latsqr - latsqr: tall-skinny QR factor

Generated automatically by Doxygen for LAPACK from the source code. Version 3.12.0 Sun Jul 20 2025 01:40:05 latsqr(3)

subroutineclatsqr(integerm,integern,integermb,integernb,complex,dimension(lda,*)a,integerlda,complex,dimension(ldt,*)t,integerldt,complex,dimension(*)work,integerlwork,integerinfo)CLATSQRPurpose: CLATSQR computes a blocked Tall-Skinny QR factorization of a complex M-by-N matrix A for M >= N: A = Q * ( R ), ( 0 ) where: Q is a M-by-M orthogonal matrix, stored on exit in an implicit form in the elements below the diagonal of the array A and in the elements of the array T; R is an upper-triangular N-by-N matrix, stored on exit in the elements on and above the diagonal of the array A. 0 is a (M-N)-by-N zero matrix, and is not stored. ParametersM M is INTEGER The number of rows of the matrix A. M >= 0. N N is INTEGER The number of columns of the matrix A. M >= N >= 0. MB MB is INTEGER The row block size to be used in the blocked QR. MB > N. NB NB is INTEGER The column block size to be used in the blocked QR. N >= NB >= 1. A A is COMPLEX array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the N-by-N upper triangular matrix R; the elements below the diagonal represent Q by the columns of blocked V (see Further Details). LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). T T is COMPLEX array, dimension (LDT, N * Number_of_row_blocks) where Number_of_row_blocks = CEIL((M-N)/(MB-N)) The blocked upper triangular block reflectors stored in compact form as a sequence of upper triangular blocks. See Further Details below. LDT LDT is INTEGER The leading dimension of the array T. LDT >= NB. WORK (workspace) COMPLEX array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the minimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. LWORK >= 1, if MIN(M,N) = 0, and LWORK >= NB*N, otherwise. If LWORK = -1, then a workspace query is assumed; the routine only calculates the minimal 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 INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value Author Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. FurtherDetails: Tall-Skinny QR (TSQR) performs QR by a sequence of orthogonal transformations, representing Q as a product of other orthogonal matrices Q = Q(1) * Q(2) * . . . * Q(k) where each Q(i) zeros out subdiagonal entries of a block of MB rows of A: Q(1) zeros out the subdiagonal entries of rows 1:MB of A Q(2) zeros out the bottom MB-N rows of rows [1:N,MB+1:2*MB-N] of A Q(3) zeros out the bottom MB-N rows of rows [1:N,2*MB-N+1:3*MB-2*N] of A . . . Q(1) is computed by GEQRT, which represents Q(1) by Householder vectors stored under the diagonal of rows 1:MB of A, and by upper triangular block reflectors, stored in array T(1:LDT,1:N). For more information see Further Details in GEQRT. Q(i) for i>1 is computed by TPQRT, which represents Q(i) by Householder vectors stored in rows [(i-1)*(MB-N)+N+1:i*(MB-N)+N] of A, and by upper triangular block reflectors, stored in array T(1:LDT,(i-1)*N+1:i*N). The last Q(k) may use fewer rows. For more information see Further Details in TPQRT. For more details of the overall algorithm, see the description of Sequential TSQR in Section 2.2 of [1]. [1] “Communication-Optimal Parallel and Sequential QR and LU Factorizations,” J. Demmel, L. Grigori, M. Hoemmen, J. Langou, SIAM J. Sci. Comput, vol. 34, no. 1, 2012 subroutinedlatsqr(integerm,integern,integermb,integernb,doubleprecision,dimension(lda,*)a,integerlda,doubleprecision,dimension(ldt,*)t,integerldt,doubleprecision,dimension(*)work,integerlwork,integerinfo)DLATSQRPurpose: DLATSQR computes a blocked Tall-Skinny QR factorization of a real M-by-N matrix A for M >= N: A = Q * ( R ), ( 0 ) where: Q is a M-by-M orthogonal matrix, stored on exit in an implicit form in the elements below the diagonal of the array A and in the elements of the array T; R is an upper-triangular N-by-N matrix, stored on exit in the elements on and above the diagonal of the array A. 0 is a (M-N)-by-N zero matrix, and is not stored. ParametersM M is INTEGER The number of rows of the matrix A. M >= 0. N N is INTEGER The number of columns of the matrix A. M >= N >= 0. MB MB is INTEGER The row block size to be used in the blocked QR. MB > 0. NB NB is INTEGER The column block size to be used in the blocked QR. N >= NB >= 1. A A is DOUBLE PRECISION array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the N-by-N upper triangular matrix R; the elements below the diagonal represent Q by the columns of blocked V (see Further Details). LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). T T is DOUBLE PRECISION array, dimension (LDT, N * Number_of_row_blocks) where Number_of_row_blocks = CEIL((M-N)/(MB-N)) The blocked upper triangular block reflectors stored in compact form as a sequence of upper triangular blocks. See Further Details below. LDT LDT is INTEGER The leading dimension of the array T. LDT >= NB. WORK (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the minimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. LWORK >= 1, if MIN(M,N) = 0, and LWORK >= NB*N, otherwise. If LWORK = -1, then a workspace query is assumed; the routine only calculates the minimal 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 INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value Author Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. FurtherDetails: Tall-Skinny QR (TSQR) performs QR by a sequence of orthogonal transformations, representing Q as a product of other orthogonal matrices Q = Q(1) * Q(2) * . . . * Q(k) where each Q(i) zeros out subdiagonal entries of a block of MB rows of A: Q(1) zeros out the subdiagonal entries of rows 1:MB of A Q(2) zeros out the bottom MB-N rows of rows [1:N,MB+1:2*MB-N] of A Q(3) zeros out the bottom MB-N rows of rows [1:N,2*MB-N+1:3*MB-2*N] of A . . . Q(1) is computed by GEQRT, which represents Q(1) by Householder vectors stored under the diagonal of rows 1:MB of A, and by upper triangular block reflectors, stored in array T(1:LDT,1:N). For more information see Further Details in GEQRT. Q(i) for i>1 is computed by TPQRT, which represents Q(i) by Householder vectors stored in rows [(i-1)*(MB-N)+N+1:i*(MB-N)+N] of A, and by upper triangular block reflectors, stored in array T(1:LDT,(i-1)*N+1:i*N). The last Q(k) may use fewer rows. For more information see Further Details in TPQRT. For more details of the overall algorithm, see the description of Sequential TSQR in Section 2.2 of [1]. [1] “Communication-Optimal Parallel and Sequential QR and LU Factorizations,” J. Demmel, L. Grigori, M. Hoemmen, J. Langou, SIAM J. Sci. Comput, vol. 34, no. 1, 2012 subroutineslatsqr(integerm,integern,integermb,integernb,real,dimension(lda,*)a,integerlda,real,dimension(ldt,*)t,integerldt,real,dimension(*)work,integerlwork,integerinfo)SLATSQRPurpose: SLATSQR computes a blocked Tall-Skinny QR factorization of a real M-by-N matrix A for M >= N: A = Q * ( R ), ( 0 ) where: Q is a M-by-M orthogonal matrix, stored on exit in an implicit form in the elements below the diagonal of the array A and in the elements of the array T; R is an upper-triangular N-by-N matrix, stored on exit in the elements on and above the diagonal of the array A. 0 is a (M-N)-by-N zero matrix, and is not stored. ParametersM M is INTEGER The number of rows of the matrix A. M >= 0. N N is INTEGER The number of columns of the matrix A. M >= N >= 0. MB MB is INTEGER The row block size to be used in the blocked QR. MB > N. NB NB is INTEGER The column block size to be used in the blocked QR. N >= NB >= 1. A A is REAL array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the N-by-N upper triangular matrix R; the elements below the diagonal represent Q by the columns of blocked V (see Further Details). LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). T T is REAL array, dimension (LDT, N * Number_of_row_blocks) where Number_of_row_blocks = CEIL((M-N)/(MB-N)) The blocked upper triangular block reflectors stored in compact form as a sequence of upper triangular blocks. See Further Details below. LDT LDT is INTEGER The leading dimension of the array T. LDT >= NB. WORK (workspace) REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the minimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. LWORK >= 1, if MIN(M,N) = 0, and LWORK >= NB*N, otherwise. If LWORK = -1, then a workspace query is assumed; the routine only calculates the minimal 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 INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value Author Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. FurtherDetails: Tall-Skinny QR (TSQR) performs QR by a sequence of orthogonal transformations, representing Q as a product of other orthogonal matrices Q = Q(1) * Q(2) * . . . * Q(k) where each Q(i) zeros out subdiagonal entries of a block of MB rows of A: Q(1) zeros out the subdiagonal entries of rows 1:MB of A Q(2) zeros out the bottom MB-N rows of rows [1:N,MB+1:2*MB-N] of A Q(3) zeros out the bottom MB-N rows of rows [1:N,2*MB-N+1:3*MB-2*N] of A . . . Q(1) is computed by GEQRT, which represents Q(1) by Householder vectors stored under the diagonal of rows 1:MB of A, and by upper triangular block reflectors, stored in array T(1:LDT,1:N). For more information see Further Details in GEQRT. Q(i) for i>1 is computed by TPQRT, which represents Q(i) by Householder vectors stored in rows [(i-1)*(MB-N)+N+1:i*(MB-N)+N] of A, and by upper triangular block reflectors, stored in array T(1:LDT,(i-1)*N+1:i*N). The last Q(k) may use fewer rows. For more information see Further Details in TPQRT. For more details of the overall algorithm, see the description of Sequential TSQR in Section 2.2 of [1]. [1] “Communication-Optimal Parallel and Sequential QR and LU Factorizations,” J. Demmel, L. Grigori, M. Hoemmen, J. Langou, SIAM J. Sci. Comput, vol. 34, no. 1, 2012 subroutinezlatsqr(integerm,integern,integermb,integernb,complex*16,dimension(lda,*)a,integerlda,complex*16,dimension(ldt,*)t,integerldt,complex*16,dimension(*)work,integerlwork,integerinfo)ZLATSQRPurpose: ZLATSQR computes a blocked Tall-Skinny QR factorization of a complex M-by-N matrix A for M >= N: A = Q * ( R ), ( 0 ) where: Q is a M-by-M orthogonal matrix, stored on exit in an implicit form in the elements below the diagonal of the array A and in the elements of the array T; R is an upper-triangular N-by-N matrix, stored on exit in the elements on and above the diagonal of the array A. 0 is a (M-N)-by-N zero matrix, and is not stored. ParametersM M is INTEGER The number of rows of the matrix A. M >= 0. N N is INTEGER The number of columns of the matrix A. M >= N >= 0. MB MB is INTEGER The row block size to be used in the blocked QR. MB > N. NB NB is INTEGER The column block size to be used in the blocked QR. N >= NB >= 1. A A is COMPLEX*16 array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the N-by-N upper triangular matrix R; the elements below the diagonal represent Q by the columns of blocked V (see Further Details). LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). T T is COMPLEX*16 array, dimension (LDT, N * Number_of_row_blocks) where Number_of_row_blocks = CEIL((M-N)/(MB-N)) The blocked upper triangular block reflectors stored in compact form as a sequence of upper triangular blocks. See Further Details below. LDT LDT is INTEGER The leading dimension of the array T. LDT >= NB. WORK (workspace) COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the minimal LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. LWORK >= 1, if MIN(M,N) = 0, and LWORK >= NB*N, otherwise. If LWORK = -1, then a workspace query is assumed; the routine only calculates the minimal 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 INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value Author Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. FurtherDetails: Tall-Skinny QR (TSQR) performs QR by a sequence of orthogonal transformations, representing Q as a product of other orthogonal matrices Q = Q(1) * Q(2) * . . . * Q(k) where each Q(i) zeros out subdiagonal entries of a block of MB rows of A: Q(1) zeros out the subdiagonal entries of rows 1:MB of A Q(2) zeros out the bottom MB-N rows of rows [1:N,MB+1:2*MB-N] of A Q(3) zeros out the bottom MB-N rows of rows [1:N,2*MB-N+1:3*MB-2*N] of A . . . Q(1) is computed by GEQRT, which represents Q(1) by Householder vectors stored under the diagonal of rows 1:MB of A, and by upper triangular block reflectors, stored in array T(1:LDT,1:N). For more information see Further Details in GEQRT. Q(i) for i>1 is computed by TPQRT, which represents Q(i) by Householder vectors stored in rows [(i-1)*(MB-N)+N+1:i*(MB-N)+N] of A, and by upper triangular block reflectors, stored in array T(1:LDT,(i-1)*N+1:i*N). The last Q(k) may use fewer rows. For more information see Further Details in TPQRT. For more details of the overall algorithm, see the description of Sequential TSQR in Section 2.2 of [1]. [1] “Communication-Optimal Parallel and Sequential QR and LU Factorizations,” J. Demmel, L. Grigori, M. Hoemmen, J. Langou, SIAM J. Sci. Comput, vol. 34, no. 1, 2012

latsqr - latsqr: tall-skinny QR factor

Functions subroutine clatsqr (m, n, mb, nb, a, lda, t, ldt, work, lwork, info) CLATSQR subroutine dlatsqr (m, n, mb, nb, a, lda, t, ldt, work, lwork, info) DLATSQR subroutine slatsqr (m, n, mb, nb, a, lda, t, ldt, work, lwork, info) SLATSQR subroutine zlatsqr (m, n, mb, nb, a, lda, t, ldt, work, lwork, info) ZLATSQR