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

C Header Interface

#include <nag.h>

void

f07cbf_ (const char *fact, const char *trans, const Integer *n, const Integer *nrhs, const double dl[], const double d[], const double du[], double dlf[], double df[], double duf[], double du2[], Integer ipiv[], const double b[], const Integer *ldb, double x[], const Integer *ldx, double *rcond, double ferr[], double berr[], double work[], Integer iwork[], Integer *info, const Charlen length_fact, const Charlen length_trans)

The routine may be called by the names f07cbf, nagf_lapacklin_dgtsvx or its LAPACK name dgtsvx.

3 Description

f07cbf performs the following steps:

1.If $fact ='N'$ , the $L U$ decomposition is used to factor the matrix $A$ as $A = L U$ , where $L$ is a product of permutation and unit lower bidiagonal matrices and $U$ is upper triangular with nonzeros in only the main diagonal and first two superdiagonals.
2.If some $u_{i i} = 0$ , so that $U$ is exactly singular, 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 https://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.

$fact ='F'$: dlf, df, duf, du2 and ipiv contain the factorized form of the matrix $A$ . dlf, df, duf, du2 and ipiv will not be modified.
$fact ='N'$: The matrix $A$ will be copied to dlf, df and duf and factorized.

Constraint:

fact ='F'

'N'

2: $trans$ – Character(1) Input

On entry: specifies the form of the system of equations.

$trans ='N'$: $A X = B$ (No transpose).
$trans ='T'$ or $'C'$: $A^{T} X = B$ (Transpose).

Constraint:

trans ='N'

'T'

'C'

3: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

4: $nrhs$ – Integer Input

On entry:

r

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

B

Constraint:

nrhs \geq 0

5: $dl (*)$ – Real (Kind=nag_wp) array Input

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

\max (1, n - 1)

On entry: the

(n - 1)

subdiagonal elements of

A

6: $d (*)$ – Real (Kind=nag_wp) array Input

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

\max (1, n)

On entry: the

n

diagonal elements of

A

7: $du (*)$ – Real (Kind=nag_wp) array Input

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

\max (1, n - 1)

On entry: the

(n - 1)

superdiagonal elements of

A

8: $dlf (*)$ – Real (Kind=nag_wp) array Input/Output

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

\max (1, n - 1)

On entry: if

fact ='F'

, dlf contains the

(n - 1)

multipliers that define the matrix

L

from the

L U

factorization of

A

On exit: if

fact ='N'

, dlf contains the

(n - 1)

multipliers that define the matrix

L

from the

L U

factorization of

A

9: $df (*)$ – Real (Kind=nag_wp) array Input/Output

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

\max (1, n)

On entry: if

fact ='F'

, df contains the

n

diagonal elements of the upper triangular matrix

U

from the

L U

factorization of

A

On exit: if

fact ='N'

, df contains the

n

diagonal elements of the upper triangular matrix

U

from the

L U

factorization of

A

10: $duf (*)$ – Real (Kind=nag_wp) array Input/Output

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

\max (1, n - 1)

On entry: if

fact ='F'

, duf contains the

(n - 1)

elements of the first superdiagonal of

U

On exit: if

fact ='N'

, duf contains the

(n - 1)

elements of the first superdiagonal of

U

11: $du2 (*)$ – Real (Kind=nag_wp) array Input/Output

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

\max (1, n - 2)

On entry: if

fact ='F'

, du2 contains the (

n - 2

) elements of the second superdiagonal of

U

On exit: if

fact ='N'

, du2 contains the (

n - 2

) elements of the second superdiagonal of

U

12: $ipiv (*)$ – Integer array Input/Output

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

\max (1, n)

On entry: if

fact ='F'

, ipiv contains the pivot indices from the

L U

factorization of

A

On exit: if

fact ='N'

, ipiv contains the pivot indices from the

L U

factorization of

A

; row

i

of the matrix was interchanged with row

ipiv (i)

ipiv (i)

will always be either

i

i + 1

;

ipiv (i) = i

indicates a row interchange was not required.

13: $b (ldb, *)$ – Real (Kind=nag_wp) array Input

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

\max (1, nrhs)

On entry: the

n \times r

right-hand side matrix

B

14: $ldb$ – Integer Input

On entry: the first dimension of the array b as declared in the (sub)program from which f07cbf is called.

Constraint:

ldb \geq \max (1, n)

15: $x (ldx, *)$ – Real (Kind=nag_wp) array Output

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

\max (1, nrhs)

On exit: if

info = 0

n + 1

, the

n \times r

solution matrix

X

16: $ldx$ – Integer Input

On entry: the first dimension of the array x as declared in the (sub)program from which f07cbf is called.

Constraint:

ldx \geq \max (1, n)

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

On exit: the estimate of the reciprocal condition number of the matrix

A

. If

rcond = 0.0

, the matrix may be exactly singular. This condition is indicated by

info > 0 and info \leq n

. Otherwise, if rcond is less than the machine precision, the matrix is singular to working precision. This condition is indicated by

info = n + 1

18: $ferr (nrhs)$ – Real (Kind=nag_wp) array Output

On exit: if

info = 0

n + 1

, an estimate of the forward error bound for each computed solution vector, such that

{‖ {\hat{x}}_{j} - x_{j} ‖}_{\infty} / {‖ 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 as reliable as the estimate for rcond, and is almost always a slight overestimate of the true error.

19: $berr (nrhs)$ – Real (Kind=nag_wp) array Output

On exit: if

info = 0

n + 1

, an 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).

20: $work (3 \times n)$ – Real (Kind=nag_wp) array Workspace

21: $iwork (n)$ – Integer array Workspace

22: $info$ – Integer Output

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.

$info > 0 and info < n$: Element $⟨ value ⟩$ of the diagonal is exactly zero. The factorization has not been completed, but the factor $U$ is exactly singular, so the solution and error bounds could not be computed. $rcond = 0.0$ is returned.

$info > 0 and info = n$: Element $⟨ value ⟩$ of the diagonal is exactly zero. The factorization has been completed, but the factor $U$ is exactly singular, so the solution and error bounds could not be computed. $rcond = 0.0$ is returned.

$info = n + 1$: $U$ 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 | L | | U |,

c (n)

is a modest linear function of

n

, and

ε

is the machine precision. See Section 9.3 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

f07cbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f07cbf 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

is proportional to

n r

The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization. The solution is then refined, and the errors estimated, using iterative refinement.

In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.

The complex analogue of this routine is f07cpf.

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, forward errors and condition number are also output.

f07cb: FL CL CPP AD

NAG FL Interfacef07cbf (dgtsvx)

▸▿ 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 FL Interface
f07cbf (dgtsvx)