NAG Library Function Document

nag_dsteqr (f08jec)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

nag_dsteqr (f08jec) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric tridiagonal matrix, or of a real symmetric matrix which has been reduced to tridiagonal form.

2 Specification

#include <nag.h>

#include <nagf08.h>

void	nag_dsteqr (Nag_OrderType order, Nag_ComputeZType compz, Integer n, double d[], double e[], double z[], Integer pdz, NagError *fail)

3 Description

nag_dsteqr (f08jec) 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 .

The function may also be used to compute all the eigenvalues and eigenvectors of a real symmetric matrix

A

which has been reduced to tridiagonal form

T

\begin{array}{l} A & = Q T Q^{T}, where ​ Q ​ is orthogonal \\ = (Q Z) Λ {(Q Z)}^{T} . \end{array}

In this case, the matrix

Q

must be formed explicitly and passed to nag_dsteqr (f08jec), which must be called with

compz = Nag_UpdateZ

. The functions which must be called to perform the reduction to tridiagonal form and form

Q

are:

full matrix	nag_dsytrd (f08fec) and nag_dorgtr (f08ffc)
full matrix, packed storage	nag_dsptrd (f08gec) and nag_dopgtr (f08gfc)
band matrix	nag_dsbtrd (f08hec) with $vect = Nag_FormQ$ .

nag_dsteqr (f08jec) uses the implicitly shifted

Q R

algorithm, switching between the

Q R

and

Q L

variants in order to handle graded matrices effectively (see Greenbaum and Dongarra (1980)). The eigenvectors are normalized so that

{‖z_{i}‖}_{2} = 1

, but are determined only to within a factor

\pm 1

If only the eigenvalues of

T

are required, it is more efficient to call nag_dsterf (f08jfc) instead. If

T

is positive definite, small eigenvalues can be computed more accurately by nag_dpteqr (f08jgc).

4 References

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

Greenbaum A and Dongarra J J (1980) Experiments with QR/QL methods for the symmetric triangular eigenproblem LAPACK Working Note No. 17 (Technical Report CS-89-92) University of Tennessee, Knoxville

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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

or Nag_ColMajor.

2: compz – Nag_ComputeZTypeInput

On entry: indicates whether the eigenvectors are to be computed.

$compz = Nag_NotZ$: Only the eigenvalues are computed (and the array z is not referenced).
$compz = Nag_InitZ$: The eigenvalues and eigenvectors of $T$ are computed (and the array z is initialized by the function).
$compz = Nag_UpdateZ$: The eigenvalues and eigenvectors of $A$ are computed (and the array z must contain the matrix $Q$ on entry).

Constraint:

compz = Nag_NotZ

Nag_UpdateZ

Nag_InitZ

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 diagonal elements of the tridiagonal matrix

T

On exit: the

n

eigenvalues in ascending order, unless

fail . code =

NE_CONVERGENCE (in which case see Section 6).

5: e[ $\dim$ ] – doubleInput/Output

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

\max (1, n - 1)

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

T

On exit: e is overwritten.

6: z[ $\dim$ ] – doubleInput/Output

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

\max (1, pdz \times n)

when

compz = Nag_UpdateZ

or Nag_InitZ.

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 entry: if

compz = Nag_UpdateZ

, z must contain the orthogonal matrix

Q

from the reduction to tridiagonal form. If

compz = Nag_InitZ

, z must be allocated, but its contents need not be set. If

compz = Nag_NotZ

, z is not referenced and may be a null pointer, i.e., (double *) 0.

On exit: if

compz = Nag_InitZ

Nag_UpdateZ

, the

n

required orthonormal eigenvectors stored as columns of

Z

; the

i

th column corresponds to the

i

th eigenvalue, where

i = 1, 2, \dots, n

, unless

fail . errnum > 0

z is not changed if

compz = Nag_NotZ

7: pdz – IntegerInput

On entry: when z is a null pointer then pdz should be set to

0

Constraints:

if $compz = Nag_NotZ$ , $pdz \geq 0$ ;
if $compz = Nag_UpdateZ$ or $Nag_InitZ$ , $pdz \geq \max (1, n)$ .

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 has failed to find all the eigenvalues after a total of $30 \times n$ iterations. In this case, d and e contain on exit the diagonal and off-diagonal elements, respectively, of a tridiagonal matrix similar to $T$ . $〈value〉$ off-diagonal elements have not converged to zero.
NE_ENUM_INT_2: On entry, $compz = 〈value〉$ , $n = 〈value〉$ and $pdz = 〈value〉$ .
Constraint: if $compz = Nag_NotZ$ , $pdz \geq 0$ ;
if $compz = Nag_UpdateZ$ or $Nag_InitZ$ , $pdz \geq \max (1, n)$ .; On entry, $compz = 〈value〉$ , $pdz = 〈value〉$ and $n = 〈value〉$ .
Constraint: if $compz = Nag_InitZ$ or $Nag_UpdateZ$ , $pdz \geq \max (1, n)$ ;
if $compz = Nag_NotZ$ , $pdz \geq 1$ .; On entry, $compz = 〈value〉$ , $pdz = 〈value〉$ , $n = 〈value〉$ .
Constraint: if $compz = Nag_UpdateZ$ or $Nag_InitZ$ , $pdz \geq \max (1, n)$ ;
if $compz = Nag_NotZ$ , $pdz \geq 1$ .
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 Further Comments

The total number of floating point operations is typically about

24 n^{2}

compz = Nag_NotZ

and about

7 n^{3}

compz = Nag_UpdateZ

Nag_InitZ

, but depends on how rapidly the algorithm converges. When

compz = Nag_NotZ

, the operations are all performed in scalar mode; the additional operations to compute the eigenvectors when

compz = Nag_UpdateZ

Nag_InitZ

can be vectorized and on some machines may be performed much faster.

The complex analogue of this function is nag_zsteqr (f08jsc).

9 Example

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

T

, where

T = (\begin{array}{r} - 6.99 & - 0.44 & 0.00 & 0.00 \\ - 0.44 & 7.92 & - 2.63 & 0.00 \\ 0.00 & - 2.63 & 2.34 & - 1.18 \\ 0.00 & 0.00 & - 1.18 & 0.32 \end{array}) .

See also the examples for nag_dorgtr (f08ffc), nag_dopgtr (f08gfc) or nag_dsbtrd (f08hec), which illustrate the use of this function to compute the eigenvalues and eigenvectors of a full or band symmetric matrix.

NAG Library Function Documentnag_dsteqr (f08jec)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Function Document

nag_dsteqr (f08jec)