f07ar computes the factorization of a complex by matrix.
- Type: System..::..Int32On entry: , the number of rows of the matrix .Constraint: .
- Type: System..::..Int32On entry: , the number of columns of the matrix .Constraint: .
- Type: array<NagLibrary..::..Complex,2>[,](,)[,][,]An array of size [dim1, dim2]Note: dim1 must satisfy the constraint:Note: the second dimension of the array a must be at least .On entry: the by matrix .On exit: the factors and from the factorization ; the unit diagonal elements of are not stored.
- Type: array<System..::..Int32>()An array of size On exit: the pivot indices that define the permutation matrix. At the th step, if then row of the matrix was interchanged with row , for . indicates that, at the th step, a row interchange was not required.
f07ar forms the factorization of a complex by matrix as , where is a permutation matrix, is lower triangular with unit diagonal elements (lower trapezoidal if ) and is upper triangular (upper trapezoidal if ). Usually is square , and both and are triangular. The method uses partial pivoting, with row interchanges.
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Some error messages may refer to parameters that are dropped from this interface (LDA) In these cases, an error in another parameter has usually caused an incorrect value to be inferred.
- If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
Element of the diagonal is exactly zero. The factorization has been completed, but the factor is exactly singular, and division by zero will occur if it is used to solve a system of equations.
The computed factors and are the exact factors of a perturbed matrix , where
is a modest linear function of , and is the machine precision.
The total number of real floating-point operations is approximately if (the usual case), if and if .
A call to this method with may be followed by calls to the methods:
- f07as to solve , or ;
- (F07AUF not in this release) to estimate the condition number of ;
- (F07AWF not in this release) to compute the inverse of .
The real analogue of this method is (F07ADF not in this release).
This example computes the factorization of the matrix , where