NAG Library Routine Document

F07ARF (ZGETRF)

+− 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

F07ARF (ZGETRF) computes the

L U

factorization of a complex

m

n

matrix.

2 Specification

SUBROUTINE F07ARF (

M, N, A, LDA, IPIV, INFO)

INTEGER	M, N, LDA, IPIV(min(M,N)), INFO
COMPLEX (KIND=nag_wp)	A(LDA,*)

The routine may be called by its LAPACK name zgetrf.

3 Description

F07ARF (ZGETRF) forms the

L U

factorization of a complex

m

n

matrix

A

A = P L U

, where

P

is a permutation matrix,

L

is lower triangular with unit diagonal elements (lower trapezoidal if

m > n

) and

U

is upper triangular (upper trapezoidal if

m < n

). Usually

A

is square

(m = n)

, and both

L

and

U

are triangular. The routine uses partial pivoting, with row interchanges.

4 References

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

5 Parameters

1: M – INTEGERInput: On entry: $m$ , the number of rows of the matrix $A$ .
Constraint: $M \geq 0$ .
2: N – INTEGERInput: On entry: $n$ , the number of columns of the matrix $A$ .
Constraint: $N \geq 0$ .
3: A(LDA, $*$ ) – COMPLEX (KIND=nag_wp) arrayInput/Output: Note: the second dimension of the array A must be at least $\max (1, N)$ .
On entry: the $m$ by $n$ matrix $A$ .

On exit: the factors $L$ and $U$ from the factorization $A = P L U$ ; the unit diagonal elements of $L$ are not stored.
4: LDA – INTEGERInput: On entry: the first dimension of the array A as declared in the (sub)program from which F07ARF (ZGETRF) is called.
Constraint: $LDA \geq \max (1, M)$ .
5: IPIV( $\min (M, N)$ ) – INTEGER arrayOutput: On exit: the pivot indices that define the permutation matrix. At the $i$ th step, if $IPIV (i) > i$ then row $i$ of the matrix $A$ was interchanged with row $IPIV (i)$ , for $i = 1, 2, \dots, \min (m, n)$ . $IPIV (i) \leq i$ indicates that, at the $i$ th step, a row interchange was not required.
6: 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 parameter 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 factors

L

and

U

are the exact factors of a perturbed matrix

A + E

, where

|E| \leq c (\min (m, n)) ε P |L| |U|,

c (n)

is a modest linear function of

n

, and

ε

is the machine precision.

8 Further Comments

The total number of real floating point operations is approximately

\frac{8}{3} n^{3}

m = n

(the usual case),

\frac{4}{3} n^{2} (3 m - n)

m > n

and

\frac{4}{3} m^{2} (3 n - m)

m < n

A call to this routine with

m = n

may be followed by calls to the routines:

F07ASF (ZGETRS) to solve $A X = B$ , $A^{T} X = B$ or $A^{H} X = B$ ;
F07AUF (ZGECON) to estimate the condition number of $A$ ;
F07AWF (ZGETRI) to compute the inverse of $A$ .

The real analogue of this routine is F07ADF (DGETRF).

9 Example

This example computes the

L U

factorization of the matrix

A

, where

A = (\begin{array}{r} - 1.34 + 2.55 i & 0.28 + 3.17 i & - 6.39 - 2.20 i & 0.72 - 0.92 i \\ - 0.17 - 1.41 i & 3.31 - 0.15 i & - 0.15 + 1.34 i & 1.29 + 1.38 i \\ - 3.29 - 2.39 i & - 1.91 + 4.42 i & - 0.14 - 1.35 i & 1.72 + 1.35 i \\ 2.41 + 0.39 i & - 0.56 + 1.47 i & - 0.83 - 0.69 i & - 1.96 + 0.67 i \end{array}) .

NAG Library Routine DocumentF07ARF (ZGETRF)

+− 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

F07ARF (ZGETRF)