tpqrt2(3) Library Functions Manual tpqrt2(3) NAME tpqrt2 - tpqrt2: QR factor, level 2 SYNOPSIS Functions subroutine ctpqrt2 (m, n, l, a, lda, b, ldb, t, ldt, info) CTPQRT2 computes a QR factorization of a real or complex 'triangular-pentagonal' matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q. subroutine dtpqrt2 (m, n, l, a, lda, b, ldb, t, ldt, info) DTPQRT2 computes a QR factorization of a real or complex 'triangular-pentagonal' matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q. subroutine stpqrt2 (m, n, l, a, lda, b, ldb, t, ldt, info) STPQRT2 computes a QR factorization of a real or complex 'triangular-pentagonal' matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q. subroutine ztpqrt2 (m, n, l, a, lda, b, ldb, t, ldt, info) ZTPQRT2 computes a QR factorization of a real or complex 'triangular-pentagonal' matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q. Detailed Description Function Documentation subroutine ctpqrt2 (integer m, integer n, integer l, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldt, * ) t, integer ldt, integer info) CTPQRT2 computes a QR factorization of a real or complex 'triangular- pentagonal' matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q. Purpose: CTPQRT2 computes a QR factorization of a complex 'triangular-pentagonal' matrix C, which is composed of a triangular block A and pentagonal block B, using the compact WY representation for Q. Parameters M M is INTEGER The total number of rows of the matrix B. M >= 0. N N is INTEGER The number of columns of the matrix B, and the order of the triangular matrix A. N >= 0. L L is INTEGER The number of rows of the upper trapezoidal part of B. MIN(M,N) >= L >= 0. See Further Details. A A is COMPLEX array, dimension (LDA,N) On entry, the upper triangular N-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the upper triangular matrix R. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). B B is COMPLEX array, dimension (LDB,N) On entry, the pentagonal M-by-N matrix B. The first M-L rows are rectangular, and the last L rows are upper trapezoidal. On exit, B contains the pentagonal matrix V. See Further Details. LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,M). T T is COMPLEX array, dimension (LDT,N) The N-by-N upper triangular factor T of the block reflector. See Further Details. LDT LDT is INTEGER The leading dimension of the array T. LDT >= max(1,N) 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. Further Details: The input matrix C is a (N+M)-by-N matrix C = [ A ] [ B ] where A is an upper triangular N-by-N matrix, and B is M-by-N pentagonal matrix consisting of a (M-L)-by-N rectangular matrix B1 on top of a L-by-N upper trapezoidal matrix B2: B = [ B1 ] <- (M-L)-by-N rectangular [ B2 ] <- L-by-N upper trapezoidal. The upper trapezoidal matrix B2 consists of the first L rows of a N-by-N upper triangular matrix, where 0 <= L <= MIN(M,N). If L=0, B is rectangular M-by-N; if M=L=N, B is upper triangular. The matrix W stores the elementary reflectors H(i) in the i-th column below the diagonal (of A) in the (N+M)-by-N input matrix C C = [ A ] <- upper triangular N-by-N [ B ] <- M-by-N pentagonal so that W can be represented as W = [ I ] <- identity, N-by-N [ V ] <- M-by-N, same form as B. Thus, all of information needed for W is contained on exit in B, which we call V above. Note that V has the same form as B; that is, V = [ V1 ] <- (M-L)-by-N rectangular [ V2 ] <- L-by-N upper trapezoidal. The columns of V represent the vectors which define the H(i)'s. The (M+N)-by-(M+N) block reflector H is then given by H = I - W * T * W**H where W**H is the conjugate transpose of W and T is the upper triangular factor of the block reflector. Definition at line 172 of file ctpqrt2.f. subroutine dtpqrt2 (integer m, integer n, integer l, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldt, * ) t, integer ldt, integer info) DTPQRT2 computes a QR factorization of a real or complex 'triangular- pentagonal' matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q. Purpose: DTPQRT2 computes a QR factorization of a real 'triangular-pentagonal' matrix C, which is composed of a triangular block A and pentagonal block B, using the compact WY representation for Q. Parameters M M is INTEGER The total number of rows of the matrix B. M >= 0. N N is INTEGER The number of columns of the matrix B, and the order of the triangular matrix A. N >= 0. L L is INTEGER The number of rows of the upper trapezoidal part of B. MIN(M,N) >= L >= 0. See Further Details. A A is DOUBLE PRECISION array, dimension (LDA,N) On entry, the upper triangular N-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the upper triangular matrix R. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). B B is DOUBLE PRECISION array, dimension (LDB,N) On entry, the pentagonal M-by-N matrix B. The first M-L rows are rectangular, and the last L rows are upper trapezoidal. On exit, B contains the pentagonal matrix V. See Further Details. LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,M). T T is DOUBLE PRECISION array, dimension (LDT,N) The N-by-N upper triangular factor T of the block reflector. See Further Details. LDT LDT is INTEGER The leading dimension of the array T. LDT >= max(1,N) 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. Further Details: The input matrix C is a (N+M)-by-N matrix C = [ A ] [ B ] where A is an upper triangular N-by-N matrix, and B is M-by-N pentagonal matrix consisting of a (M-L)-by-N rectangular matrix B1 on top of a L-by-N upper trapezoidal matrix B2: B = [ B1 ] <- (M-L)-by-N rectangular [ B2 ] <- L-by-N upper trapezoidal. The upper trapezoidal matrix B2 consists of the first L rows of a N-by-N upper triangular matrix, where 0 <= L <= MIN(M,N). If L=0, B is rectangular M-by-N; if M=L=N, B is upper triangular. The matrix W stores the elementary reflectors H(i) in the i-th column below the diagonal (of A) in the (N+M)-by-N input matrix C C = [ A ] <- upper triangular N-by-N [ B ] <- M-by-N pentagonal so that W can be represented as W = [ I ] <- identity, N-by-N [ V ] <- M-by-N, same form as B. Thus, all of information needed for W is contained on exit in B, which we call V above. Note that V has the same form as B; that is, V = [ V1 ] <- (M-L)-by-N rectangular [ V2 ] <- L-by-N upper trapezoidal. The columns of V represent the vectors which define the H(i)'s. The (M+N)-by-(M+N) block reflector H is then given by H = I - W * T * W**T where W^H is the conjugate transpose of W and T is the upper triangular factor of the block reflector. Definition at line 172 of file dtpqrt2.f. subroutine stpqrt2 (integer m, integer n, integer l, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldt, * ) t, integer ldt, integer info) STPQRT2 computes a QR factorization of a real or complex 'triangular- pentagonal' matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q. Purpose: STPQRT2 computes a QR factorization of a real 'triangular-pentagonal' matrix C, which is composed of a triangular block A and pentagonal block B, using the compact WY representation for Q. Parameters M M is INTEGER The total number of rows of the matrix B. M >= 0. N N is INTEGER The number of columns of the matrix B, and the order of the triangular matrix A. N >= 0. L L is INTEGER The number of rows of the upper trapezoidal part of B. MIN(M,N) >= L >= 0. See Further Details. A A is REAL array, dimension (LDA,N) On entry, the upper triangular N-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the upper triangular matrix R. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). B B is REAL array, dimension (LDB,N) On entry, the pentagonal M-by-N matrix B. The first M-L rows are rectangular, and the last L rows are upper trapezoidal. On exit, B contains the pentagonal matrix V. See Further Details. LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,M). T T is REAL array, dimension (LDT,N) The N-by-N upper triangular factor T of the block reflector. See Further Details. LDT LDT is INTEGER The leading dimension of the array T. LDT >= max(1,N) 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. Further Details: The input matrix C is a (N+M)-by-N matrix C = [ A ] [ B ] where A is an upper triangular N-by-N matrix, and B is M-by-N pentagonal matrix consisting of a (M-L)-by-N rectangular matrix B1 on top of a L-by-N upper trapezoidal matrix B2: B = [ B1 ] <- (M-L)-by-N rectangular [ B2 ] <- L-by-N upper trapezoidal. The upper trapezoidal matrix B2 consists of the first L rows of a N-by-N upper triangular matrix, where 0 <= L <= MIN(M,N). If L=0, B is rectangular M-by-N; if M=L=N, B is upper triangular. The matrix W stores the elementary reflectors H(i) in the i-th column below the diagonal (of A) in the (N+M)-by-N input matrix C C = [ A ] <- upper triangular N-by-N [ B ] <- M-by-N pentagonal so that W can be represented as W = [ I ] <- identity, N-by-N [ V ] <- M-by-N, same form as B. Thus, all of information needed for W is contained on exit in B, which we call V above. Note that V has the same form as B; that is, V = [ V1 ] <- (M-L)-by-N rectangular [ V2 ] <- L-by-N upper trapezoidal. The columns of V represent the vectors which define the H(i)'s. The (M+N)-by-(M+N) block reflector H is then given by H = I - W * T * W^H where W^H is the conjugate transpose of W and T is the upper triangular factor of the block reflector. Definition at line 172 of file stpqrt2.f. subroutine ztpqrt2 (integer m, integer n, integer l, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldt, * ) t, integer ldt, integer info) ZTPQRT2 computes a QR factorization of a real or complex 'triangular- pentagonal' matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q. Purpose: ZTPQRT2 computes a QR factorization of a complex 'triangular-pentagonal' matrix C, which is composed of a triangular block A and pentagonal block B, using the compact WY representation for Q. Parameters M M is INTEGER The total number of rows of the matrix B. M >= 0. N N is INTEGER The number of columns of the matrix B, and the order of the triangular matrix A. N >= 0. L L is INTEGER The number of rows of the upper trapezoidal part of B. MIN(M,N) >= L >= 0. See Further Details. A A is COMPLEX*16 array, dimension (LDA,N) On entry, the upper triangular N-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the upper triangular matrix R. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). B B is COMPLEX*16 array, dimension (LDB,N) On entry, the pentagonal M-by-N matrix B. The first M-L rows are rectangular, and the last L rows are upper trapezoidal. On exit, B contains the pentagonal matrix V. See Further Details. LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,M). T T is COMPLEX*16 array, dimension (LDT,N) The N-by-N upper triangular factor T of the block reflector. See Further Details. LDT LDT is INTEGER The leading dimension of the array T. LDT >= max(1,N) 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. Further Details: The input matrix C is a (N+M)-by-N matrix C = [ A ] [ B ] where A is an upper triangular N-by-N matrix, and B is M-by-N pentagonal matrix consisting of a (M-L)-by-N rectangular matrix B1 on top of a L-by-N upper trapezoidal matrix B2: B = [ B1 ] <- (M-L)-by-N rectangular [ B2 ] <- L-by-N upper trapezoidal. The upper trapezoidal matrix B2 consists of the first L rows of a N-by-N upper triangular matrix, where 0 <= L <= MIN(M,N). If L=0, B is rectangular M-by-N; if M=L=N, B is upper triangular. The matrix W stores the elementary reflectors H(i) in the i-th column below the diagonal (of A) in the (N+M)-by-N input matrix C C = [ A ] <- upper triangular N-by-N [ B ] <- M-by-N pentagonal so that W can be represented as W = [ I ] <- identity, N-by-N [ V ] <- M-by-N, same form as B. Thus, all of information needed for W is contained on exit in B, which we call V above. Note that V has the same form as B; that is, V = [ V1 ] <- (M-L)-by-N rectangular [ V2 ] <- L-by-N upper trapezoidal. The columns of V represent the vectors which define the H(i)'s. The (M+N)-by-(M+N) block reflector H is then given by H = I - W * T * W**H where W**H is the conjugate transpose of W and T is the upper triangular factor of the block reflector. Definition at line 172 of file ztpqrt2.f. Author Generated automatically by Doxygen for LAPACK from the source code. LAPACK Version 3.12.0 tpqrt2(3)