NAG Library Function Document

nag_dstevd (f08jcc)

+− 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

nag_dstevd (f08jcc) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric tridiagonal matrix. If the eigenvectors are requested, then it uses a divide-and-conquer algorithm to compute eigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal–Walker–Kahan variant of the

Q L

Q R

algorithm.

2 Specification

#include <nag.h>

#include <nagf08.h>

void	nag_dstevd (Nag_OrderType order, Nag_JobType job, Integer n, double d[], double e[], double z[], Integer pdz, NagError *fail)

3 Description

nag_dstevd (f08jcc) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric tridiagonal matrix

T

. In other words, it can compute the spectral factorization of

T

T = Z Λ Z^{T},

where

Λ

is a diagonal matrix whose diagonal elements are the eigenvalues

λ_{i}

, and

Z

is the orthogonal matrix whose columns are the eigenvectors

z_{i}

. Thus

T z_{i} = λ_{i} z_{i}, i = 1, 2, \dots, n .

4 References

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

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: job – Nag_JobTypeInput

On entry: indicates whether eigenvectors are computed.

$job = Nag_DoNothing$: Only eigenvalues are computed.
$job = Nag_EigVecs$: Eigenvalues and eigenvectors are computed.

Constraint:

job = Nag_DoNothing

Nag_EigVecs

3: n – IntegerInput

On entry:

n

, the order of the matrix

T

Constraint:

n \geq 0

4: d[ $\dim$ ] – doubleInput/Output

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

\max (1, n)

On entry: the

n

diagonal elements of the tridiagonal matrix

T

On exit: the eigenvalues of the matrix

T

in ascending order.

5: e[ $\dim$ ] – doubleInput/Output

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

\max (1, n)

On entry: the

n - 1

off-diagonal elements of the tridiagonal matrix

T

. The

n

th element of this array is used as workspace.

On exit: e is overwritten with intermediate results.

6: z[ $\dim$ ] – doubleOutput

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

$\max (1, pdz \times n)$ when $job = Nag_EigVecs$ ;
$1$ when $job = Nag_DoNothing$ .

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_EigVecs

, z is overwritten by the orthogonal matrix

Z

which contains the eigenvectors of

T

job = Nag_DoNothing

, z is not referenced.

7: pdz – IntegerInput

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

Constraints:

if $job = Nag_EigVecs$ , $pdz \geq \max (1, n)$ ;
if $job = Nag_DoNothing$ , $pdz \geq 1$ .

8: 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_CONVERGENCE: The algorithm failed to converge; $⟨value⟩$ eigenvectors did not converge.
NE_ENUM_INT_2: On entry, $job = ⟨value⟩$ , $pdz = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: if $job = Nag_EigVecs$ , $pdz \geq \max (1, n)$ ;
if $job = Nag_DoNothing$ , $pdz \geq 1$ .
NE_INT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .
On entry, $pdz = ⟨value⟩$ .
Constraint: $pdz > 0$ .
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

The computed eigenvalues and eigenvectors are exact for a nearby matrix

(T + E)

, where

{‖E‖}_{2} = O (ε) {‖T‖}_{2},

and

ε

is the machine precision.

λ_{i}

is an exact eigenvalue and

{\tilde{λ}}_{i}

is the corresponding computed value, then

|{\tilde{λ}}_{i} - λ_{i}| \leq c (n) ε {‖T‖}_{2},

where

c (n)

is a modestly increasing function of

n

z_{i}

is the corresponding exact eigenvector, and

{\tilde{z}}_{i}

is the corresponding computed eigenvector, then the angle

θ ({\tilde{z}}_{i}, z_{i})

between them is bounded as follows:

θ ({\tilde{z}}_{i}, z_{i}) \leq \frac{c (n) ε {‖T‖}_{2}}{\min_{i \neq j} |λ_{i} - λ_{j}|} .

Thus the accuracy of a computed eigenvector depends on the gap between its eigenvalue and all the other eigenvalues.

8 Parallelism and Performance

nag_dstevd (f08jcc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_dstevd (f08jcc) 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.

9 Further Comments

There is no complex analogue of this function.

10 Example

This example computes all the eigenvalues and eigenvectors of the symmetric tridiagonal matrix

T

, where

T = (\begin{array}{r} 1.0 & 1.0 & 0.0 & 0.0 \\ 1.0 & 4.0 & 2.0 & 0.0 \\ 0.0 & 2.0 & 9.0 & 3.0 \\ 0.0 & 0.0 & 3.0 & 16.0 \end{array}) .

NAG Library Function Documentnag_dstevd (f08jcc)

+− 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_dstevd (f08jcc)