NAG Library Routine Document

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

5

Arguments

1: $n$ – IntegerInput: On entry: $n$ , 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: $d (*)$ – Real (Kind=nag_wp) arrayInput/Output: Note: the dimension of the array d must be at least $\max (1, n)$ .

On entry: the $n$ diagonal elements of the tridiagonal matrix $A$ .

On exit: the $n$ diagonal elements of the diagonal matrix $D$ from the factorization $A = L D L^{T}$ .
4: $e (*)$ – Real (Kind=nag_wp) arrayInput/Output: 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 $A$ .

On exit: the $(n - 1)$ subdiagonal elements of the unit bidiagonal factor $L$ from the $L D L^{T}$ factorization of $A$ . (e can also be regarded as the superdiagonal of the unit bidiagonal factor $U$ from the $U^{T} D U$ factorization of $A$ .)
5: $b (ldb *)$ – Real (Kind=nag_wp) arrayInput/Output: Note: the second dimension of the array b must be at least $\max (1, nrhs)$ .

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

On exit: if $info = 0$ , the $n$ by $r$ solution matrix $X$ .
6: $ldb$ – IntegerInput: On entry: the first dimension of the array b as declared in the (sub)program from which f07jaf (dptsv) is called.

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

$info > 0$: The leading minor of order $〈value〉$ is not positive definite, 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.

f07jbf (dptsvx) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, f04bgf solves

A x = b

and returns a forward error bound and condition estimate. f04bgf calls f07jaf (dptsv) to solve the equations.

8

Parallelism and Performance

f07jaf (dptsv) 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 of

A

is proportional to

n

, and the number of floating-point operations required for the solution of the equations is proportional to

n r

, where

r

is the number of right-hand sides.

The complex analogue of this routine is f07jnf (zptsv).

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 \\ 9.0 \\ 2.0 \\ 14.0 \\ 7.0 \end{matrix}) .

Details of the

L D L^{T}

factorization of

A

are also output.

NAG Library Routine Document

f07jaf (dptsv)

▸▿ 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