**Purpose**

To compute a rank-revealing RQ factorization of a complex general M-by-N matrix A, which may be rank-deficient, and estimate its effective rank using incremental condition estimation. The routine uses a truncated RQ factorization with row pivoting: [ R11 R12 ] P * A = R * Q, where R = [ ], [ 0 R22 ] with R22 defined as the largest trailing upper triangular submatrix whose estimated condition number is less than 1/RCOND. The order of R22, RANK, is the effective rank of A. Condition estimation is performed during the RQ factorization process. Matrix R11 is full (but of small norm), or empty. MB3PYZ does not perform any scaling of the matrix A.

SUBROUTINE MB3PYZ( M, N, A, LDA, RCOND, SVLMAX, RANK, SVAL, JPVT, $ TAU, DWORK, ZWORK, INFO ) C .. Scalar Arguments .. INTEGER INFO, LDA, M, N, RANK DOUBLE PRECISION RCOND, SVLMAX C .. Array Arguments .. INTEGER JPVT( * ) COMPLEX*16 A( LDA, * ), TAU( * ), ZWORK( * ) DOUBLE PRECISION DWORK( * ), SVAL( 3 )

**Input/Output Parameters**

M (input) INTEGER The number of rows of the matrix A. M >= 0. N (input) INTEGER The number of columns of the matrix A. N >= 0. A (input/output) COMPLEX*16 array, dimension ( LDA, N ) On entry, the leading M-by-N part of this array must contain the given matrix A. On exit, the upper triangle of the subarray A(M-RANK+1:M,N-RANK+1:N) contains the RANK-by-RANK upper triangular matrix R22; the remaining elements in the last RANK rows, with the array TAU, represent the unitary matrix Q as a product of RANK elementary reflectors (see METHOD). The first M-RANK rows contain the result of the RQ factorization process used. LDA INTEGER The leading dimension of the array A. LDA >= max(1,M). RCOND (input) DOUBLE PRECISION RCOND is used to determine the effective rank of A, which is defined as the order of the largest trailing triangular submatrix R22 in the RQ factorization with pivoting of A, whose estimated condition number is less than 1/RCOND. 0 <= RCOND <= 1. NOTE that when SVLMAX > 0, the estimated rank could be less than that defined above (see SVLMAX). SVLMAX (input) DOUBLE PRECISION If A is a submatrix of another matrix B, and the rank decision should be related to that matrix, then SVLMAX should be an estimate of the largest singular value of B (for instance, the Frobenius norm of B). If this is not the case, the input value SVLMAX = 0 should work. SVLMAX >= 0. RANK (output) INTEGER The effective (estimated) rank of A, i.e., the order of the submatrix R22. SVAL (output) DOUBLE PRECISION array, dimension ( 3 ) The estimates of some of the singular values of the triangular factor R: SVAL(1): largest singular value of R(M-RANK+1:M,N-RANK+1:N); SVAL(2): smallest singular value of R(M-RANK+1:M,N-RANK+1:N); SVAL(3): smallest singular value of R(M-RANK:M,N-RANK:N), if RANK < MIN( M, N ), or of R(M-RANK+1:M,N-RANK+1:N), otherwise. If the triangular factorization is a rank-revealing one (which will be the case if the trailing rows were well- conditioned), then SVAL(1) will also be an estimate for the largest singular value of A, and SVAL(2) and SVAL(3) will be estimates for the RANK-th and (RANK+1)-st singular values of A, respectively. By examining these values, one can confirm that the rank is well defined with respect to the chosen value of RCOND. The ratio SVAL(1)/SVAL(2) is an estimate of the condition number of R(M-RANK+1:M,N-RANK+1:N). JPVT (output) INTEGER array, dimension ( M ) If JPVT(i) = k, then the i-th row of P*A was the k-th row of A. TAU (output) COMPLEX*16 array, dimension ( MIN( M, N ) ) The trailing RANK elements of TAU contain the scalar factors of the elementary reflectors.

DWORK DOUBLE PRECISION array, dimension ( 2*M ) ZWORK COMPLEX*16 array, dimension ( 3*M-1 )

INFO INTEGER = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value.

The routine computes a truncated RQ factorization with row pivoting of A, P * A = R * Q, with R defined above, and, during this process, finds the largest trailing submatrix whose estimated condition number is less than 1/RCOND, taking the possible positive value of SVLMAX into account. This is performed using an adaptation of the LAPACK incremental condition estimation scheme and a slightly modified rank decision test. The factorization process stops when RANK has been determined. The matrix Q is represented as a product of elementary reflectors Q = H(k-rank+1)' H(k-rank+2)' . . . H(k)', where k = min(m,n). Each H(i) has the form H = I - tau * v * v' where tau is a complex scalar, and v is a complex vector with v(n-k+i+1:n) = 0 and v(n-k+i) = 1; conjg(v(1:n-k+i-1)) is stored on exit in A(m-k+i,1:n-k+i-1), and tau in TAU(i). The matrix P is represented in jpvt as follows: If jpvt(j) = i then the jth row of P is the ith canonical unit vector.

[1] Bischof, C.H. and P. Tang. Generalizing Incremental Condition Estimation. LAPACK Working Notes 32, Mathematics and Computer Science Division, Argonne National Laboratory, UT, CS-91-132, May 1991. [2] Bischof, C.H. and P. Tang. Robust Incremental Condition Estimation. LAPACK Working Notes 33, Mathematics and Computer Science Division, Argonne National Laboratory, UT, CS-91-133, May 1991.

The algorithm is backward stable.

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**Program Text**

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