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The IMSL_QRFAC procedure computes the QR factorization of a real matrix A.

The IMSL_QRFAC procedure computes a QR factorization of the matrix AP, where P is the permutation matrix defined by the pivoting and computes the smallest integer k satisfying |rk,k| < TOLERANCE*|r0,0| to the keyword BASIS.

Householder transformations:

Qk = I – τkukukT, k = 0, ..., min(m – 1, n) – 1

compute the factorization. The decomposition is computed in the form Qmin (m – 1, n) – 1 ... Q0AP = R, so AP = QR where Q = Q0 ... Qmin (m – 1, n) – 1. Since each Householder vector uk has zeros in the first k + 1 entries, it is stored as part of column k of QR. The upper-trapezoidal matrix R is stored in the upper-trapezoidal part of the first min(m, n) rows of QR.

When computing the factorization, the procedure computes the QR factorization of AP with P defined by the input pivot and by column pivoting among “free” columns. Before the factorization, initial columns are moved to the beginning of the array A and the final columns to the end. Neither initial nor final columns are permuted further during the computation. Only the free columns are moved.

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