NAG Library Routine Document

F08JGF (DPTEQR)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

F08JGF (DPTEQR) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric positive definite tridiagonal matrix, or of a real symmetric positive definite matrix which has been reduced to tridiagonal form.

2 Specification

SUBROUTINE F08JGF (

COMPZ, N, D, E, Z, LDZ, WORK, INFO)

INTEGER	N, LDZ, INFO
REAL (KIND=nag_wp)	D(), E(), Z(LDZ,), WORK(4N)
CHARACTER(1)	COMPZ

The routine may be called by its LAPACK name dpteqr.

3 Description

F08JGF (DPTEQR) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric positive definite 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 routine may also be used to compute all the eigenvalues and eigenvectors of a real symmetric positive definite 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 F08JGF (DPTEQR), which must be called with

COMPZ ='V'

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

Q

are:

full matrix	F08FEF (DSYTRD) and F08FFF (DORGTR)
full matrix, packed storage	F08GEF (DSPTRD) and F08GFF (DOPGTR)
band matrix	F08HEF (DSBTRD) with $VECT ='V'$ .

F08JGF (DPTEQR) first factorizes

T

L D L^{T}

where

L

is unit lower bidiagonal and

D

is diagonal. It forms the bidiagonal matrix

B = L D^{\frac{1}{2}}

, and then calls F08MEF (DBDSQR) to compute the singular values of

B

which are the same as the eigenvalues of

T

. The method used by the routine allows high relative accuracy to be achieved in the small eigenvalues of

T

. The eigenvectors are normalized so that

{‖z_{i}‖}_{2} = 1

, but are determined only to within a factor

\pm 1

4 References

Barlow J and Demmel J W (1990) Computing accurate eigensystems of scaled diagonally dominant matrices SIAM J. Numer. Anal. 27 762–791

5 Parameters

1: COMPZ – CHARACTER(1)Input

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

$COMPZ ='N'$: Only the eigenvalues are computed (and the array Z is not referenced).
$COMPZ ='I'$: The eigenvalues and eigenvectors of $T$ are computed (and the array Z is initialized by the routine).
$COMPZ ='V'$: The eigenvalues and eigenvectors of $A$ are computed (and the array Z must contain the matrix $Q$ on entry).

Constraint:

COMPZ ='N'

'V'

'I'

2: N – INTEGERInput

On entry:

n

, the order of the matrix

T

Constraint:

N \geq 0

3: D( $*$ ) – REAL (KIND=nag_wp) arrayInput/Output

Note: the dimension 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 descending order, unless

INFO > 0

, in which case D is overwritten.

4: E( $*$ ) – REAL (KIND=nag_wp) arrayInput/Output

Note: the dimension 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.

5: Z(LDZ, $*$ ) – REAL (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array Z must be at least

\max (1, N)

COMPZ ='V'

'I'

and at least

1

COMPZ ='N'

On entry: if

COMPZ ='V'

, Z must contain the orthogonal matrix

Q

from the reduction to tridiagonal form.

COMPZ ='I'

, Z need not be set.

On exit: if

COMPZ ='I'

'V'

, 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

INFO > 0

COMPZ ='N'

, Z is not referenced.

6: LDZ – INTEGERInput

On entry: the first dimension of the array Z as declared in the (sub)program from which F08JGF (DPTEQR) is called.

Constraints:

if $COMPZ ='I'$ or $'V'$ , $LDZ \geq \max (1, N)$ ;
if $COMPZ ='N'$ , $LDZ \geq 1$ .

7: WORK( $4 \times N$ ) – REAL (KIND=nag_wp) arrayWorkspace

8: INFO – INTEGEROutput

On exit:

INFO = 0

unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

Errors or warnings detected by the routine:

$INFO < 0$: If $INFO = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

$INFO > 0$: If $INFO = i$ , the leading minor of order $i$ is not positive definite and the Cholesky factorization of $T$ could not be completed. Hence $T$ itself is not positive definite.

If $INFO = N + i$ , the algorithm to compute the singular values of the Cholesky factor $B$ failed to converge; $i$ off-diagonal elements did not converge to zero.

7 Accuracy

The eigenvalues and eigenvectors of

T

are computed to high relative accuracy which means that if they vary widely in magnitude, then any small eigenvalues (and corresponding eigenvectors) will be computed more accurately than, for example, with the standard

Q R

method. However, the reduction to tridiagonal form (prior to calling the routine) may exclude the possibility of obtaining high relative accuracy in the small eigenvalues of the original matrix if its eigenvalues vary widely in magnitude.

To be more precise, let

H

be the tridiagonal matrix defined by

H = D T D

, where

D

is diagonal with

d_{i i} = t_{i i}^{- \frac{1}{2}}

, and

h_{i i} = 1

for all

i

. If

λ_{i}

is an exact eigenvalue of

T

and

{\tilde{λ}}_{i}

is the corresponding computed value, then

|{\tilde{λ}}_{i} - λ_{i}| \leq c (n) ε κ_{2} (H) λ_{i}

where

c (n)

is a modestly increasing function of

n

ε

is the machine precision, and

κ_{2} (H)

is the condition number of

H

with respect to inversion defined by:

κ_{2} (H) = ‖H‖ \cdot ‖H^{- 1}‖

z_{i}

is the corresponding exact eigenvector of

T

, 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 κ_{2} (H)}{{relgap}_{i}}

where

{relgap}_{i}

is the relative gap between

λ_{i}

and the other eigenvalues, defined by

{relgap}_{i} = \min_{i \neq j} \frac{|λ_{i} - λ_{j}|}{(λ_{i} + λ_{j})} .

8 Further Comments

The total number of floating point operations is typically about

30 n^{2}

COMPZ ='N'

and about

6 n^{3}

COMPZ ='V'

'I'

, but depends on how rapidly the algorithm converges. When

COMPZ ='N'

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

COMPZ ='V'

'I'

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

The complex analogue of this routine is F08JUF (ZPTEQR).

9 Example

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

T

, where

T = (\begin{array}{r} 4.16 & 3.17 & 0.00 & 0.00 \\ 3.17 & 5.25 & - 0.97 & 0.00 \\ 0.00 & - 0.97 & 1.09 & 0.55 \\ 0.00 & 0.00 & 0.55 & 0.62 \end{array}) .

NAG Library Routine DocumentF08JGF (DPTEQR)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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 Routine Document

F08JGF (DPTEQR)