dggqrf.f.html

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      SUBROUTINE <a name="DGGQRF.1"></a><a href="dggqrf.f.html#DGGQRF.1">DGGQRF</a>( N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK,
     $                   LWORK, INFO )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  -- LAPACK routine (version 3.1) --
</span><span class="comment">*</span><span class="comment">     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment">     November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Scalar Arguments ..
</span>      INTEGER            INFO, LDA, LDB, LWORK, M, N, P
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Array Arguments ..
</span>      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
     $                   WORK( * )
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Purpose
</span><span class="comment">*</span><span class="comment">  =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  <a name="DGGQRF.19"></a><a href="dggqrf.f.html#DGGQRF.1">DGGQRF</a> computes a generalized QR factorization of an N-by-M matrix A
</span><span class="comment">*</span><span class="comment">  and an N-by-P matrix B:
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">              A = Q*R,        B = Q*T*Z,
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal
</span><span class="comment">*</span><span class="comment">  matrix, and R and T assume one of the forms:
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  if N &gt;= M,  R = ( R11 ) M  ,   or if N &lt; M,  R = ( R11  R12 ) N,
</span><span class="comment">*</span><span class="comment">                  (  0  ) N-M                         N   M-N
</span><span class="comment">*</span><span class="comment">                     M
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  where R11 is upper triangular, and
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  if N &lt;= P,  T = ( 0  T12 ) N,   or if N &gt; P,  T = ( T11 ) N-P,
</span><span class="comment">*</span><span class="comment">                   P-N  N                           ( T21 ) P
</span><span class="comment">*</span><span class="comment">                                                       P
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  where T12 or T21 is upper triangular.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  In particular, if B is square and nonsingular, the GQR factorization
</span><span class="comment">*</span><span class="comment">  of A and B implicitly gives the QR factorization of inv(B)*A:
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">               inv(B)*A = Z'*(inv(T)*R)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  where inv(B) denotes the inverse of the matrix B, and Z' denotes the
</span><span class="comment">*</span><span class="comment">  transpose of the matrix Z.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Arguments
</span><span class="comment">*</span><span class="comment">  =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  N       (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The number of rows of the matrices A and B. N &gt;= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  M       (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The number of columns of the matrix A.  M &gt;= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  P       (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The number of columns of the matrix B.  P &gt;= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  A       (input/output) DOUBLE PRECISION array, dimension (LDA,M)
</span><span class="comment">*</span><span class="comment">          On entry, the N-by-M matrix A.
</span><span class="comment">*</span><span class="comment">          On exit, the elements on and above the diagonal of the array
</span><span class="comment">*</span><span class="comment">          contain the min(N,M)-by-M upper trapezoidal matrix R (R is
</span><span class="comment">*</span><span class="comment">          upper triangular if N &gt;= M); the elements below the diagonal,
</span><span class="comment">*</span><span class="comment">          with the array TAUA, represent the orthogonal matrix Q as a
</span><span class="comment">*</span><span class="comment">          product of min(N,M) elementary reflectors (see Further
</span><span class="comment">*</span><span class="comment">          Details).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDA     (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array A. LDA &gt;= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  TAUA    (output) DOUBLE PRECISION array, dimension (min(N,M))
</span><span class="comment">*</span><span class="comment">          The scalar factors of the elementary reflectors which
</span><span class="comment">*</span><span class="comment">          represent the orthogonal matrix Q (see Further Details).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  B       (input/output) DOUBLE PRECISION array, dimension (LDB,P)
</span><span class="comment">*</span><span class="comment">          On entry, the N-by-P matrix B.
</span><span class="comment">*</span><span class="comment">          On exit, if N &lt;= P, the upper triangle of the subarray
</span><span class="comment">*</span><span class="comment">          B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
</span><span class="comment">*</span><span class="comment">          if N &gt; P, the elements on and above the (N-P)-th subdiagonal
</span><span class="comment">*</span><span class="comment">          contain the N-by-P upper trapezoidal matrix T; the remaining
</span><span class="comment">*</span><span class="comment">          elements, with the array TAUB, represent the orthogonal
</span><span class="comment">*</span><span class="comment">          matrix Z as a product of elementary reflectors (see Further
</span><span class="comment">*</span><span class="comment">          Details).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDB     (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array B. LDB &gt;= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  TAUB    (output) DOUBLE PRECISION array, dimension (min(N,P))
</span><span class="comment">*</span><span class="comment">          The scalar factors of the elementary reflectors which
</span><span class="comment">*</span><span class="comment">          represent the orthogonal matrix Z (see Further Details).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
</span><span class="comment">*</span><span class="comment">          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LWORK   (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The dimension of the array WORK. LWORK &gt;= max(1,N,M,P).
</span><span class="comment">*</span><span class="comment">          For optimum performance LWORK &gt;= max(N,M,P)*max(NB1,NB2,NB3),
</span><span class="comment">*</span><span class="comment">          where NB1 is the optimal blocksize for the QR factorization
</span><span class="comment">*</span><span class="comment">          of an N-by-M matrix, NB2 is the optimal blocksize for the
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