NAG Library Routine Document

F07JNF (ZPTSV)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

F07JNF (ZPTSV) computes the solution to a complex system of linear equations

A X = B,

where

A

is an

n

n

Hermitian positive definite tridiagonal matrix, and

X

and

B

are

n

r

matrices.

2 Specification

SUBROUTINE F07JNF (

N, NRHS, D, E, B, LDB, INFO)

INTEGER	N, NRHS, LDB, INFO
REAL (KIND=nag_wp)	D(*)
COMPLEX (KIND=nag_wp)	E(), B(LDB,)

The routine may be called by its LAPACK name zptsv.

3 Description

F07JNF (ZPTSV) factors

A

A = L D L^{H}

. The factored form of

A

is then used to solve the system of equations.

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

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Parameters

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 $A = L D L^{H}$ .
4: E( $*$ ) – COMPLEX (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 $L D L^{H}$ factorization of $A$ . (E can also be regarded as the superdiagonal of the unit bidiagonal factor $U$ from the $U^{H} D U$ factorization of $A$ .)
5: B(LDB, $*$ ) – COMPLEX (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 F07JNF (ZPTSV) 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

Errors or warnings detected by the routine:

$INFO < 0$: If $INFO = - i$ , the $i$ th argument had an illegal value. An explanatory message is output, and execution of the program is terminated.

$INFO > 0$: If $INFO = i$ , the leading minor of order $i$ is not positive definite, and the solution has not been computed. The factorization has not been completed unless $i = N$ .

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.

F07JPF (ZPTSVX) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, F04CGF solves

A x = b

and returns a forward error bound and condition estimate. F04CGF calls F07JNF (ZPTSV) to solve the equations.

8 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 real analogue of this routine is F07JAF (DPTSV).

9 Example

This example solves the equations

A x = b,

where

A

is the Hermitian positive definite tridiagonal matrix

A = (\begin{array}{r} 16.0 & 16.0 - 16.0 i & 0 & 0 \\ 16.0 + 16.0 i & 41.0 & 18.0 + 9.0 i & 0 \\ 0 & 18.0 - 9.0 i & 46.0 & 1.0 + 4.0 i \\ 0 & 0 & 1.0 - 4.0 i & 21.0 \end{array})

and

b = (\begin{array}{r} 64.0 + 16.0 i \\ 93.0 + 62.0 i \\ 78.0 - 80.0 i \\ 14.0 - 27.0 i \end{array}) .

Details of the

L D L^{H}

factorization of

A

are also output.

NAG Library Routine DocumentF07JNF (ZPTSV)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

F07JNF (ZPTSV)