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CGGQRF(l)) CGGQRF(l)NAMECGGQRF - compute a generalized QR factorization of an N-by-M matrix A and an N-by-P matrix BSYNOPSISSUBROUTINE CGGQRF( N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO ) INTEGER INFO, LDA, LDB, LWORK, M, N, P COMPLEX A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ), WORK( * )PURPOSECGGQRF computes a generalized QR factorization of an N-by-M matrix A and an N-by-P matrix B: A = Q*R, B = Q*T*Z, where Q is an N-by-N unitary matrix, Z is a P-by-P unitary matrix, and R and T assume one of the forms: if N >= M, R = ( R11 ) M , or if N < M, R = ( R11 R12 ) N, ( 0 ) N-M N M-N M where R11 is upper triangular, and if N <= P, T = ( 0 T12 ) N, or if N > P, T = ( T11 ) N-P, P-N N ( T21 ) P P where T12 or T21 is upper triangular. In particular, if B is square and nonsingular, the GQR factorization of A and B implicitly gives the QR factorization of inv(B)*A: inv(B)*A = Z'*(inv(T)*R) where inv(B) denotes the inverse of the matrix B, and Z' denotes the conjugate transpose of matrix Z.ARGUMENTSN (input) INTEGER The number of rows of the matrices A and B. N >= 0. M (input) INTEGER The number of columns of the matrix A. M >= 0. P (input) INTEGER The number of columns of the matrix B. P >= 0. A (input/output) COMPLEX array, dimension (LDA,M) On entry, the N-by-M matrix A. On exit, the elements on and above the diagonal of the array contain the min(N,M)-by-M upper trapezoidal matrix R (R is upper trian- gular if N >= M); the elements below the diagonal, with the array TAUA, represent the unitary matrix Q as a product of min(N,M) elementary reflectors (see Further Details). LDA (input) INTEGER The leading dimension of the array A. LDA >= max(1,N). TAUA (output) COMPLEX array, dimension (min(N,M)) The scalar factors of the elementary reflectors which represent the unitary matrix Q (see Further Details). B (input/output) COMPLEX array, dimension (LDB,P) On entry, the N-by-P matrix B. On exit, if N <= P, the upper triangle of the sub- array B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T; if N > P, the elements on and above the (N-P)-th subdiagonal contain the N-by-P upper trape- zoidal matrix T; the remaining elements, with the array TAUB, represent the uni- tary matrix Z as a product of elementary reflectors (see Further Details). LDB (input) INTEGER The leading dimension of the array B. LDB >= max(1,N). TAUB (output) COMPLEX array, dimension (min(N,P)) The scalar factors of the elementary reflectors which represent the unitary matrix Z (see Further Details). WORK (workspace/output) COMPLEX array, dimension (LWORK) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK (input) INTEGER The dimension of the array WORK. LWORK >= max(1,N,M,P). For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3), where NB1 is the optimal blocksize for the QR factorization of an N-by-M matrix, NB2 is the optimal blocksize for the RQ fac- torization of an N-by-P matrix, and NB3 is the optimal blocksize for a call of CUNMQR. If LWORK =, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. INFO (output) INTEGER = 0: successful exit < 0: if INFO =-1, the i-th argument had an illegal value.-iFURTHER DETAILSThe matrix Q is represented as a product of elementary reflectors Q = H(1) H(2) . . . H(k), where k = min(n,m). Each H(i) has the form H(i) = I - taua * v * v' where taua is a complex scalar, and v is a complex vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), and taua in TAUA(i). To form Q explicitly, use LAPACK subroutine CUNGQR. To use Q to update another matrix, use LAPACK subroutine CUNMQR. The matrix Z is represented as a product of elementary reflectors Z = H(1) H(2) . . . H(k), where k = min(n,p). Each H(i) has the form H(i) = I - taub * v * v' where taub is a complex scalar, and v is a complex vector with v(p-k+i+1:p) = 0 and v(p- k+i) = 1; v(1:p-k+i-1) is stored on exit in B(n-k+i,1:p-k+i-1), and taub in TAUB(i). To form Z explicitly, use LAPACK subroutine CUNGRQ. To use Z to update another matrix, use LAPACK subroutine CUNMRQ.LAPACK version 3.015 June 2000 CGGQRF(l)

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