void	f08hac (Nag_OrderType order, Nag_JobType job, Nag_UploType uplo, Integer n, Integer kd, double ab[], Integer pdab, double w[], double z[], Integer pdz, NagError *fail)

The function may be called by the names: f08hac, nag_lapackeig_dsbev or nag_dsbev.

3 Description

The symmetric band matrix

A

is first reduced to tridiagonal form, using orthogonal similarity transformations, and then the

Q R

algorithm is applied to the tridiagonal matrix to compute the eigenvalues and (optionally) the eigenvectors.

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 https://www.netlib.org/lapack/lug

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

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: $job$ – Nag_JobType Input

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: $uplo$ – Nag_UploType Input

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

4: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

5: $kd$ – Integer Input

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: $ab [\dim]$ – 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 $ab [k_{d} + i - j + (j - 1) \times pdab]$ , 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: $pdab$ – 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 kd + 1

8: $w [n]$ – double Output

On exit: the eigenvalues in ascending order.

9: $z [\dim]$ – double Output

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

, z contains the orthonormal eigenvectors of the matrix

A

, with the

i

th column of

Z

holding the eigenvector associated with

w [i - 1]

job = Nag_EigVals

, z is not referenced.

10: $pdz$ – Integer Input

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 $pdz \geq 1$ .

11: $fail$ – 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_CONVERGENCE: The algorithm failed to converge; $⟨ value ⟩$ off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
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_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, $pdz = ⟨ value ⟩$ .
Constraint: $pdz > 0$ .
NE_INT_2: On entry, $pdab = ⟨ value ⟩$ and $kd = ⟨ value ⟩$ .
Constraint: $pdab \geq kd + 1$ .
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 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

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

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

f08hac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08hac 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}

job = Nag_DoBoth

and is proportional to

k_{d} n^{2}

otherwise.

The complex analogue of this function is f08hnc.

10 Example

This example finds all the eigenvalues and eigenvectors of the symmetric band matrix

A = (\begin{matrix} 1 & 2 & 3 & 0 & 0 \\ 2 & 2 & 3 & 4 & 0 \\ 3 & 3 & 3 & 4 & 5 \\ 0 & 4 & 4 & 4 & 5 \\ 0 & 0 & 5 & 5 & 5 \end{matrix}),

together with approximate error bounds for the computed eigenvalues and eigenvectors.

f08ha: FL CL CPP AD PY MB

NAG CL Interfacef08hac (dsbev)

▸▿ 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
f08hac (dsbev)