NAG Library Function Document

nag_dpteqr (f08jgc)

+− 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_dpteqr (f08jgc) 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

#include <nag.h>

#include <nagf08.h>

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

3 Description

nag_dpteqr (f08jgc) 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 function 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 nag_dpteqr (f08jgc), 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_dpteqr (f08jgc) 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 nag_dbdsqr (f08mec) to compute the singular values of

B

which are the same as the eigenvalues of

T

. The method used by the function 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 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: 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 descending order, unless

fail . code =

NE_CONVERGENCE or NE_POS_DEF, in which case d is overwritten.

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 $Nag_InitZ$ and $order = Nag_ColMajor$ ;
$\max (1, \times pdz)$ when $compz = Nag_UpdateZ$ or $Nag_InitZ$ and $order = Nag_RowMajor$ ;
$1$ when $compz = Nag_NotZ$ .

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.

compz = Nag_InitZ

, z need not be set.

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 . code =

NE_CONVERGENCE or NE_POS_DEF.

compz = Nag_NotZ

, 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 order=Nag_ColMajor,
- if $compz = Nag_InitZ$ or $Nag_UpdateZ$ , $pdz \geq \max (1, n)$ ;
- if $compz = Nag_NotZ$ , $pdz \geq 1$ ;
if order=Nag_RowMajor,
- if $compz = Nag_UpdateZ$ or $Nag_InitZ$ , $pdz \geq \max (1, n)$ ;
- if $compz = Nag_NotZ$ , $pdz \geq 1$ .

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 to compute the singular values of the Cholesky factor $B$ failed to converge; $⟨value⟩$ off-diagonal elements did not converge to zero.
NE_ENUM_INT_2: 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.
NE_POS_DEF: The leading minor of order $⟨value⟩$ is not positive definite and the Cholesky factorization of $T$ could not be completed. Hence $T$ itself is not positive definite.

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 function) 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 Parallelism and Performance

nag_dpteqr (f08jgc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_dpteqr (f08jgc) 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

The total number of floating-point operations is typically about

30 n^{2}

compz = Nag_NotZ

and about

6 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_zpteqr (f08juc).

10 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 Function Documentnag_dpteqr (f08jgc)

+− 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_dpteqr (f08jgc)