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NAG Library Routine Document

f07caf (dgtsv)

▸▿ 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

f07caf (dgtsv) computes the solution to a real system of linear equations

A X = B,

where

A

is an

n

n

tridiagonal matrix and

X

and

B

are

n

r

matrices.

2

Specification

Fortran Interface

Subroutine f07caf (

n, nrhs, dl, d, du, b, ldb, info)

Integer, Intent (In)	::	n, nrhs, ldb
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (Inout)	::	dl(), d(), du(), b(ldb,)

C Header Interface

#include nagmk26.h

void	f07caf_ ( const Integer n, const Integer nrhs, double dl[], double d[], double du[], double b[], const Integer ldb, Integer info)

The routine may be called by its LAPACK name dgtsv.

3

Description

f07caf (dgtsv) uses Gaussian elimination with partial pivoting and row interchanges to solve the equations

A X = B

. The matrix

A

is factorized as

A = P L U

, where

P

is a permutation matrix,

L

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

U

is an upper triangular band matrix, with two superdiagonals.

Note that equations

A^{T} X = B

may be solved by interchanging the order of the arguments du and dl.

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: $n$ – IntegerInput: On entry: $n$ , the number of linear equations, i.e., the order of the matrix $A$ .

Constraint: $n \geq 0$ .
2: $nrhs$ – IntegerInput: On entry: $r$ , the number of right-hand sides, i.e., the number of columns of the matrix $B$ .

Constraint: $nrhs \geq 0$ .
3: $dl (*)$ – Real (Kind=nag_wp) arrayInput/Output: 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 $A$ .

On exit: if no constraints are violated, dl is overwritten by the ( $n - 2$ ) elements of the second superdiagonal of the upper triangular matrix $U$ from the $L U$ factorization of $A$ , in $dl (1), dl (2), \dots, dl (n - 2)$ .
4: $d (*)$ – Real (Kind=nag_wp) arrayInput/Output: 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 $A$ .

On exit: if no constraints are violated, d is overwritten by the $n$ diagonal elements of the upper triangular matrix $U$ from the $L U$ factorization of $A$ .
5: $du (*)$ – Real (Kind=nag_wp) arrayInput/Output: 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 $A$ .

On exit: if no constraints are violated, du is overwritten by the $(n - 1)$ elements of the first superdiagonal of $U$ .
6: $b (ldb *)$ – Real (Kind=nag_wp) arrayInput/Output: Note: the second dimension of the array b must be at least $\max (1, nrhs)$ .

To solve the equations $A x = b$ , where $b$ is a single right-hand side, b may be supplied as a one-dimensional array with length $ldb = \max (1, n)$ .

On entry: the $n$ by $r$ right-hand side matrix $B$ .

On exit: if $info = 0$ , the $n$ by $r$ solution matrix $X$ .
7: $ldb$ – IntegerInput: On entry: the first dimension of the array b as declared in the (sub)program from which f07caf (dgtsv) is called.

Constraint: $ldb \geq \max (1, n)$ .
8: $info$ – 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.

$info > 0$: Element $〈value〉$ of the diagonal is exactly zero, and the solution has not been computed. The factorization has not been completed unless $n = 〈value〉$ .

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‖}_{1} = O (ε) {‖A‖}_{1}

and

ε

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

\frac{{‖\hat{x} - x‖}_{1}}{{‖x‖}_{1}} \leq κ (A) \frac{{‖E‖}_{1}}{{‖A‖}_{1}},

where

κ (A) = {‖A^{- 1}‖}_{1} {‖A‖}_{1}

, 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.

Alternatives to f07caf (dgtsv), which return condition and error estimates are f04bcf and f07cbf (dgtsvx).

8

Parallelism and Performance

f07caf (dgtsv) is not threaded in any implementation.

9

Further Comments

The total number of floating-point operations required to solve the equations

A X = B

is proportional to

n r

The complex analogue of this routine is f07cnf (zgtsv).

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 \\ - 0.5 \\ 2.6 \\ 0.6 \\ 2.7 \end{matrix}) .

NAG Library Routine Document

f07caf (dgtsv)

▸▿ 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