F010: Linear Equations, Matrix Inversion

Author(s): G.A. Erskine	Library: KERNLIB
Submitter:	Submitted: 18.12.1979
Language: Fortran	Revised: 27.11.1984

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

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

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, KERMTR, ABEND

Usage:

For (type REAL), (type DOUBLE PRECISION), (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 ).

IR

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

IFAIL

(INTEGER) On exit, IFAIL will be set to 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 .

These subroutines must be called with matrix A in array A and matrix B in array B. Then, provided the matrix A is non-singular, IFAIL will be set to 0 and arrays A and B will be set as follows:

tEQN

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

tINV

The inverse of A replaces A.

tEQINV

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

If the matrix A is singular, IFAIL will be set to . In this case the contents of A is unpredictable and the contents of B is unchanged.

Method:

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

Examples:

Assume that the matrix A and the 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)

To replace B by the solution matrix X of the system of equations and to replace A by , with a jump to label 100 if A is singular:

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