NAG Library Routine Document

Integer, Intent (In)	::	n, nrhs, ldb, ldx
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (In)	::	d(), e(), b(ldb,*)
Real (Kind=nag_wp), Intent (Inout)	::	df(), ef(), x(ldx,*)
Real (Kind=nag_wp), Intent (Out)	::	rcond, ferr(nrhs), berr(nrhs), work(2*n)
Character (1), Intent (In)	::	fact

C Header Interface

#include <nagmk26.h>

void	f07jbf_ (const char fact, const Integer n, const Integer nrhs, const double d[], const double e[], double df[], double ef[], const double b[], const Integer ldb, double x[], const Integer ldx, double rcond, double ferr[], double berr[], double work[], Integer *info, const Charlen length_fact)

The routine may be called by its LAPACK name dptsvx.

3

Description

f07jbf (dptsvx) performs the following steps:

1.	If $fact ='N'$ , the matrix $A$ is factorized as $A = L D L^{T}$ , where $L$ is a unit lower bidiagonal matrix and $D$ is diagonal. The factorization can also be regarded as having the form $A = U^{T} D U$ .
2.	If the leading $i$ by $i$ principal minor is not positive definite, then the routine returns with $info = i$ . Otherwise, the factored form of $A$ is used to estimate the condition number of the matrix $A$ . If the reciprocal of the condition number is less than machine precision, $info = n + 1$ is returned as a warning, but the routine still goes on to solve for $X$ and compute error bounds as described below.
3.	The system of equations is solved for $X$ using the factored form of $A$ .
4.	Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for it.

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

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

Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

5

Arguments

1: $fact$ – Character(1)Input

On entry: specifies whether or not the factorized form of the matrix

A

has been supplied.

$fact ='F'$: df and ef contain the factorized form of the matrix $A$ . df and ef will not be modified.
$fact ='N'$: The matrix $A$ will be copied to df and ef and factorized.

Constraint:

fact ='F'

'N'

2: $n$ – IntegerInput

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

3: $nrhs$ – IntegerInput

On entry:

r

, the number of right-hand sides, i.e., the number of columns of the matrix

B

Constraint:

nrhs \geq 0

4: $d (*)$ – Real (Kind=nag_wp) arrayInput

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

\max (1, n)

On entry: the

n

diagonal elements of the tridiagonal matrix

A

5: $e (*)$ – Real (Kind=nag_wp) arrayInput

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

\max (1, n - 1)

On entry: the

(n - 1)

subdiagonal elements of the tridiagonal matrix

A

6: $df (*)$ – Real (Kind=nag_wp) arrayInput/Output

Note: the dimension of the array df must be at least

\max (1, n)

On entry: if

fact ='F'

, df must contain the

n

diagonal elements of the diagonal matrix

D

from the

L D L^{T}

factorization of

A

On exit: if

fact ='N'

, df contains the

n

diagonal elements of the diagonal matrix

D

from the

L D L^{T}

factorization of

A

7: $ef (*)$ – Real (Kind=nag_wp) arrayInput/Output

Note: the dimension of the array ef must be at least

\max (1, n - 1)

On entry: if

fact ='F'

, ef must contain the

(n - 1)

subdiagonal elements of the unit bidiagonal factor

L

from the

L D L^{T}

factorization of

A

On exit: if

fact ='N'

, ef contains the

(n - 1)

subdiagonal elements of the unit bidiagonal factor

L

from the

L D L^{T}

factorization of

A

8: $b (ldb *)$ – Real (Kind=nag_wp) arrayInput

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

\max (1, nrhs)

On entry: the

n

r

right-hand side matrix

B

9: $ldb$ – IntegerInput

On entry: the first dimension of the array b as declared in the (sub)program from which f07jbf (dptsvx) is called.

Constraint:

ldb \geq \max (1, n)

10: $x (ldx *)$ – Real (Kind=nag_wp) arrayOutput

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

\max (1, nrhs)

On exit: if

info = 0

n + 1

, the

n

r

solution matrix

X

11: $ldx$ – IntegerInput

On entry: the first dimension of the array x as declared in the (sub)program from which f07jbf (dptsvx) is called.

Constraint:

ldx \geq \max (1, n)

12: $rcond$ – Real (Kind=nag_wp)Output

On exit: the reciprocal condition number of the matrix

A

. If rcond is less than the machine precision (in particular, if

rcond = 0.0

), the matrix is singular to working precision. This condition is indicated by a return code of

info = n + 1

13: $ferr (nrhs)$ – Real (Kind=nag_wp) arrayOutput

On exit: the forward error bound for each solution vector

{\hat{x}}_{j}

(the

j

th column of the solution matrix

X

). If

x_{j}

is the true solution corresponding to

{\hat{x}}_{j}

ferr (j)

is an estimated upper bound for the magnitude of the largest element in (

{\hat{x}}_{j} - x_{j}

) divided by the magnitude of the largest element in

{\hat{x}}_{j}

14: $berr (nrhs)$ – Real (Kind=nag_wp) arrayOutput

On exit: the component-wise relative backward error of each solution vector

{\hat{x}}_{j}

(i.e., the smallest relative change in any element of

A

B

that makes

{\hat{x}}_{j}

an exact solution).

15: $work (2 \times n)$ – Real (Kind=nag_wp) arrayWorkspace

16: $info$ – IntegerOutput

On exit:

info = 0

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

6

Error Indicators and Warnings

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

$info > 0 and info \leq n$: The leading minor of order $〈value〉$ of $A$ is not positive definite, so the factorization could not be completed, and the solution has not been computed. $rcond = 0.0$ is returned.

$info = n + 1$: $D$ is nonsingular, but rcond is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of rcond would suggest.

7

Accuracy

For each right-hand side vector

b

, the computed solution

\hat{x}

is the exact solution of a perturbed system of equations

(A + E) \hat{x} = b

, where

|E| \leq c (n) ε |R| |R^{T}|, where ​ R = L D^{\frac{1}{2}},

c (n)

is a modest linear function of

n

, and

ε

is the machine precision. See Section 10.1 of Higham (2002) for further details.

x

is the true solution, then the computed solution

\hat{x}

satisfies a forward error bound of the form

\frac{{‖x - \hat{x}‖}_{\infty}}{{‖\hat{x}‖}_{\infty}} \leq w_{c} cond (A, \hat{x}, b)

where

cond (A, \hat{x}, b) = {‖|A^{- 1}| (|A| |\hat{x}| + |b|)‖}_{\infty} / {‖\hat{x}‖}_{\infty} \leq cond (A) = {‖|A^{- 1}| |A|‖}_{\infty} \leq κ_{\infty} (A)

. If

\hat{x}

is the

j

th column of

X

, then

w_{c}

is returned in

berr (j)

and a bound on

{‖x - \hat{x}‖}_{\infty} / {‖\hat{x}‖}_{\infty}

is returned in

ferr (j)

. See Section 4.4 of Anderson et al. (1999) for further details.

8

Parallelism and Performance

f07jbf (dptsvx) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f07jbf (dptsvx) 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9

Further Comments

The number of floating-point operations required for the factorization, and for the estimation of the condition number of

A

is proportional to

n

. The number of floating-point operations required for the solution of the equations, and for the estimation of the forward and backward error is proportional to

n r

, where

r

is the number of right-hand sides.

The condition estimation is based upon Equation (15.11) of Higham (2002). For further details of the error estimation, see Section 4.4 of Anderson et al. (1999).

The complex analogue of this routine is f07jpf (zptsvx).

10

Example

This example solves the equations

A X = B,

where

A

is the symmetric positive definite tridiagonal matrix

A = (\begin{matrix} 4.0 & - 2.0 & 0 & 0 & 0 \\ - 2.0 & 10.0 & - 6.0 & 0 & 0 \\ 0 & - 6.0 & 29.0 & 15.0 & 0 \\ 0 & 0 & 15.0 & 25.0 & 8.0 \\ 0 & 0 & 0 & 8.0 & 5.0 \end{matrix})

and

B = (\begin{matrix} 6.0 & 10.0 \\ 9.0 & 4.0 \\ 2.0 & 9.0 \\ 14.0 & 65.0 \\ 7.0 & 23.0 \end{matrix}) .

Error estimates for the solutions and an estimate of the reciprocal of the condition number of

A

are also output.

NAG Library Routine Document

f07jbf (dptsvx)

▸▿ 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

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