SUBROUTINE MB04TB( TRANA, TRANB, N, ILO, A, LDA, B, LDB, G, LDG,
$ Q, LDQ, CSL, CSR, TAUL, TAUR, DWORK, LDWORK,
$ INFO )
C
C PURPOSE
C
C To compute a symplectic URV (SURV) decomposition of a real
C 2N-by-2N matrix H,
C
C [ op(A) G ] [ op(R11) R12 ]
C H = [ ] = U R V' = U * [ ] * V' ,
C [ Q op(B) ] [ 0 op(R22) ]
C
C where A, B, G, Q, R12 are real N-by-N matrices, op(R11) is a real
C N-by-N upper triangular matrix, op(R22) is a real N-by-N lower
C Hessenberg matrix and U, V are 2N-by-2N orthogonal symplectic
C matrices. Blocked version.
C
C ARGUMENTS
C
C Mode Parameters
C
C TRANA CHARACTER*1
C Specifies the form of op( A ) as follows:
C = 'N': op( A ) = A;
C = 'T': op( A ) = A';
C = 'C': op( A ) = A'.
C
C TRANB CHARACTER*1
C Specifies the form of op( B ) as follows:
C = 'N': op( B ) = B;
C = 'T': op( B ) = B';
C = 'C': op( B ) = B'.
C
C Input/Output Parameters
C
C N (input) INTEGER
C The order of the matrix A. N >= 0.
C
C ILO (input) INTEGER
C It is assumed that op(A) is already upper triangular,
C op(B) is lower triangular and Q is zero in rows and
C columns 1:ILO-1. ILO is normally set by a previous call
C to MB04DD; otherwise it should be set to 1.
C 1 <= ILO <= N+1, if N > 0; ILO = 1, if N = 0.
C
C A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
C On entry, the leading N-by-N part of this array must
C contain the matrix A.
C On exit, the leading N-by-N part of this array contains
C the triangular matrix R11, and in the zero part
C information about the elementary reflectors used to
C compute the SURV decomposition.
C
C LDA INTEGER
C The leading dimension of the array A. LDA >= MAX(1,N).
C
C B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
C On entry, the leading N-by-N part of this array must
C contain the matrix B.
C On exit, the leading N-by-N part of this array contains
C the Hessenberg matrix R22, and in the zero part
C information about the elementary reflectors used to
C compute the SURV decomposition.
C
C LDB INTEGER
C The leading dimension of the array B. LDB >= MAX(1,N).
C
C G (input/output) DOUBLE PRECISION array, dimension (LDG,N)
C On entry, the leading N-by-N part of this array must
C contain the matrix G.
C On exit, the leading N-by-N part of this array contains
C the matrix R12.
C
C LDG INTEGER
C The leading dimension of the array G. LDG >= MAX(1,N).
C
C Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
C On entry, the leading N-by-N part of this array must
C contain the matrix Q.
C On exit, the leading N-by-N part of this array contains
C information about the elementary reflectors used to
C compute the SURV decomposition.
C
C LDQ INTEGER
C The leading dimension of the array Q. LDQ >= MAX(1,N).
C
C CSL (output) DOUBLE PRECISION array, dimension (2N)
C On exit, the first 2N elements of this array contain the
C cosines and sines of the symplectic Givens rotations
C applied from the left-hand side used to compute the SURV
C decomposition.
C
C CSR (output) DOUBLE PRECISION array, dimension (2N-2)
C On exit, the first 2N-2 elements of this array contain the
C cosines and sines of the symplectic Givens rotations
C applied from the right-hand side used to compute the SURV
C decomposition.
C
C TAUL (output) DOUBLE PRECISION array, dimension (N)
C On exit, the first N elements of this array contain the
C scalar factors of some of the elementary reflectors
C applied form the left-hand side.
C
C TAUR (output) DOUBLE PRECISION array, dimension (N-1)
C On exit, the first N-1 elements of this array contain the
C scalar factors of some of the elementary reflectors
C applied form the right-hand side.
C
C Workspace
C
C DWORK DOUBLE PRECISION array, dimension (LDWORK)
C On exit, if INFO = 0, DWORK(1) returns the optimal
C value of LDWORK, (16*N + 5)*NB, where NB is the optimal
C block size determined by the function UE01MD.
C On exit, if INFO = -16, DWORK(1) returns the minimum
C value of LDWORK.
C
C LDWORK INTEGER
C The length of the array DWORK. LDWORK >= MAX(1,N).
C
C If LDWORK = -1, then a workspace query is assumed;
C the routine only calculates the optimal size of the
C DWORK array, returns this value as the first entry of
C the DWORK array, and no error message related to LDWORK
C is issued by XERBLA.
C
C Error Indicator
C
C INFO INTEGER
C = 0: successful exit;
C < 0: if INFO = -i, the i-th argument had an illegal
C value.
C
C METHOD
C
C The matrices U and V are represented as products of symplectic
C reflectors and Givens rotations
C
C U = diag( HU(1),HU(1) ) GU(1) diag( FU(1),FU(1) )
C diag( HU(2),HU(2) ) GU(2) diag( FU(2),FU(2) )
C ....
C diag( HU(n),HU(n) ) GU(n) diag( FU(n),FU(n) ),
C
C V = diag( HV(1),HV(1) ) GV(1) diag( FV(1),FV(1) )
C diag( HV(2),HV(2) ) GV(2) diag( FV(2),FV(2) )
C ....
C diag( HV(n-1),HV(n-1) ) GV(n-1) diag( FV(n-1),FV(n-1) ).
C
C Each HU(i) has the form
C
C HU(i) = I - tau * v * v'
C
C where tau is a real scalar, and v is a real vector with
C v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in
C Q(i+1:n,i), and tau in Q(i,i).
C
C Each FU(i) has the form
C
C FU(i) = I - nu * w * w'
C
C where nu is a real scalar, and w is a real vector with
C w(1:i-1) = 0 and w(i) = 1; w(i+1:n) is stored on exit in
C A(i+1:n,i), if op(A) = 'N', and in A(i,i+1:n), otherwise. The
C scalar nu is stored in TAUL(i).
C
C Each GU(i) is a Givens rotation acting on rows i and n+i,
C where the cosine is stored in CSL(2*i-1) and the sine in
C CSL(2*i).
C
C Each HV(i) has the form
C
C HV(i) = I - tau * v * v'
C
C where tau is a real scalar, and v is a real vector with
C v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in
C Q(i,i+2:n), and tau in Q(i,i+1).
C
C Each FV(i) has the form
C
C FV(i) = I - nu * w * w'
C
C where nu is a real scalar, and w is a real vector with
C w(1:i) = 0 and w(i+1) = 1; w(i+2:n) is stored on exit in
C B(i,i+2:n), if op(B) = 'N', and in B(i+2:n,i), otherwise.
C The scalar nu is stored in TAUR(i).
C
C Each GV(i) is a Givens rotation acting on columns i+1 and n+i+1,
C where the cosine is stored in CSR(2*i-1) and the sine in
C CSR(2*i).
C
C NUMERICAL ASPECTS
C
C The algorithm requires 80/3*N**3 + ( 64*NB + 77 )*N**2 +
C ( -16*NB + 48 )*NB*N + O(N) floating point operations, where
C NB is the used block size, and is numerically backward stable.
C
C REFERENCES
C
C [1] Benner, P., Mehrmann, V., and Xu, H.
C A numerically stable, structure preserving method for
C computing the eigenvalues of real Hamiltonian or symplectic
C pencils. Numer. Math., Vol 78 (3), pp. 329-358, 1998.
C
C [2] Kressner, D.
C Block algorithms for orthogonal symplectic factorizations.
C BIT, 43 (4), pp. 775-790, 2003.
C
C CONTRIBUTORS
C
C D. Kressner, Technical Univ. Berlin, Germany, and
C P. Benner, Technical Univ. Chemnitz, Germany, December 2003.
C
C REVISIONS
C
C V. Sima, June 2008 (SLICOT version of the HAPACK routine DGESUB).
C V. Sima, Aug. 2011, Oct. 2011.
C
C KEYWORDS
C
C Elementary matrix operations, Matrix decompositions, Hamiltonian
C matrix
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
C .. Scalar Arguments ..
CHARACTER TRANA, TRANB
INTEGER ILO, INFO, LDA, LDB, LDG, LDQ, LDWORK, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), CSL(*), CSR(*), DWORK(*),
$ G(LDG,*), Q(LDQ,*), TAUL(*), TAUR(*)
C .. Local Scalars ..
LOGICAL LQUERY, LTRA, LTRB
INTEGER I, IB, IERR, MINWRK, NB, NBMIN, NH, NIB, NNB,
$ NX, PDW, PXA, PXB, PXG, PXQ, PYA, PYB, PYG, PYQ,
$ WRKOPT
C .. External Functions ..
LOGICAL LSAME
INTEGER UE01MD
EXTERNAL LSAME, UE01MD
C .. External Subroutines ..
EXTERNAL DGEQRF, DGEMM, MB03XU, MB04TS, XERBLA
C .. Intrinsic Functions ..
INTRINSIC DBLE, INT, MAX, MIN
C
C .. Executable Statements ..
C
C Check the scalar input parameters.
C
INFO = 0
LTRA = LSAME( TRANA, 'T' ) .OR. LSAME( TRANA, 'C' )
LTRB = LSAME( TRANB, 'T' ) .OR. LSAME( TRANB, 'C' )
C
MINWRK = MAX( 1, N )
C
IF ( .NOT.LTRA .AND. .NOT.LSAME( TRANA, 'N' ) ) THEN
INFO = -1
ELSE IF ( .NOT.LTRB .AND. .NOT.LSAME( TRANB, 'N' ) ) THEN
INFO = -2
ELSE IF ( N.LT.0 ) THEN
INFO = -3
ELSE IF ( ILO.LT.1 .OR. ILO.GT.N+1 ) THEN
INFO = -4
ELSE IF ( LDA.LT.MINWRK ) THEN
INFO = -6
ELSE IF ( LDB.LT.MINWRK ) THEN
INFO = -8
ELSE IF ( LDG.LT.MINWRK ) THEN
INFO = -10
ELSE IF ( LDQ.LT.MINWRK ) THEN
INFO = -12
ELSE
LQUERY = LDWORK.EQ.-1
IF ( LDWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
DWORK(1) = DBLE( MINWRK )
INFO = -18
ELSE
IF ( N.EQ.0 ) THEN
WRKOPT = ONE
ELSE
CALL DGEQRF( N, ILO, DWORK, MINWRK, DWORK, DWORK, -1,
$ INFO )
WRKOPT = MAX( MINWRK, INT( DWORK(1) ) )
NB = INT( WRKOPT/ILO )
WRKOPT = MAX( WRKOPT, 16*N*NB + 5*NB )
END IF
IF ( LQUERY ) THEN
DWORK(1) = DBLE( WRKOPT )
RETURN
END IF
END IF
END IF
C
C Return if there were illegal values.
C
IF ( INFO.NE.0 ) THEN
CALL XERBLA( 'MB04TB', -INFO )
RETURN
END IF
C
C Set elements 1:ILO-1 of CSL, CSR, TAUL and TAUR to their default
C values.
C
DO 10 I = 1, ILO - 1
CSL(2*I-1) = ONE
CSL(2*I) = ZERO
CSR(2*I-1) = ONE
CSR(2*I) = ZERO
TAUL(I) = ZERO
TAUR(I) = ZERO
10 CONTINUE
C
C Quick return if possible.
C
NH = N - ILO + 1
IF ( NH.EQ.0 ) THEN
DWORK(1) = ONE
RETURN
END IF
C
C Determine the block size.
C
NBMIN = 2
IF ( NB.GT.1 .AND. NB.LT.NH ) THEN
C
C Determine when to cross over from blocked to unblocked code.
C
NX = MAX( NB, UE01MD( 3, 'MB04TB', TRANA // TRANB, N, ILO, -1 )
$ )
IF ( NX.LT.NH ) THEN
C
C Check whether workspace is large enough for blocked code.
C
IF ( LDWORK.LT.WRKOPT ) THEN
C
C Not enough workspace available. Determine minimum value
C of NB, and reduce NB.
C
NBMIN = MAX( 2, UE01MD( 2, 'MB04TB', TRANA // TRANB, N,
$ ILO, -1 ) )
NB = LDWORK / ( 16*N + 5 )
END IF
END IF
END IF
C
NNB = N*NB
PYB = 1
PYQ = PYB + 2*NNB
PYA = PYQ + 2*NNB
PYG = PYA + 2*NNB
PXQ = PYG + 2*NNB
PXA = PXQ + 2*NNB
PXG = PXA + 2*NNB
PXB = PXG + 2*NNB
PDW = PXB + 2*NNB
C
IF ( NB.LT.NBMIN .OR. NB.GE.NH ) THEN
C
C Use unblocked code.
C
I = ILO
C
ELSE IF ( LTRA .AND. LTRB ) THEN
DO 20 I = ILO, N-NX-1, NB
IB = MIN( NB, N-I )
NIB = N*IB
C
C Reduce rows and columns i:i+nb-1 to symplectic URV form and
C return the matrices XA, XB, XG, XQ, YA, YB, YG and YQ which
C are needed to update the unreduced parts of the matrices.
C
CALL MB03XU( LTRA, LTRB, N-I+1, I-1, IB, A(I,1), LDA,
$ B(1,I), LDB, G, LDG, Q(I,I), LDQ, DWORK(PXA),
$ N, DWORK(PXB), N, DWORK(PXG), N, DWORK(PXQ), N,
$ DWORK(PYA), N, DWORK(PYB), N, DWORK(PYG), N,
$ DWORK(PYQ), N, CSL(2*I-1), CSR(2*I-1), TAUL(I),
$ TAUR(I), DWORK(PDW) )
C
C Update the submatrix A(i+1+ib:n,1:n).
C
CALL DGEMM( 'No transpose', 'Transpose', N-I-IB, N-I-IB+1,
$ IB, ONE, DWORK(PXA+NB+1), N, Q(I+IB,I), LDQ,
$ ONE, A(I+IB+1,I+IB), LDA )
CALL DGEMM( 'No transpose', 'No transpose', N-I-IB,
$ N-I-IB+1, IB, ONE, DWORK(PXA+NIB+NB+1), N,
$ A(I,I+IB), LDA, ONE, A(I+IB+1,I+IB), LDA )
CALL DGEMM( 'Transpose', 'Transpose', N-I-IB, N, IB,
$ ONE, Q(I,I+IB+1), LDQ, DWORK(PYA), N, ONE,
$ A(I+IB+1,1), LDA )
CALL DGEMM( 'No transpose', 'Transpose', N-I-IB, N, IB,
$ ONE, B(I+IB+1,I), LDB, DWORK(PYA+NIB), N, ONE,
$ A(I+IB+1,1), LDA )
C
C Update the submatrix Q(i+ib:n,i+1+ib:n).
C
CALL DGEMM( 'No transpose', 'Transpose', N-I-IB+1, N-I-IB,
$ IB, ONE, Q(I+IB,I), LDQ, DWORK(PXQ+NB+1), N,
$ ONE, Q(I+IB,I+IB+1), LDQ )
CALL DGEMM( 'Transpose', 'Transpose', N-I-IB+1, N-I-IB,
$ IB, ONE, A(I,I+IB), LDA, DWORK(PXQ+NIB+NB+1), N,
$ ONE, Q(I+IB,I+IB+1), LDQ )
CALL DGEMM( 'No transpose', 'No transpose', N-I-IB+1,
$ N-I-IB, IB, ONE, DWORK(PYQ+NB), N,
$ Q(I,I+IB+1), LDQ, ONE, Q(I+IB,I+IB+1), LDQ )
CALL DGEMM( 'No transpose', 'Transpose', N-I-IB+1,
$ N-I-IB, IB, ONE, DWORK(PYQ+NIB+NB), N,
$ B(I+IB+1,I), LDB, ONE, Q(I+IB,I+IB+1), LDQ )
C
C Update the matrix G.
C
CALL DGEMM( 'No transpose', 'Transpose', N-I-IB+1, N, IB,
$ ONE, Q(I+IB,I), LDQ, DWORK(PXG), N, ONE,
$ G(I+IB,1), LDG )
CALL DGEMM( 'Transpose', 'Transpose', N-I-IB+1, N, IB,
$ ONE, A(I,I+IB), LDA, DWORK(PXG+NIB), N, ONE,
$ G(I+IB,1), LDG )
CALL DGEMM( 'No transpose', 'No transpose', N, N-I-IB, IB,
$ ONE, DWORK(PYG), N, Q(I,I+IB+1), LDQ, ONE,
$ G(1,I+IB+1), LDG )
CALL DGEMM( 'No transpose', 'Transpose', N, N-I-IB, IB,
$ ONE, DWORK(PYG+NIB), N, B(I+IB+1,I), LDB, ONE,
$ G(1,I+IB+1), LDG )
C
C Update the submatrix B(1:n,i+ib:n).
C
CALL DGEMM( 'No transpose', 'Transpose', N, N-I-IB+1,
$ IB, ONE, DWORK(PXB), N, Q(I+IB,I), LDQ,
$ ONE, B(1,I+IB), LDB )
CALL DGEMM( 'No transpose', 'No transpose', N, N-I-IB+1, IB,
$ ONE, DWORK(PXB+NIB), N, A(I,I+IB), LDA, ONE,
$ B(1,I+IB), LDB )
CALL DGEMM( 'Transpose', 'Transpose', N-I-IB, N-I-IB+1,
$ IB, ONE, Q(I,I+IB+1), LDQ, DWORK(PYB+NB), N,
$ ONE, B(I+IB+1,I+IB), LDB )
CALL DGEMM( 'No transpose', 'Transpose', N-I-IB, N-I-IB+1,
$ IB, ONE, B(I+IB+1,I), LDB, DWORK(PYB+NIB+NB), N,
$ ONE, B(I+IB+1,I+IB), LDB )
20 CONTINUE
C
ELSE IF ( LTRA ) THEN
DO 30 I = ILO, N-NX-1, NB
IB = MIN( NB, N-I )
NIB = N*IB
C
C Reduce rows and columns i:i+nb-1 to symplectic URV form and
C return the matrices XA, XB, XG, XQ, YA, YB, YG and YQ which
C are needed to update the unreduced parts of the matrices.
C
CALL MB03XU( LTRA, LTRB, N-I+1, I-1, IB, A(I,1), LDA,
$ B(I,1), LDB, G, LDG, Q(I,I), LDQ, DWORK(PXA),
$ N, DWORK(PXB), N, DWORK(PXG), N, DWORK(PXQ), N,
$ DWORK(PYA), N, DWORK(PYB), N, DWORK(PYG), N,
$ DWORK(PYQ), N, CSL(2*I-1), CSR(2*I-1), TAUL(I),
$ TAUR(I), DWORK(PDW) )
C
C Update the submatrix A(i+1+ib:n,1:n).
C
CALL DGEMM( 'No transpose', 'Transpose', N-I-IB, N-I-IB+1,
$ IB, ONE, DWORK(PXA+NB+1), N, Q(I+IB,I), LDQ,
$ ONE, A(I+IB+1,I+IB), LDA )
CALL DGEMM( 'No transpose', 'No transpose', N-I-IB,
$ N-I-IB+1, IB, ONE, DWORK(PXA+NIB+NB+1), N,
$ A(I,I+IB), LDA, ONE, A(I+IB+1,I+IB), LDA )
CALL DGEMM( 'Transpose', 'Transpose', N-I-IB, N, IB,
$ ONE, Q(I,I+IB+1), LDQ, DWORK(PYA), N, ONE,
$ A(I+IB+1,1), LDA )
CALL DGEMM( 'Transpose', 'Transpose', N-I-IB, N, IB,
$ ONE, B(I,I+IB+1), LDB, DWORK(PYA+NIB), N, ONE,
$ A(I+IB+1,1), LDA )
C
C Update the submatrix Q(i+ib:n,i+1+ib:n).
C
CALL DGEMM( 'No transpose', 'Transpose', N-I-IB+1, N-I-IB,
$ IB, ONE, Q(I+IB,I), LDQ, DWORK(PXQ+NB+1), N,
$ ONE, Q(I+IB,I+IB+1), LDQ )
CALL DGEMM( 'Transpose', 'Transpose', N-I-IB+1, N-I-IB,
$ IB, ONE, A(I,I+IB), LDA, DWORK(PXQ+NIB+NB+1), N,
$ ONE, Q(I+IB,I+IB+1), LDQ )
CALL DGEMM( 'No transpose', 'No transpose', N-I-IB+1,
$ N-I-IB, IB, ONE, DWORK(PYQ+NB), N,
$ Q(I,I+IB+1), LDQ, ONE, Q(I+IB,I+IB+1), LDQ )
CALL DGEMM( 'No transpose', 'No transpose', N-I-IB+1,
$ N-I-IB, IB, ONE, DWORK(PYQ+NIB+NB), N,
$ B(I,I+IB+1), LDB, ONE, Q(I+IB,I+IB+1), LDQ )
C
C Update the matrix G.
C
CALL DGEMM( 'No transpose', 'Transpose', N-I-IB+1, N, IB,
$ ONE, Q(I+IB,I), LDQ, DWORK(PXG), N, ONE,
$ G(I+IB,1), LDG )
CALL DGEMM( 'Transpose', 'Transpose', N-I-IB+1, N, IB,
$ ONE, A(I,I+IB), LDA, DWORK(PXG+NIB), N, ONE,
$ G(I+IB,1), LDG )
CALL DGEMM( 'No transpose', 'No transpose', N, N-I-IB, IB,
$ ONE, DWORK(PYG), N, Q(I,I+IB+1), LDQ, ONE,
$ G(1,I+IB+1), LDG )
CALL DGEMM( 'No transpose', 'No transpose', N, N-I-IB, IB,
$ ONE, DWORK(PYG+NIB), N, B(I,I+IB+1), LDB, ONE,
$ G(1,I+IB+1), LDG )
C
C Update the submatrix B(i+ib:n,1:n).
C
CALL DGEMM( 'No transpose', 'Transpose', N-I-IB+1, N,
$ IB, ONE, Q(I+IB,I), LDQ, DWORK(PXB), N,
$ ONE, B(I+IB,1), LDB )
CALL DGEMM( 'Transpose', 'Transpose', N-I-IB+1, N, IB,
$ ONE, A(I,I+IB), LDA, DWORK(PXB+NIB), N, ONE,
$ B(I+IB,1), LDB )
CALL DGEMM( 'No transpose', 'No transpose', N-I-IB+1,
$ N-I-IB, IB, ONE, DWORK(PYB+NB), N, Q(I,I+IB+1),
$ LDQ, ONE, B(I+IB,I+IB+1), LDB )
CALL DGEMM( 'No transpose', 'No transpose', N-I-IB+1,
$ N-I-IB, IB, ONE, DWORK(PYB+NIB+NB), N,
$ B(I,I+IB+1), LDB, ONE, B(I+IB,I+IB+1), LDB )
30 CONTINUE
C
ELSE IF ( LTRB ) THEN
DO 40 I = ILO, N-NX-1, NB
IB = MIN( NB, N-I )
NIB = N*IB
C
C Reduce rows and columns i:i+nb-1 to symplectic URV form and
C return the matrices XA, XB, XG, XQ, YA, YB, YG and YQ which
C are needed to update the unreduced parts of the matrices.
C
CALL MB03XU( LTRA, LTRB, N-I+1, I-1, IB, A(1,I), LDA,
$ B(1,I), LDB, G, LDG, Q(I,I), LDQ, DWORK(PXA),
$ N, DWORK(PXB), N, DWORK(PXG), N, DWORK(PXQ), N,
$ DWORK(PYA), N, DWORK(PYB), N, DWORK(PYG), N,
$ DWORK(PYQ), N, CSL(2*I-1), CSR(2*I-1), TAUL(I),
$ TAUR(I), DWORK(PDW) )
C
C Update the submatrix A(1:n,i+1+ib:n).
C
CALL DGEMM( 'No transpose', 'Transpose', N-I-IB+1, N-I-IB,
$ IB, ONE, Q(I+IB,I), LDQ, DWORK(PXA+NB+1), N,
$ ONE, A(I+IB,I+IB+1), LDA )
CALL DGEMM( 'No transpose', 'Transpose', N-I-IB+1, N-I-IB,
$ IB, ONE, A(I+IB,I), LDA, DWORK(PXA+NIB+NB+1), N,
$ ONE, A(I+IB,I+IB+1), LDA )
CALL DGEMM( 'No transpose', 'No transpose', N, N-I-IB, IB,
$ ONE, DWORK(PYA), N, Q(I,I+IB+1), LDQ, ONE,
$ A(1,I+IB+1), LDA )
CALL DGEMM( 'No transpose', 'Transpose', N, N-I-IB, IB,
$ ONE, DWORK(PYA+NIB), N, B(I+IB+1,I), LDB, ONE,
$ A(1,I+IB+1), LDA )
C
C Update the submatrix Q(i+ib:n,i+1+ib:n).
C
CALL DGEMM( 'No transpose', 'Transpose', N-I-IB+1, N-I-IB,
$ IB, ONE, Q(I+IB,I), LDQ, DWORK(PXQ+NB+1), N,
$ ONE, Q(I+IB,I+IB+1), LDQ )
CALL DGEMM( 'No transpose', 'Transpose', N-I-IB+1, N-I-IB,
$ IB, ONE, A(I+IB,I), LDA, DWORK(PXQ+NIB+NB+1), N,
$ ONE, Q(I+IB,I+IB+1), LDQ )
CALL DGEMM( 'No transpose', 'No transpose', N-I-IB+1,
$ N-I-IB, IB, ONE, DWORK(PYQ+NB), N,
$ Q(I,I+IB+1), LDQ, ONE, Q(I+IB,I+IB+1), LDQ )
CALL DGEMM( 'No Transpose', 'Transpose', N-I-IB+1,
$ N-I-IB, IB, ONE, DWORK(PYQ+NIB+NB), N,
$ B(I+IB+1,I), LDB, ONE, Q(I+IB,I+IB+1), LDQ )
C
C Update the matrix G.
C
CALL DGEMM( 'No transpose', 'Transpose', N-I-IB+1, N, IB,
$ ONE, Q(I+IB,I), LDQ, DWORK(PXG), N, ONE,
$ G(I+IB,1), LDG )
CALL DGEMM( 'No transpose', 'Transpose', N-I-IB+1, N, IB,
$ ONE, A(I+IB,I), LDA, DWORK(PXG+NIB), N, ONE,
$ G(I+IB,1), LDG )
CALL DGEMM( 'No transpose', 'No transpose', N, N-I-IB, IB,
$ ONE, DWORK(PYG), N, Q(I,I+IB+1), LDQ, ONE,
$ G(1,I+IB+1), LDG )
CALL DGEMM( 'No transpose', 'Transpose', N, N-I-IB, IB,
$ ONE, DWORK(PYG+NIB), N, B(I+IB+1,I), LDB, ONE,
$ G(1,I+IB+1), LDG )
C
C Update the submatrix B(1:n,i+ib:n).
C
CALL DGEMM( 'No transpose', 'Transpose', N, N-I-IB+1,
$ IB, ONE, DWORK(PXB), N, Q(I+IB,I), LDQ,
$ ONE, B(1,I+IB), LDB )
CALL DGEMM( 'No transpose', 'Transpose', N, N-I-IB+1, IB,
$ ONE, DWORK(PXB+NIB), N, A(I+IB,I), LDA, ONE,
$ B(1,I+IB), LDB )
CALL DGEMM( 'Transpose', 'Transpose', N-I-IB, N-I-IB+1,
$ IB, ONE, Q(I,I+IB+1), LDQ, DWORK(PYB+NB), N,
$ ONE, B(I+IB+1,I+IB), LDB )
CALL DGEMM( 'No transpose', 'Transpose', N-I-IB, N-I-IB+1,
$ IB, ONE, B(I+IB+1,I), LDB, DWORK(PYB+NIB+NB), N,
$ ONE, B(I+IB+1,I+IB), LDB )
40 CONTINUE
C
ELSE
DO 50 I = ILO, N-NX-1, NB
IB = MIN( NB, N-I )
NIB = N*IB
C
C Reduce rows and columns i:i+nb-1 to symplectic URV form and
C return the matrices XA, XB, XG, XQ, YA, YB, YG and YQ which
C are needed to update the unreduced parts of the matrices.
C
CALL MB03XU( LTRA, LTRB, N-I+1, I-1, IB, A(1,I), LDA,
$ B(I,1), LDB, G, LDG, Q(I,I), LDQ, DWORK(PXA),
$ N, DWORK(PXB), N, DWORK(PXG), N, DWORK(PXQ), N,
$ DWORK(PYA), N, DWORK(PYB), N, DWORK(PYG), N,
$ DWORK(PYQ), N, CSL(2*I-1), CSR(2*I-1), TAUL(I),
$ TAUR(I), DWORK(PDW) )
C
C Update the submatrix A(1:n,i+1+ib:n).
C
CALL DGEMM( 'No transpose', 'Transpose', N-I-IB+1, N-I-IB,
$ IB, ONE, Q(I+IB,I), LDQ, DWORK(PXA+NB+1), N,
$ ONE, A(I+IB,I+IB+1), LDA )
CALL DGEMM( 'No transpose', 'Transpose', N-I-IB+1, N-I-IB,
$ IB, ONE, A(I+IB,I), LDA, DWORK(PXA+NIB+NB+1), N,
$ ONE, A(I+IB,I+IB+1), LDA )
CALL DGEMM( 'No transpose', 'No transpose', N, N-I-IB, IB,
$ ONE, DWORK(PYA), N, Q(I,I+IB+1), LDQ, ONE,
$ A(1,I+IB+1), LDA )
CALL DGEMM( 'No transpose', 'No transpose', N, N-I-IB, IB,
$ ONE, DWORK(PYA+NIB), N, B(I,I+IB+1), LDB, ONE,
$ A(1,I+IB+1), LDA )
C
C Update the submatrix Q(i+ib:n,i+1+ib:n).
C
CALL DGEMM( 'No transpose', 'Transpose', N-I-IB+1, N-I-IB,
$ IB, ONE, Q(I+IB,I), LDQ, DWORK(PXQ+NB+1), N,
$ ONE, Q(I+IB,I+IB+1), LDQ )
CALL DGEMM( 'No transpose', 'Transpose', N-I-IB+1, N-I-IB,
$ IB, ONE, A(I+IB,I), LDA, DWORK(PXQ+NIB+NB+1), N,
$ ONE, Q(I+IB,I+IB+1), LDQ )
CALL DGEMM( 'No transpose', 'No transpose', N-I-IB+1,
$ N-I-IB, IB, ONE, DWORK(PYQ+NB), N,
$ Q(I,I+IB+1), LDQ, ONE, Q(I+IB,I+IB+1), LDQ )
CALL DGEMM( 'No transpose', 'No transpose', N-I-IB+1,
$ N-I-IB, IB, ONE, DWORK(PYQ+NIB+NB), N,
$ B(I,I+IB+1), LDB, ONE, Q(I+IB,I+IB+1), LDQ )
C
C Update the matrix G.
C
CALL DGEMM( 'No transpose', 'Transpose', N-I-IB+1, N, IB,
$ ONE, Q(I+IB,I), LDQ, DWORK(PXG), N, ONE,
$ G(I+IB,1), LDG )
CALL DGEMM( 'No transpose', 'Transpose', N-I-IB+1, N, IB,
$ ONE, A(I+IB,I), LDA, DWORK(PXG+NIB), N, ONE,
$ G(I+IB,1), LDG )
CALL DGEMM( 'No transpose', 'No transpose', N, N-I-IB, IB,
$ ONE, DWORK(PYG), N, Q(I,I+IB+1), LDQ, ONE,
$ G(1,I+IB+1), LDG )
CALL DGEMM( 'No transpose', 'No transpose', N, N-I-IB, IB,
$ ONE, DWORK(PYG+NIB), N, B(I,I+IB+1), LDB, ONE,
$ G(1,I+IB+1), LDG )
C
C Update the submatrix B(i+ib:n,1:n).
C
CALL DGEMM( 'No transpose', 'Transpose', N-I-IB+1, N,
$ IB, ONE, Q(I+IB,I), LDQ, DWORK(PXB), N,
$ ONE, B(I+IB,1), LDB )
CALL DGEMM( 'No transpose', 'Transpose', N-I-IB+1, N, IB,
$ ONE, A(I+IB,I), LDA, DWORK(PXB+NIB), N, ONE,
$ B(I+IB,1), LDB )
CALL DGEMM( 'No transpose', 'No transpose', N-I-IB+1,
$ N-I-IB, IB, ONE, DWORK(PYB+NB), N, Q(I,I+IB+1),
$ LDQ, ONE, B(I+IB,I+IB+1), LDB )
CALL DGEMM( 'No transpose', 'No transpose', N-I-IB+1,
$ N-I-IB, IB, ONE, DWORK(PYB+NIB+NB), N,
$ B(I,I+IB+1), LDB, ONE, B(I+IB,I+IB+1), LDB )
50 CONTINUE
END IF
C
C Unblocked code to reduce the rest of the matrices.
C
CALL MB04TS( TRANA, TRANB, N, I, A, LDA, B, LDB, G, LDG, Q, LDQ,
$ CSL, CSR, TAUL, TAUR, DWORK, LDWORK, IERR )
C
DWORK(1) = DBLE( WRKOPT )
C
RETURN
C *** Last line of MB04TB ***
END