NAG Library Routine Document

F07CRF (ZGTTRF)

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 order of the matrix $A$ .
Constraint: $N \geq 0$ .
2: 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: is overwritten by the $(n - 1)$ multipliers that define the matrix $L$ of the $L U$ factorization of $A$ .
3: 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: is overwritten by the $n$ diagonal elements of the upper triangular matrix $U$ from the $L U$ factorization of $A$ .
4: 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: is overwritten by the $(n - 1)$ elements of the first superdiagonal of $U$ .
5: DU2( $N - 2$ ) – COMPLEX (KIND=nag_wp) arrayOutput: On exit: contains the $(n - 2)$ elements of the second superdiagonal of $U$ .
6: IPIV(N) – INTEGER arrayOutput: On exit: contains the $n$ pivot indices that define the permutation matrix $P$ . At the $i$ th step, row $i$ of the matrix was interchanged with row $IPIV (i)$ . $IPIV (i)$ will always be either $i$ or $(i + 1)$ , $IPIV (i) = i$ indicating that a row interchange was not performed.
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 $INFO = i$ , $U (i, i)$ is exactly zero. The factorization has been completed, but the factor $U$ is exactly singular, and division by zero will occur if it is used to solve a system of equations.

7 Accuracy

The computed factorization satisfies an equation of the form

A + E = P L U,

where

{‖E‖}_{\infty} = O (ε) {‖A‖}_{\infty}

and

ε

is the machine precision.

Following the use of this routine, F07CSF (ZGTTRS) can be used to solve systems of equations

A X = B

A^{T} X = B

A^{H} X = B

, and F07CUF (ZGTCON) can be used to estimate the condition number of

A

8 Further Comments

The total number of floating point operations required to factorize the matrix

A

is proportional to

n

The real analogue of this routine is F07CDF (DGTTRF).

9 Example

This example factorizes the tridiagonal matrix

A

given by

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}) .

NAG Library Routine DocumentF07CRF (ZGTTRF)

+− 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

F07CRF (ZGTTRF)