.TH "tpqrt2" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME tpqrt2 \- tpqrt2: QR factor, level 2 .SH SYNOPSIS .br .PP .SS "Functions" .in +1c .ti -1c .RI "subroutine \fBctpqrt2\fP (m, n, l, a, lda, b, ldb, t, ldt, info)" .br .RI "\fBCTPQRT2\fP 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\&. " .ti -1c .RI "subroutine \fBdtpqrt2\fP (m, n, l, a, lda, b, ldb, t, ldt, info)" .br .RI "\fBDTPQRT2\fP 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\&. " .ti -1c .RI "subroutine \fBstpqrt2\fP (m, n, l, a, lda, b, ldb, t, ldt, info)" .br .RI "\fBSTPQRT2\fP 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\&. " .ti -1c .RI "subroutine \fBztpqrt2\fP (m, n, l, a, lda, b, ldb, t, ldt, info)" .br .RI "\fBZTPQRT2\fP 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\&. " .in -1c .SH "Detailed Description" .PP .SH "Function Documentation" .PP .SS "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)" .PP \fBCTPQRT2\fP 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\&. .PP \fBPurpose:\fP .RS 4 .PP .nf 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\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIM\fP .PP .nf M is INTEGER The total number of rows of the matrix B\&. M >= 0\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The number of columns of the matrix B, and the order of the triangular matrix A\&. N >= 0\&. .fi .PP .br \fIL\fP .PP .nf L is INTEGER The number of rows of the upper trapezoidal part of B\&. MIN(M,N) >= L >= 0\&. See Further Details\&. .fi .PP .br \fIA\fP .PP .nf 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\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= max(1,N)\&. .fi .PP .br \fIB\fP .PP .nf 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\&. .fi .PP .br \fILDB\fP .PP .nf LDB is INTEGER The leading dimension of the array B\&. LDB >= max(1,M)\&. .fi .PP .br \fIT\fP .PP .nf T is COMPLEX array, dimension (LDT,N) The N-by-N upper triangular factor T of the block reflector\&. See Further Details\&. .fi .PP .br \fILDT\fP .PP .nf LDT is INTEGER The leading dimension of the array T\&. LDT >= max(1,N) .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf 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\&. .fi .PP .RE .PP .SS "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)" .PP \fBDTPQRT2\fP 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\&. .PP \fBPurpose:\fP .RS 4 .PP .nf 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\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIM\fP .PP .nf M is INTEGER The total number of rows of the matrix B\&. M >= 0\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The number of columns of the matrix B, and the order of the triangular matrix A\&. N >= 0\&. .fi .PP .br \fIL\fP .PP .nf L is INTEGER The number of rows of the upper trapezoidal part of B\&. MIN(M,N) >= L >= 0\&. See Further Details\&. .fi .PP .br \fIA\fP .PP .nf 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\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= max(1,N)\&. .fi .PP .br \fIB\fP .PP .nf 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\&. .fi .PP .br \fILDB\fP .PP .nf LDB is INTEGER The leading dimension of the array B\&. LDB >= max(1,M)\&. .fi .PP .br \fIT\fP .PP .nf T is DOUBLE PRECISION array, dimension (LDT,N) The N-by-N upper triangular factor T of the block reflector\&. See Further Details\&. .fi .PP .br \fILDT\fP .PP .nf LDT is INTEGER The leading dimension of the array T\&. LDT >= max(1,N) .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf 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\&. .fi .PP .RE .PP .SS "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)" .PP \fBSTPQRT2\fP 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\&. .PP \fBPurpose:\fP .RS 4 .PP .nf 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\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIM\fP .PP .nf M is INTEGER The total number of rows of the matrix B\&. M >= 0\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The number of columns of the matrix B, and the order of the triangular matrix A\&. N >= 0\&. .fi .PP .br \fIL\fP .PP .nf L is INTEGER The number of rows of the upper trapezoidal part of B\&. MIN(M,N) >= L >= 0\&. See Further Details\&. .fi .PP .br \fIA\fP .PP .nf 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\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= max(1,N)\&. .fi .PP .br \fIB\fP .PP .nf 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\&. .fi .PP .br \fILDB\fP .PP .nf LDB is INTEGER The leading dimension of the array B\&. LDB >= max(1,M)\&. .fi .PP .br \fIT\fP .PP .nf T is REAL array, dimension (LDT,N) The N-by-N upper triangular factor T of the block reflector\&. See Further Details\&. .fi .PP .br \fILDT\fP .PP .nf LDT is INTEGER The leading dimension of the array T\&. LDT >= max(1,N) .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf 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\&. .fi .PP .RE .PP .SS "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)" .PP \fBZTPQRT2\fP 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\&. .PP \fBPurpose:\fP .RS 4 .PP .nf 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\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIM\fP .PP .nf M is INTEGER The total number of rows of the matrix B\&. M >= 0\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The number of columns of the matrix B, and the order of the triangular matrix A\&. N >= 0\&. .fi .PP .br \fIL\fP .PP .nf L is INTEGER The number of rows of the upper trapezoidal part of B\&. MIN(M,N) >= L >= 0\&. See Further Details\&. .fi .PP .br \fIA\fP .PP .nf 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\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= max(1,N)\&. .fi .PP .br \fIB\fP .PP .nf 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\&. .fi .PP .br \fILDB\fP .PP .nf LDB is INTEGER The leading dimension of the array B\&. LDB >= max(1,M)\&. .fi .PP .br \fIT\fP .PP .nf T is COMPLEX*16 array, dimension (LDT,N) The N-by-N upper triangular factor T of the block reflector\&. See Further Details\&. .fi .PP .br \fILDT\fP .PP .nf LDT is INTEGER The leading dimension of the array T\&. LDT >= max(1,N) .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf 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\&. .fi .PP .RE .PP .SH "Author" .PP Generated automatically by Doxygen for LAPACK from the source code\&.