CTPQRT2(3)     LAPACK routine of NEC Numeric Library Collection     CTPQRT2(3)



NAME
       CTPQRT2

SYNOPSIS
       SUBROUTINE CTPQRT2 (M, N, L, A, LDA, B, LDB, T, LDT, INFO)



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.




ARGUMENTS
           M         (input)
                     M is INTEGER
                     The total number of rows of the matrix B.
                     M >= 0.

           N         (input)
                     N is INTEGER
                     The number of columns of the matrix B, and the order of
                     the triangular matrix A.
                     N >= 0.

           L         (input)
                     L is INTEGER
                     The number of rows of the upper trapezoidal part of B.
                     MIN(M,N) >= L >= 0.  See Further Details.

           A         (input/output)
                     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       (input)
                     LDA is INTEGER
                     The leading dimension of the array A.  LDA >= max(1,N).

           B         (input/output)
                     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       (input)
                     LDB is INTEGER
                     The leading dimension of the array B.  LDB >= max(1,M).

           T         (output)
                     T is COMPLEX array, dimension (LDT,N)
                     The N-by-N upper triangular factor T of the block reflector.
                     See Further Details.

           LDT       (input)
                     LDT is INTEGER
                     The leading dimension of the array T.  LDT >= max(1,N)

           INFO      (output)
                     INFO is INTEGER
                     = 0: successful exit
                     < 0: if INFO = -i, the i-th argument had an illegal value






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.



LAPACK routine                  31 October 2017                     CTPQRT2(3)