Inverse of a matrix
PROBLEM
Calculation of the matrix X. which is the matrix inverse of a square matrix A., i. e., A.*X.=X.*A.=I., where I. is the identity matrix and operator * denotes matrix multiplications.
ALGORITHM
To find the inverse of a matrix column by column, we can use
LUDCMP and
LUBKSB routines. The total operation count is
N**3
IMPLEMENTATION
Unit: fragment of a program
Interface: LUDCMP and LUBKSB
Result: Matrix X. will now contain the inverse of the original matrix A.
/* read an input N by N matrix A. */
...
call LUDCMP N
do J = 1 to N
B. = 0; B.J = 1; call LUBKSB N
do I = 1 to N; X.I.J = B.I; end
end
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EXAMPLE
N = 3
A.1.1 = 1; A.1.2 = -4; A.1.3 = -3
A.2.1 = 1; A.2.2 = -5; A.2.3 = -3
A.3.1 = -1; A.3.2 = 6; A.3.3 = 4
/* Matrix A. will destroyed by LUDCMP */
do I = 1 to N
do J = 1 to N
Y.I.J = A.I.J
end
end
call LUDCMP N
do J = 1 to N
B. = 0; B.J = 1; call LUBKSB N
do I = 1 to N; X.I.J = B.I; end
end
/* X. is the inverse of the original A., proof?
X.*A.=Identity matrix
*/
do I = 1 to N
Row = ""
do K = 1 to N
C.I.K = 0
do J = 1 to N
C.I.K = C.I.K + Y.I.J * X.J.K
end
Row = Row C.I.K
end
say Row
end
exit
...
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CONNECTIONS
Literature
Press W.H., Teukolsky S.A., Vetterling W.T., Flannery B.P. Numerical Recipes in C : the art of scientific computing
- 2nd ed. University Press, Cambridge, 1992
Faddejev A.K., Sominskij J.S. Sbornik zadac po vyssej algebre
Nauka, Moskva 1964
