NAG Library Routine Document

F07CSF (ZGTTRS)

F07CSF (ZGTTRS) should be preceded by a call to F07CRF (ZGTTRF), which uses Gaussian elimination with partial pivoting and row interchanges to factorize the matrix

A

A = P L U,

where

P

is a permutation matrix,

L

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

U

is an upper triangular band matrix, with two superdiagonals. F07CSF (ZGTTRS) then utilizes the factorization to solve the required 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: TRANS – CHARACTER(1)Input

On entry: specifies the equations to be solved as follows:

$TRANS ='N'$: Solve $A X = B$ for $X$ .
$TRANS ='T'$: Solve $A^{T} X = B$ for $X$ .
$TRANS ='C'$: Solve $A^{H} X = B$ for $X$ .

Constraint:

TRANS ='N'

'T'

'C'

2: N – INTEGERInput

On entry:

n

, the order of the matrix

A

Constraint:

N \geq 0

3: 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

4: DL( $*$ ) – COMPLEX (KIND=nag_wp) arrayInput

Note: the dimension of the array DL must be at least

\max (1, N - 1)

On entry: must contain the

(n - 1)

multipliers that define the matrix

L

of the

L U

factorization of

A

5: D( $*$ ) – COMPLEX (KIND=nag_wp) arrayInput

Note: the dimension of the array D must be at least

\max (1, N)

On entry: must contain the

n

diagonal elements of the upper triangular matrix

U

from the

L U

factorization of

A

6: DU( $*$ ) – COMPLEX (KIND=nag_wp) arrayInput

Note: the dimension of the array DU must be at least

\max (1, N - 1)

On entry: must contain the

(n - 1)

elements of the first superdiagonal of

U

7: DU2( $*$ ) – COMPLEX (KIND=nag_wp) arrayInput

Note: the dimension of the array DU2 must be at least

\max (1, N - 2)

On entry: must contain the

(n - 2)

elements of the second superdiagonal of

U

8: IPIV( $*$ ) – INTEGER arrayInput

Note: the dimension of the array IPIV must be at least

\max (1, N)

On entry: must contain the

n

pivot indices that define the permutation matrix

P

. At the

i

th step, row

i

of the matrix was interchanged with row

IPIV (i)

, and

IPIV (i)

must always be either

i

(i + 1)

IPIV (i) = i

indicating that a row interchange was not performed.

9: 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

r

matrix of right-hand sides

B

On exit: the

n

r

solution matrix

X

10: LDB – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F07CSF (ZGTTRS) is called.

Constraint:

LDB \geq \max (1, N)

11: 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.

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.

Following the use of this routine F07CUF (ZGTCON) can be used to estimate the condition number of

A

and F07CVF (ZGTRFS) can be used to obtain approximate error bounds.

8 Further Comments

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

A X = B

A^{T} X = B

A^{H} X = B

is proportional to

n r

The real analogue of this routine is F07CEF (DGTTRS).

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 & 2.7 + 6.9 i \\ 3.4 + 18.2 i & - 6.9 - 5.3 i \\ - 14.7 + 9.7 i & - 6.0 - 0.6 i \\ 31.9 - 7.7 i & - 3.9 + 9.3 i \\ - 1.0 + 1.6 i & - 3.0 + 12.2 i \end{array}) .

NAG Library Routine DocumentF07CSF (ZGTTRS)

+− 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

F07CSF (ZGTTRS)