void	nag_dsbtrd (Nag_OrderType order, Nag_VectType vect, Nag_UploType uplo, Integer n, Integer kd, double ab[], Integer pdab, double d[], double e[], double q[], Integer pdq, NagError *fail)

3

Description

nag_dsbtrd (f08hec) reduces a symmetric band matrix

A

to symmetric tridiagonal form

T

by an orthogonal similarity transformation:

T = Q^{T} A Q .

The orthogonal matrix

Q

is determined as a product of Givens rotation matrices, and may be formed explicitly by the function if required.

The function uses a vectorizable form of the reduction, due to Kaufman (1984).

4

References

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

Parlett B N (1998) The Symmetric Eigenvalue Problem SIAM, Philadelphia

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

Q

is to be returned.

$vect = Nag_FormQ$: $Q$ is returned.
$vect = Nag_UpdateQ$: $Q$ is updated (and the array q must contain a matrix on entry).
$vect = Nag_DoNotForm$: $Q$ is not required.

Constraint:

vect = Nag_FormQ

Nag_UpdateQ

Nag_DoNotForm

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 matrix

A

Constraint:

n \geq 0

5: $kd$ – IntegerInput

On entry: if

uplo = Nag_Upper

, the number of superdiagonals,

k_{d}

, of the matrix

A

uplo = Nag_Lower

, the number of subdiagonals,

k_{d}

, of the matrix

A

Constraint:

kd \geq 0

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

On exit: ab is overwritten by values generated during the reduction to tridiagonal form.

The first superdiagonal or subdiagonal and the diagonal of the tridiagonal matrix

T

are returned in ab using the same storage format as described above.

7: $pdab$ – 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 \max (1, kd + 1)

8: $d [n]$ – doubleOutput

On exit: the diagonal elements of the tridiagonal matrix

T

9: $e [n - 1]$ – doubleOutput

On exit: the off-diagonal elements of the tridiagonal matrix

T

10: $q [\dim]$ – doubleInput/Output

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

$\max (1, pdq \times n)$ when $vect = Nag_FormQ$ or $Nag_UpdateQ$ ;
$1$ when $vect = Nag_DoNotForm$ .

The

(i, j)

th element of the matrix

Q

is stored in

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

On entry: if

vect = Nag_UpdateQ

, q must contain the matrix formed in a previous stage of the reduction (for example, the reduction of a banded symmetric-definite generalized eigenproblem); otherwise q need not be set.

On exit: if

vect = Nag_FormQ

Nag_UpdateQ

, the

n

n

matrix

Q

vect = Nag_DoNotForm

, q is not referenced.

11: $pdq$ – IntegerInput

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

Constraints:

if $vect = Nag_FormQ$ or $Nag_UpdateQ$ , $pdq \geq \max (1, n)$ ;
if $vect = Nag_DoNotForm$ , $pdq \geq 1$ .

12: $fail$ – 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, $vect = 〈value〉$ , $pdq = 〈value〉$ and $n = 〈value〉$ .
Constraint: if $vect = Nag_FormQ$ or $Nag_UpdateQ$ , $pdq \geq \max (1, n)$ ;
if $vect = Nag_DoNotForm$ , $pdq \geq 1$ .
NE_INT: On entry, $kd = 〈value〉$ .
Constraint: $kd \geq 0$ .

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

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

On entry, $pdq = 〈value〉$ .
Constraint: $pdq > 0$ .
NE_INT_2: On entry, $pdab = 〈value〉$ and $kd = 〈value〉$ .
Constraint: $pdab \geq \max (1, kd + 1)$ .
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

The computed tridiagonal matrix

T

is exactly similar to a nearby matrix

(A + E)

, where

{‖E‖}_{2} \leq c (n) ε {‖A‖}_{2},

c (n)

is a modestly increasing function of

n

, and

ε

is the machine precision.

The elements of

T

themselves may be sensitive to small perturbations in

A

or to rounding errors in the computation, but this does not affect the stability of the eigenvalues and eigenvectors.

The computed matrix

Q

differs from an exactly orthogonal matrix by a matrix

E

such that

{‖E‖}_{2} = O (ε),

where

ε

is the machine precision.

8

Parallelism and Performance

nag_dsbtrd (f08hec) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_dsbtrd (f08hec) 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

vect = Nag_DoNotForm

with

3 n^{3} (k - 1) / k

additional operations if

vect = Nag_FormQ

The complex analogue of this function is nag_zhbtrd (f08hsc).

10

Example

This example computes all the eigenvalues and eigenvectors of the matrix

A

, where

A = (\begin{array}{r} 4.99 & 0.04 & 0.22 & 0.00 \\ 0.04 & 1.05 & - 0.79 & 1.04 \\ 0.22 & - 0.79 & - 2.31 & - 1.30 \\ 0.00 & 1.04 & - 1.30 & - 0.43 \end{array}) .

Here

A

is symmetric and is treated as a band matrix. The program first calls nag_dsbtrd (f08hec) to reduce

A

to tridiagonal form

T

, and to form the orthogonal matrix

Q

; the results are then passed to nag_dsteqr (f08jec) which computes the eigenvalues and eigenvectors of

A

NAG Library Function Document

nag_dsbtrd (f08hec)

▸▿ 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