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cggqrf (3)
  • >> cggqrf (3) ( Solaris man: Библиотечные вызовы )
  • 
    NAME
         cggqrf - compute a generalized QR factorization of an N-by-M
         matrix A and an N-by-P matrix B
    
    SYNOPSIS
         SUBROUTINE 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( * )
    
    
    
         #include <sunperf.h>
    
         void cggqrf(int n, int m, int p, complex *ca, int lda,  com-
                   plex  *taua,  complex *cb, int ldb, complex *taub,
                   int *info) ;
    
    PURPOSE
         CGGQRF 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  fac-
         torization  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.
    
    
    ARGUMENTS
         N         (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  ele-
                   ments  on and above the diagonal of the array con-
                   tain the min(N,M)-by-M upper trapezoidal matrix  R
                   (R  is  upper  triangular if N >= M); the elements
                   below the diagonal, with the array TAUA, represent
                   the unitary matrix Q as a product of min(N,M) ele-
                   mentary 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
                   subarray  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 trapezoidal matrix  T;  the  remaining  ele-
                   ments,  with the array TAUB, represent the unitary
                   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 factorization of an N-by-P matrix, and NB3
                   is the optimal blocksize for a call of CUNMQR.
    
         INFO      (output) INTEGER
                   = 0:  successful exit
                   < 0:  if INFO = -i, the i-th argument had an ille-
                   gal value.
    
    FURTHER DETAILS
         The 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.
    
    
    
    


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