OrderType order, Nag_VectType vect, Nag_UploType uplo, Integer n, Integer ka, Integer kb, double ab[], Integer pdab, const double bb[], Integer pdbb, double x[], Integer pdx, NagError *fail)

The function may be called by the names: f08uec, nag_lapackeig_dsbgst or nag_dsbgst.

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, f08uec must be preceded by a call to 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.

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_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: $vect$ – Nag_VectType Input

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_UploType Input

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: $n$ – Integer Input

On entry:

n

, the order of the matrices

A

and

B

Constraint:

n \geq 0

5: $ka$ – Integer Input

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: $kb$ – Integer Input

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: $ab [\dim]$ – double Input/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 \times 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: $pdab$ – Integer Input

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 double Input

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 f08ufc.

10: $pdbb$ – Integer Input

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: $x [\dim]$ – double Output

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 \times n

matrix

X = S^{- 1} Q

, if

vect = Nag_FormX

vect = Nag_DoNotForm

, x is not referenced.

12: $pdx$ – Integer Input

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 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_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 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

Forming the reduced matrix

C

is a stable procedure. However it involves implicit multiplication by

B^{- 1}

. When 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

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

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 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 approximately

6 n^{2} k_{B}

, when

vect = Nag_DoNotForm

, assuming

n ≫ k_{A}, k_{B}

; there are an additional

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

operations when

vect = Nag_FormX

The complex analogue of this function is f08usc.

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 f08ufc. The program calls f08uec to reduce the problem to the standard form

C y = λ y

, then f08hec to reduce

C

to tridiagonal form, and f08jfc to compute the eigenvalues.

f08ue: FL CL CPP AD PY MB

NAG CL Interfacef08uec (dsbgst)

▸▿ 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 CL Interface
f08uec (dsbgst)