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

5 Arguments

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

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

Constraint: $nrhs \geq 0$ .
3: $dl (*)$ – Real (Kind=nag_wp) array Input/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) array Input/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) array Input/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) array Input/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: $ldb$ – Integer Input: On entry: the first dimension of the array b as declared in the (sub)program from which f07caf is called.

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

$info > 0$: 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, which return condition and error estimates are f04bcf and f07cbf.

8 Parallelism and Performance

f07caf 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.

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}) .

Interfaces: FL CL AD

NAG FL Interface Introduction

F07 (Lapacklin) Chapter Contents

F07 (Lapacklin) Chapter Introduction

f07ca: FL CL AD

NAG FL Interfacef07caf (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

NAG FL Interface
f07caf (dgtsv)