Integrating Doxygen in comments

1 files changed, 302 insertions, 239 deletions

diff --git a/SRC/dtgsja.f b/SRC/dtgsja.f
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--- a/SRC/dtgsja.f
+++ b/SRC/dtgsja.f
@@ -1,11 +1,311 @@
+*> \brief \b DTGSJA
+*
+*  =========== DOCUMENTATION ===========
+*
+* Online html documentation available at 
+*            http://www.netlib.org/lapack/explore-html/ 
+*
+*  Definition
+*  ==========
+*
+*       SUBROUTINE DTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B,
+*                          LDB, TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV,
+*                          Q, LDQ, WORK, NCYCLE, INFO )
+* 
+*       .. Scalar Arguments ..
+*       CHARACTER          JOBQ, JOBU, JOBV
+*       INTEGER            INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N,
+*      $                   NCYCLE, P
+*       DOUBLE PRECISION   TOLA, TOLB
+*       ..
+*       .. Array Arguments ..
+*       DOUBLE PRECISION   A( LDA, * ), ALPHA( * ), B( LDB, * ),
+*      $                   BETA( * ), Q( LDQ, * ), U( LDU, * ),
+*      $                   V( LDV, * ), WORK( * )
+*       ..
+*  
+*  Purpose
+*  =======
+*
+*>\details \b Purpose:
+*>\verbatim
+*>
+*> DTGSJA computes the generalized singular value decomposition (GSVD)
+*> of two real upper triangular (or trapezoidal) matrices A and B.
+*>
+*> On entry, it is assumed that matrices A and B have the following
+*> forms, which may be obtained by the preprocessing subroutine DGGSVP
+*> from a general M-by-N matrix A and P-by-N matrix B:
+*>
+*>              N-K-L  K    L
+*>    A =    K ( 0    A12  A13 ) if M-K-L >= 0;
+*>           L ( 0     0   A23 )
+*>       M-K-L ( 0     0    0  )
+*>
+*>            N-K-L  K    L
+*>    A =  K ( 0    A12  A13 ) if M-K-L < 0;
+*>       M-K ( 0     0   A23 )
+*>
+*>            N-K-L  K    L
+*>    B =  L ( 0     0   B13 )
+*>       P-L ( 0     0    0  )
+*>
+*> where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
+*> upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
+*> otherwise A23 is (M-K)-by-L upper trapezoidal.
+*>
+*> On exit,
+*>
+*>        U**T *A*Q = D1*( 0 R ),    V**T *B*Q = D2*( 0 R ),
+*>
+*> where U, V and Q are orthogonal matrices.
+*> R is a nonsingular upper triangular matrix, and D1 and D2 are
+*> ``diagonal'' matrices, which are of the following structures:
+*>
+*> If M-K-L >= 0,
+*>
+*>                     K  L
+*>        D1 =     K ( I  0 )
+*>                 L ( 0  C )
+*>             M-K-L ( 0  0 )
+*>
+*>                   K  L
+*>        D2 = L   ( 0  S )
+*>             P-L ( 0  0 )
+*>
+*>                N-K-L  K    L
+*>   ( 0 R ) = K (  0   R11  R12 ) K
+*>             L (  0    0   R22 ) L
+*>
+*> where
+*>
+*>   C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
+*>   S = diag( BETA(K+1),  ... , BETA(K+L) ),
+*>   C**2 + S**2 = I.
+*>
+*>   R is stored in A(1:K+L,N-K-L+1:N) on exit.
+*>
+*> If M-K-L < 0,
+*>
+*>                K M-K K+L-M
+*>     D1 =   K ( I  0    0   )
+*>          M-K ( 0  C    0   )
+*>
+*>                  K M-K K+L-M
+*>     D2 =   M-K ( 0  S    0   )
+*>          K+L-M ( 0  0    I   )
+*>            P-L ( 0  0    0   )
+*>
+*>                N-K-L  K   M-K  K+L-M
+*> ( 0 R ) =    K ( 0    R11  R12  R13  )
+*>           M-K ( 0     0   R22  R23  )
+*>         K+L-M ( 0     0    0   R33  )
+*>
+*> where
+*> C = diag( ALPHA(K+1), ... , ALPHA(M) ),
+*> S = diag( BETA(K+1),  ... , BETA(M) ),
+*> C**2 + S**2 = I.
+*>
+*> R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored
+*>     (  0  R22 R23 )
+*> in B(M-K+1:L,N+M-K-L+1:N) on exit.
+*>
+*> The computation of the orthogonal transformation matrices U, V or Q
+*> is optional.  These matrices may either be formed explicitly, or they
+*> may be postmultiplied into input matrices U1, V1, or Q1.
+*>
+*>\endverbatim
+*
+*  Arguments
+*  =========
+*
+*> \param[in] JOBU
+*> \verbatim
+*>          JOBU is CHARACTER*1
+*>          = 'U':  U must contain an orthogonal matrix U1 on entry, and
+*>                  the product U1*U is returned;
+*>          = 'I':  U is initialized to the unit matrix, and the
+*>                  orthogonal matrix U is returned;
+*>          = 'N':  U is not computed.
+*> \endverbatim
+*>
+*> \param[in] JOBV
+*> \verbatim
+*>          JOBV is CHARACTER*1
+*>          = 'V':  V must contain an orthogonal matrix V1 on entry, and
+*>                  the product V1*V is returned;
+*>          = 'I':  V is initialized to the unit matrix, and the
+*>                  orthogonal matrix V is returned;
+*>          = 'N':  V is not computed.
+*> \endverbatim
+*>
+*> \param[in] JOBQ
+*> \verbatim
+*>          JOBQ is CHARACTER*1
+*>          = 'Q':  Q must contain an orthogonal matrix Q1 on entry, and
+*>                  the product Q1*Q is returned;
+*>          = 'I':  Q is initialized to the unit matrix, and the
+*>                  orthogonal matrix Q is returned;
+*>          = 'N':  Q is not computed.
+*> \endverbatim
+*>
+*> \param[in] M
+*> \verbatim
+*>          M is INTEGER
+*>          The number of rows of the matrix A.  M >= 0.
+*> \endverbatim
+*>
+*> \param[in] P
+*> \verbatim
+*>          P is INTEGER
+*>          The number of rows of the matrix B.  P >= 0.
+*> \endverbatim
+*>
+*> \param[in] N
+*> \verbatim
+*>          N is INTEGER
+*>          The number of columns of the matrices A and B.  N >= 0.
+*> \endverbatim
+*>
+*
+*  Authors
+*  =======
+*
+*> \author Univ. of Tennessee 
+*> \author Univ. of California Berkeley 
+*> \author Univ. of Colorado Denver 
+*> \author NAG Ltd. 
+*
+*> \date November 2011
+*
+*> \ingroup doubleOTHERcomputational
+*
+*
+*  Further Details
+*  ===============
+*>\details \b Further \b Details
+*> \verbatim
+*          See Further Details.
+*>
+*>  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
+*>          On entry, the M-by-N matrix A.
+*>          On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular
+*>          matrix R or part of R.  See Purpose for details.
+*>
+*>  LDA     (input) INTEGER
+*>          The leading dimension of the array A. LDA >= max(1,M).
+*>
+*>  B       (input/output) DOUBLE PRECISION array, dimension (LDB,N)
+*>          On entry, the P-by-N matrix B.
+*>          On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains
+*>          a part of R.  See Purpose for details.
+*>
+*>  LDB     (input) INTEGER
+*>          The leading dimension of the array B. LDB >= max(1,P).
+*>
+*>  TOLA    (input) DOUBLE PRECISION
+*>  TOLB    (input) DOUBLE PRECISION
+*>          TOLA and TOLB are the convergence criteria for the Jacobi-
+*>          Kogbetliantz iteration procedure. Generally, they are the
+*>          same as used in the preprocessing step, say
+*>              TOLA = max(M,N)*norm(A)*MAZHEPS,
+*>              TOLB = max(P,N)*norm(B)*MAZHEPS.
+*>
+*>  ALPHA   (output) DOUBLE PRECISION array, dimension (N)
+*>  BETA    (output) DOUBLE PRECISION array, dimension (N)
+*>          On exit, ALPHA and BETA contain the generalized singular
+*>          value pairs of A and B;
+*>            ALPHA(1:K) = 1,
+*>            BETA(1:K)  = 0,
+*>          and if M-K-L >= 0,
+*>            ALPHA(K+1:K+L) = diag(C),
+*>            BETA(K+1:K+L)  = diag(S),
+*>          or if M-K-L < 0,
+*>            ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0
+*>            BETA(K+1:M) = S, BETA(M+1:K+L) = 1.
+*>          Furthermore, if K+L < N,
+*>            ALPHA(K+L+1:N) = 0 and
+*>            BETA(K+L+1:N)  = 0.
+*>
+*>  U       (input/output) DOUBLE PRECISION array, dimension (LDU,M)
+*>          On entry, if JOBU = 'U', U must contain a matrix U1 (usually
+*>          the orthogonal matrix returned by DGGSVP).
+*>          On exit,
+*>          if JOBU = 'I', U contains the orthogonal matrix U;
+*>          if JOBU = 'U', U contains the product U1*U.
+*>          If JOBU = 'N', U is not referenced.
+*>
+*>  LDU     (input) INTEGER
+*>          The leading dimension of the array U. LDU >= max(1,M) if
+*>          JOBU = 'U'; LDU >= 1 otherwise.
+*>
+*>  V       (input/output) DOUBLE PRECISION array, dimension (LDV,P)
+*>          On entry, if JOBV = 'V', V must contain a matrix V1 (usually
+*>          the orthogonal matrix returned by DGGSVP).
+*>          On exit,
+*>          if JOBV = 'I', V contains the orthogonal matrix V;
+*>          if JOBV = 'V', V contains the product V1*V.
+*>          If JOBV = 'N', V is not referenced.
+*>
+*>  LDV     (input) INTEGER
+*>          The leading dimension of the array V. LDV >= max(1,P) if
+*>          JOBV = 'V'; LDV >= 1 otherwise.
+*>
+*>  Q       (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
+*>          On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
+*>          the orthogonal matrix returned by DGGSVP).
+*>          On exit,
+*>          if JOBQ = 'I', Q contains the orthogonal matrix Q;
+*>          if JOBQ = 'Q', Q contains the product Q1*Q.
+*>          If JOBQ = 'N', Q is not referenced.
+*>
+*>  LDQ     (input) INTEGER
+*>          The leading dimension of the array Q. LDQ >= max(1,N) if
+*>          JOBQ = 'Q'; LDQ >= 1 otherwise.
+*>
+*>  WORK    (workspace) DOUBLE PRECISION array, dimension (2*N)
+*>
+*>  NCYCLE  (output) INTEGER
+*>          The number of cycles required for convergence.
+*>
+*>  INFO    (output) INTEGER
+*>          = 0:  successful exit
+*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
+*>          = 1:  the procedure does not converge after MAXIT cycles.
+*>
+*>  Internal Parameters
+*>  ===================
+*>
+*>  MAXIT   INTEGER
+*>          MAXIT specifies the total loops that the iterative procedure
+*>          may take. If after MAXIT cycles, the routine fails to
+*>          converge, we return INFO = 1.
+*>
+*>
+*>  DTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce
+*>  min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L
+*>  matrix B13 to the form:
+*>
+*>           U1**T *A13*Q1 = C1*R1; V1**T *B13*Q1 = S1*R1,
+*>
+*>  where U1, V1 and Q1 are orthogonal matrix, and Z**T is the transpose
+*>  of Z.  C1 and S1 are diagonal matrices satisfying
+*>
+*>                C1**2 + S1**2 = I,
+*>
+*>  and R1 is an L-by-L nonsingular upper triangular matrix.
+*>
+*> \endverbatim
+*>
+*  =====================================================================
       SUBROUTINE DTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B,
      $                   LDB, TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV,
      $                   Q, LDQ, WORK, NCYCLE, INFO )
 *
-*  -- LAPACK routine (version 3.3.1)                                  --
+*  -- LAPACK computational routine (version 3.3.1) --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
-*  -- April 2009                                                      --
+*     November 2011
 *
 *     .. Scalar Arguments ..
       CHARACTER          JOBQ, JOBU, JOBV
@@ -19,243 +319,6 @@
      $                   V( LDV, * ), WORK( * )
 *     ..
 *
-*  Purpose
-*  =======
-*
-*  DTGSJA computes the generalized singular value decomposition (GSVD)
-*  of two real upper triangular (or trapezoidal) matrices A and B.
-*
-*  On entry, it is assumed that matrices A and B have the following
-*  forms, which may be obtained by the preprocessing subroutine DGGSVP
-*  from a general M-by-N matrix A and P-by-N matrix B:
-*
-*               N-K-L  K    L
-*     A =    K ( 0    A12  A13 ) if M-K-L >= 0;
-*            L ( 0     0   A23 )
-*        M-K-L ( 0     0    0  )
-*
-*             N-K-L  K    L
-*     A =  K ( 0    A12  A13 ) if M-K-L < 0;
-*        M-K ( 0     0   A23 )
-*
-*             N-K-L  K    L
-*     B =  L ( 0     0   B13 )
-*        P-L ( 0     0    0  )
-*
-*  where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
-*  upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
-*  otherwise A23 is (M-K)-by-L upper trapezoidal.
-*
-*  On exit,
-*
-*         U**T *A*Q = D1*( 0 R ),    V**T *B*Q = D2*( 0 R ),
-*
-*  where U, V and Q are orthogonal matrices.
-*  R is a nonsingular upper triangular matrix, and D1 and D2 are
-*  ``diagonal'' matrices, which are of the following structures:
-*
-*  If M-K-L >= 0,
-*
-*                      K  L
-*         D1 =     K ( I  0 )
-*                  L ( 0  C )
-*              M-K-L ( 0  0 )
-*
-*                    K  L
-*         D2 = L   ( 0  S )
-*              P-L ( 0  0 )
-*
-*                 N-K-L  K    L
-*    ( 0 R ) = K (  0   R11  R12 ) K
-*              L (  0    0   R22 ) L
-*
-*  where
-*
-*    C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
-*    S = diag( BETA(K+1),  ... , BETA(K+L) ),
-*    C**2 + S**2 = I.
-*
-*    R is stored in A(1:K+L,N-K-L+1:N) on exit.
-*
-*  If M-K-L < 0,
-*
-*                 K M-K K+L-M
-*      D1 =   K ( I  0    0   )
-*           M-K ( 0  C    0   )
-*
-*                   K M-K K+L-M
-*      D2 =   M-K ( 0  S    0   )
-*           K+L-M ( 0  0    I   )
-*             P-L ( 0  0    0   )
-*
-*                 N-K-L  K   M-K  K+L-M
-* ( 0 R ) =    K ( 0    R11  R12  R13  )
-*            M-K ( 0     0   R22  R23  )
-*          K+L-M ( 0     0    0   R33  )
-*
-*  where
-*  C = diag( ALPHA(K+1), ... , ALPHA(M) ),
-*  S = diag( BETA(K+1),  ... , BETA(M) ),
-*  C**2 + S**2 = I.
-*
-*  R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored
-*      (  0  R22 R23 )
-*  in B(M-K+1:L,N+M-K-L+1:N) on exit.
-*
-*  The computation of the orthogonal transformation matrices U, V or Q
-*  is optional.  These matrices may either be formed explicitly, or they
-*  may be postmultiplied into input matrices U1, V1, or Q1.
-*
-*  Arguments
-*  =========
-*
-*  JOBU    (input) CHARACTER*1
-*          = 'U':  U must contain an orthogonal matrix U1 on entry, and
-*                  the product U1*U is returned;
-*          = 'I':  U is initialized to the unit matrix, and the
-*                  orthogonal matrix U is returned;
-*          = 'N':  U is not computed.
-*
-*  JOBV    (input) CHARACTER*1
-*          = 'V':  V must contain an orthogonal matrix V1 on entry, and
-*                  the product V1*V is returned;
-*          = 'I':  V is initialized to the unit matrix, and the
-*                  orthogonal matrix V is returned;
-*          = 'N':  V is not computed.
-*
-*  JOBQ    (input) CHARACTER*1
-*          = 'Q':  Q must contain an orthogonal matrix Q1 on entry, and
-*                  the product Q1*Q is returned;
-*          = 'I':  Q is initialized to the unit matrix, and the
-*                  orthogonal matrix Q is returned;
-*          = 'N':  Q is not computed.
-*
-*  M       (input) INTEGER
-*          The number of rows of the matrix A.  M >= 0.
-*
-*  P       (input) INTEGER
-*          The number of rows of the matrix B.  P >= 0.
-*
-*  N       (input) INTEGER
-*          The number of columns of the matrices A and B.  N >= 0.
-*
-*  K       (input) INTEGER
-*  L       (input) INTEGER
-*          K and L specify the subblocks in the input matrices A and B:
-*          A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N)
-*          of A and B, whose GSVD is going to be computed by DTGSJA.
-*          See Further Details.
-*
-*  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
-*          On entry, the M-by-N matrix A.
-*          On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular
-*          matrix R or part of R.  See Purpose for details.
-*
-*  LDA     (input) INTEGER
-*          The leading dimension of the array A. LDA >= max(1,M).
-*
-*  B       (input/output) DOUBLE PRECISION array, dimension (LDB,N)
-*          On entry, the P-by-N matrix B.
-*          On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains
-*          a part of R.  See Purpose for details.
-*
-*  LDB     (input) INTEGER
-*          The leading dimension of the array B. LDB >= max(1,P).
-*
-*  TOLA    (input) DOUBLE PRECISION
-*  TOLB    (input) DOUBLE PRECISION
-*          TOLA and TOLB are the convergence criteria for the Jacobi-
-*          Kogbetliantz iteration procedure. Generally, they are the
-*          same as used in the preprocessing step, say
-*              TOLA = max(M,N)*norm(A)*MAZHEPS,
-*              TOLB = max(P,N)*norm(B)*MAZHEPS.
-*
-*  ALPHA   (output) DOUBLE PRECISION array, dimension (N)
-*  BETA    (output) DOUBLE PRECISION array, dimension (N)
-*          On exit, ALPHA and BETA contain the generalized singular
-*          value pairs of A and B;
-*            ALPHA(1:K) = 1,
-*            BETA(1:K)  = 0,
-*          and if M-K-L >= 0,
-*            ALPHA(K+1:K+L) = diag(C),
-*            BETA(K+1:K+L)  = diag(S),
-*          or if M-K-L < 0,
-*            ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0
-*            BETA(K+1:M) = S, BETA(M+1:K+L) = 1.
-*          Furthermore, if K+L < N,
-*            ALPHA(K+L+1:N) = 0 and
-*            BETA(K+L+1:N)  = 0.
-*
-*  U       (input/output) DOUBLE PRECISION array, dimension (LDU,M)
-*          On entry, if JOBU = 'U', U must contain a matrix U1 (usually
-*          the orthogonal matrix returned by DGGSVP).
-*          On exit,
-*          if JOBU = 'I', U contains the orthogonal matrix U;
-*          if JOBU = 'U', U contains the product U1*U.
-*          If JOBU = 'N', U is not referenced.
-*
-*  LDU     (input) INTEGER
-*          The leading dimension of the array U. LDU >= max(1,M) if
-*          JOBU = 'U'; LDU >= 1 otherwise.
-*
-*  V       (input/output) DOUBLE PRECISION array, dimension (LDV,P)
-*          On entry, if JOBV = 'V', V must contain a matrix V1 (usually
-*          the orthogonal matrix returned by DGGSVP).
-*          On exit,
-*          if JOBV = 'I', V contains the orthogonal matrix V;
-*          if JOBV = 'V', V contains the product V1*V.
-*          If JOBV = 'N', V is not referenced.
-*
-*  LDV     (input) INTEGER
-*          The leading dimension of the array V. LDV >= max(1,P) if
-*          JOBV = 'V'; LDV >= 1 otherwise.
-*
-*  Q       (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
-*          On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
-*          the orthogonal matrix returned by DGGSVP).
-*          On exit,
-*          if JOBQ = 'I', Q contains the orthogonal matrix Q;
-*          if JOBQ = 'Q', Q contains the product Q1*Q.
-*          If JOBQ = 'N', Q is not referenced.
-*
-*  LDQ     (input) INTEGER
-*          The leading dimension of the array Q. LDQ >= max(1,N) if
-*          JOBQ = 'Q'; LDQ >= 1 otherwise.
-*
-*  WORK    (workspace) DOUBLE PRECISION array, dimension (2*N)
-*
-*  NCYCLE  (output) INTEGER
-*          The number of cycles required for convergence.
-*
-*  INFO    (output) INTEGER
-*          = 0:  successful exit
-*          < 0:  if INFO = -i, the i-th argument had an illegal value.
-*          = 1:  the procedure does not converge after MAXIT cycles.
-*
-*  Internal Parameters
-*  ===================
-*
-*  MAXIT   INTEGER
-*          MAXIT specifies the total loops that the iterative procedure
-*          may take. If after MAXIT cycles, the routine fails to
-*          converge, we return INFO = 1.
-*
-*  Further Details
-*  ===============
-*
-*  DTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce
-*  min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L
-*  matrix B13 to the form:
-*
-*           U1**T *A13*Q1 = C1*R1; V1**T *B13*Q1 = S1*R1,
-*
-*  where U1, V1 and Q1 are orthogonal matrix, and Z**T is the transpose
-*  of Z.  C1 and S1 are diagonal matrices satisfying
-*
-*                C1**2 + S1**2 = I,
-*
-*  and R1 is an L-by-L nonsingular upper triangular matrix.
-*
 *  =====================================================================
 *
 *     .. Parameters ..