f08be computes the factorization, with column pivoting, of a real 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<System..::..Double,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: if , the elements below the diagonal are overwritten by details of the orthogonal matrix and the upper triangle is overwritten by the corresponding elements of the by upper triangular matrix .If , the strictly lower triangular part is overwritten by details of the orthogonal matrix and the remaining elements are overwritten by the corresponding elements of the by upper trapezoidal matrix .
- Type: array<System..::..Int32>()An array of size [dim1]Note: the dimension of the array jpvt must be at least .On entry: if , then the th column of is moved to the beginning of before the decomposition is computed and is fixed in place during the computation. Otherwise, the th column of is a free column (i.e., one which may be interchanged during the computation with any other free column).On exit: details of the permutation matrix . More precisely, if , then the th column of is moved to become the th column of ; in other words, the columns of are the columns of in the order .
- Type: array<System..::..Double>()An array of size On exit: further details of the orthogonal matrix .
f08be forms the factorization, with column pivoting, of an arbitrary rectangular real by matrix.
If , the factorization is given by:
where is an by upper triangular matrix, is an by orthogonal matrix and is an by permutation matrix. It is sometimes more convenient to write the factorization as
which reduces to
where consists of the first columns of , and the remaining columns.
If , is trapezoidal, and the factorization can be written
where is upper triangular and is rectangular.
The matrix is not formed explicitly but is represented as a product of elementary reflectors (see the F08 class for details). Methods are provided to work with in this representation (see [Further Comments]).
Note also that for any , the information returned in the first columns of the array a represents a factorization of the first columns of the permuted matrix .
The method allows specified columns of to be moved to the leading columns of at the start of the factorization and fixed there. The remaining columns are free to be interchanged so that at the th stage the pivot column is chosen to be the column which maximizes the -norm of elements to over columns to .
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.
The computed factorization is the exact factorization of a nearby matrix , where
and is the machine precision.
The total number of floating-point operations is approximately if or if .
To form the orthogonal matrix f08be may be followed by a call to (F08AFF not in this release): but note that the second dimension of the array a must be at least m, which may be larger than was required by f08be.
When , it is often only the first columns of that are required, and they may be formed by the call:
To apply to an arbitrary real rectangular matrix , f08be may be followed by a call to f08ag. For example, forms , where is by .
To compute a factorization without column pivoting, use (F08AEF not in this release).
The complex analogue of this method is (F08BSF not in this release).