F010 Linear Equations, Matrix Inversion

Subroutine tEQN (where $t=R$ , D or C as described below) solves the matrix equation

which represents a system of N simultaneous linear equations with K right-hand sides:

$\sum$_j=1^Na_ijx_jk=b_ik,&sp;(i=1,2,...,N, k=1,2,...,K).

Subroutine tINV computes the inverse of a square matrix A. Subroutine tEQINV solves the system (*) and also computes the inverse of A, but is appreciably slower than tEQN.

If the determinant of A is also required, or if several systems of the form (*) are to be solved sequentially with the same coefficient matrix A but differing right-hand sides B, the subroutines in RFACT (F011) should be used.

Structure:

SUBROUTINE subprograms
User Entry Names: RINV, REQN, REQINV, DINV, DEQN, DEQINV, CINV, CEQN, CEQINV
Internal Entry Names: F010PR
Files Refeenced: Printer
External References: RFACT, RFEQN, RFINV, DFACT, DFEQN, DFINV, CFACT, CFEQN, CFINV,
TMPRNT (F011), KERMTR (N001), ABEND (Z035)

Usage:

For $t=R$ (type REAL), $t=D$ (type DOUBLE PRECISION), $t=C$ (type COMPLEX):

    CALL tEQN  (N,A,IDIM,IR,IFAIL,K,B)
    CALL tINV  (N,A,IDIM,IR,IFAIL)
    CALL tEQINV(N,A,IDIM,IR,IFAIL,K,B)

N

( INTEGER) Order of the square matrix A.

A

(Type according to t) Two-dimensional array whose first dimension has the value IDIM.

IDIM

( INTEGER) First dimension of array A (and of array B if $K\; >\; 1$ ).

IR

( INTEGER) Array of at least N elements, required as working space.

IFAIL

( INTEGER) On exit, IFAIL will be set to $-1$ if A is found to be singular, and to 0 otherwise. (Singularity will often go undetected because of rounding errors during factorization even if the elements of A have integral values.)

K

( INTEGER) Number of columns of the matrices B and X.

B

(Type according to t) In general, a two-dimensional array whose first dimension has the value IDIM. B may be one-dimensional if $K\; =\; 1$ .

tEQN

The solution X replaces B. The matrix A is destroyed.

tINV

The inverse $A-1$ of A replaces A.

tEQINV

The solution X replaces B, and the inverse $A-1$ of A replaces A.

Method:

Triangular factorization with row interchanges, implemented by in-line code if $N\; \le 3$ and by calls to library program RFACT (F011) if $N\; >\; 3$ . If or or , a message is printed and program execution is terminated by calling ABEND (Z035).

Examples:

Assume that the $10\; x10$ matrix A and the $10\; x3$

matrix B are stored according to the Fortran convention in arrays A and B respectively of a program containing declarations

    DIMENSION IR(25)
    DOUBLE PRECISION A(25,30),B(25,10)

    CALL DEQINV (10,A,25,IR,IFAIL,3,B)
    IF(IFAIL .NE. 0) GO TO 100

F011