NAG Library Routine Document

F07CNF (ZGTSV)

+− 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

F07CNF (ZGTSV) computes the solution to a complex 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

SUBROUTINE F07CNF (

N, NRHS, DL, D, DU, B, LDB, INFO)

INTEGER	N, NRHS, LDB, INFO
COMPLEX (KIND=nag_wp)	DL(), D(), DU(), B(LDB,)

The routine may be called by its LAPACK name zgtsv.

3 Description

F07CNF (ZGTSV) 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 the 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 Parameters

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( $*$ ) – COMPLEX (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 constrains 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( $*$ ) – COMPLEX (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( $*$ ) – COMPLEX (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, $*$ ) – COMPLEX (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 F07CNF (ZGTSV) 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

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 $INFO = i$ , $u_{i i}$ is exactly zero, 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.

Alternatives to F07CNF (ZGTSV), which return condition and error estimates are F04CCF and F07CPF (ZGTSVX).

8 Further Comments

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

A X = B

is proportional to

n r

The real analogue of this routine is F07CAF (DGTSV).

9 Example

This example solves the equations

A x = b,

where

A

is the tridiagonal matrix

A = (\begin{array}{r} - 1.3 + 1.3 i & 2.0 - 1.0 i & 0 & 0 & 0 \\ 1.0 - 2.0 i & - 1.3 + 1.3 i & 2.0 + 1.0 i & 0 & 0 \\ 0 & 1.0 + 1.0 i & - 1.3 + 3.3 i & - 1.0 + 1.0 i & 0 \\ 0 & 0 & 2.0 - 3.0 i & - 0.3 + 4.3 i & 1.0 - 1.0 i \\ 0 & 0 & 0 & 1.0 + 1.0 i & - 3.3 + 1.3 i \end{array})

and

b = (\begin{array}{r} 2.4 - 5.0 i \\ 3.4 + 18.2 i \\ - 14.7 + 9.7 i \\ 31.9 - 7.7 i \\ - 1.0 + 1.6 i \end{array}) .

NAG Library Routine DocumentF07CNF (ZGTSV)

+− 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

F07CNF (ZGTSV)