factorization returned by f07cdf (dgttrf) and an initial solution returned by f07cef (dgttrs). Iterative refinement is used to reduce the backward error as much as possible.

2

Specification

Fortran Interface

Subroutine f07chf (

trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info)

Integer, Intent (In)	::	n, nrhs, ipiv(*), ldb, ldx
Integer, Intent (Out)	::	iwork(n), info
Real (Kind=nag_wp), Intent (In)	::	dl(), d(), du(), dlf(), df(), duf(), du2(), b(ldb,)
Real (Kind=nag_wp), Intent (Inout)	::	x(ldx,*)
Real (Kind=nag_wp), Intent (Out)	::	ferr(nrhs), berr(nrhs), work(3*n)
Character (1), Intent (In)	::	trans

C Header Interface

#include nagmk26.h

void

f07chf_ ( const char *trans, const Integer *n, const Integer *nrhs, const double dl[], const double d[], const double du[], const double dlf[], const double df[], const double duf[], const double du2[], const Integer ipiv[], const double b[], const Integer *ldb, double x[], const Integer *ldx, double ferr[], double berr[], double work[], Integer iwork[], Integer *info, const Charlen length_trans)

The routine may be called by its LAPACK name dgtrfs.

3

Description

f07chf (dgtrfs) should normally be preceded by calls to f07cdf (dgttrf) and f07cef (dgttrs). f07cdf (dgttrf) uses Gaussian elimination with partial pivoting and row interchanges to factorize the matrix

A

A = P L U,

where

P

is a permutation matrix,

L

is unit lower triangular with at most one nonzero subdiagonal element in each column, and

U

is an upper triangular band matrix, with two superdiagonals. f07cef (dgttrs) then utilizes the factorization to compute a solution,

\hat{X}

, to the required equations. Letting

\hat{x}

denote a column of

\hat{X}

, f07chf (dgtrfs) computes a component-wise backward error,

β

, the smallest relative perturbation in each element of

A

and

b

such that

\hat{x}

is the exact solution of a perturbed system

(A + E) \hat{x} = b + f, with |e_{i j}| \leq β |a_{i j}|, and |f_{j}| \leq β |b_{j}| .

The routine also estimates a bound for the component-wise forward error in the computed solution defined by

\max |x_{i} - \hat{x_{i}}| / \max |\hat{x_{i}}|

, where

x

is the corresponding column of the exact solution,

X

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

5

Arguments

1: $trans$ – Character(1)Input

On entry: specifies the equations to be solved as follows:

$trans ='N'$: Solve $A X = B$ for $X$ .
$trans ='T'$ or $'C'$: Solve $A^{T} X = B$ for $X$ .

Constraint:

trans ='N'

'T'

'C'

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: $dl (*)$ – Real (Kind=nag_wp) arrayInput

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

\max (1, n - 1)

On entry: must contain the

(n - 1)

subdiagonal elements of the matrix

A

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

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

\max (1, n)

On entry: must contain the

n

diagonal elements of the matrix

A

6: $du (*)$ – Real (Kind=nag_wp) arrayInput

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

\max (1, n - 1)

On entry: must contain the

(n - 1)

superdiagonal elements of the matrix

A

7: $dlf (*)$ – Real (Kind=nag_wp) arrayInput

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

\max (1, n - 1)

On entry: must contain the

(n - 1)

multipliers that define the matrix

L

of the

L U

factorization of

A

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

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

\max (1, n)

On entry: must contain the

n

diagonal elements of the upper triangular matrix

U

from the

L U

factorization of

A

9: $duf (*)$ – Real (Kind=nag_wp) arrayInput

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

\max (1, n - 1)

On entry: must contain the

(n - 1)

elements of the first superdiagonal of

U

10: $du2 (*)$ – Real (Kind=nag_wp) arrayInput

Note: the dimension of the array du2 must be at least

\max (1, n - 2)

On entry: must contain the

(n - 2)

elements of the second superdiagonal of

U

11: $ipiv (*)$ – Integer arrayInput

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

\max (1, n)

On entry: must contain the

n

pivot indices that define the permutation matrix

P

. At the

i

th step, row

i

of the matrix was interchanged with row

ipiv (i)

, and

ipiv (i)

must always be either

i

(i + 1)

ipiv (i) = i

indicating that a row interchange was not performed.

12: $b (ldb *)$ – 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

matrix of right-hand sides

B

13: $ldb$ – IntegerInput

On entry: the first dimension of the array b as declared in the (sub)program from which f07chf (dgtrfs) is called.

Constraint:

ldb \geq \max (1, n)

14: $x (ldx *)$ – Real (Kind=nag_wp) arrayInput/Output

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

\max (1, nrhs)

On entry: the

n

r

initial solution matrix

X

On exit: the

n

r

refined solution matrix

X

15: $ldx$ – IntegerInput

On entry: the first dimension of the array x as declared in the (sub)program from which f07chf (dgtrfs) is called.

Constraint:

ldx \geq \max (1, n)

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

On exit: estimate of the forward error bound for each computed solution vector, such that

{‖{\hat{x}}_{j} - x_{j}‖}_{\infty} / {‖{\hat{x}}_{j}‖}_{\infty} \leq ferr (j)

, where

{\hat{x}}_{j}

is the

j

th column of the computed solution returned in the array x and

x_{j}

is the corresponding column of the exact solution

X

. The estimate is almost always a slight overestimate of the true error.

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

On exit: estimate of the component-wise relative backward error of each computed 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).

18: $work (3 \times n)$ – Real (Kind=nag_wp) arrayWorkspace

19: $iwork (n)$ – Integer arrayWorkspace

20: $info$ – 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.

7

Accuracy

The computed solution for a single right-hand side,

\hat{x}

, satisfies an equation of the form

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

where

{‖E‖}_{\infty} = O (ε) {‖A‖}_{\infty}

and

ε

is the machine precision. An approximate error bound for the computed solution is given by

\frac{{‖\hat{x} - x‖}_{\infty}}{{‖x‖}_{\infty}} \leq κ (A) \frac{{‖E‖}_{\infty}}{{‖A‖}_{\infty}},

where

κ (A) = {‖A^{- 1}‖}_{\infty} {‖A‖}_{\infty}

, the condition number of

A

with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.

Routine f07cgf (dgtcon) can be used to estimate the condition number of

A

8

Parallelism and Performance

f07chf (dgtrfs) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f07chf (dgtrfs) 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 total number of floating-point operations required to solve the equations

A X = B

A^{T} X = B

is proportional to

n r

. At most five steps of iterative refinement are performed, but usually only one or two steps are required.

The complex analogue of this routine is f07cvf (zgtrfs).

10

Example

This example solves the equations

A X = B,

where

A

is the tridiagonal matrix

A = (\begin{array}{r} 3.0 & 2.1 & 0 & 0 & 0 \\ 3.4 & 2.3 & - 1.0 & 0 & 0 \\ 0 & 3.6 & - 5.0 & 1.9 & 0 \\ 0 & 0 & 7.0 & - 0.9 & 8.0 \\ 0 & 0 & 0 & - 6.0 & 7.1 \end{array}) and B = (\begin{matrix} 2.7 & 6.6 \\ - 0.5 & 10.8 \\ 2.6 & - 3.2 \\ 0.6 & - 11.2 \\ 2.7 & 19.1 \end{matrix}) .

Estimates for the backward errors and forward errors are also output.

NAG Library Routine Document

f07chf (dgtrfs)

▸▿ 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