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diff --git a/SRC/slatdf.f b/SRC/slatdf.f new file mode 100644 index 00000000..43156c7c --- /dev/null +++ b/SRC/slatdf.f @@ -0,0 +1,237 @@ + SUBROUTINE SLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV, + $ JPIV ) +* +* -- LAPACK auxiliary routine (version 3.1) -- +* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. +* November 2006 +* +* .. Scalar Arguments .. + INTEGER IJOB, LDZ, N + REAL RDSCAL, RDSUM +* .. +* .. Array Arguments .. + INTEGER IPIV( * ), JPIV( * ) + REAL RHS( * ), Z( LDZ, * ) +* .. +* +* Purpose +* ======= +* +* SLATDF uses the LU factorization of the n-by-n matrix Z computed by +* SGETC2 and computes a contribution to the reciprocal Dif-estimate +* by solving Z * x = b for x, and choosing the r.h.s. b such that +* the norm of x is as large as possible. On entry RHS = b holds the +* contribution from earlier solved sub-systems, and on return RHS = x. +* +* The factorization of Z returned by SGETC2 has the form Z = P*L*U*Q, +* where P and Q are permutation matrices. L is lower triangular with +* unit diagonal elements and U is upper triangular. +* +* Arguments +* ========= +* +* IJOB (input) INTEGER +* IJOB = 2: First compute an approximative null-vector e +* of Z using SGECON, e is normalized and solve for +* Zx = +-e - f with the sign giving the greater value +* of 2-norm(x). About 5 times as expensive as Default. +* IJOB .ne. 2: Local look ahead strategy where all entries of +* the r.h.s. b is choosen as either +1 or -1 (Default). +* +* N (input) INTEGER +* The number of columns of the matrix Z. +* +* Z (input) REAL array, dimension (LDZ, N) +* On entry, the LU part of the factorization of the n-by-n +* matrix Z computed by SGETC2: Z = P * L * U * Q +* +* LDZ (input) INTEGER +* The leading dimension of the array Z. LDA >= max(1, N). +* +* RHS (input/output) REAL array, dimension N. +* On entry, RHS contains contributions from other subsystems. +* On exit, RHS contains the solution of the subsystem with +* entries acoording to the value of IJOB (see above). +* +* RDSUM (input/output) REAL +* On entry, the sum of squares of computed contributions to +* the Dif-estimate under computation by STGSYL, where the +* scaling factor RDSCAL (see below) has been factored out. +* On exit, the corresponding sum of squares updated with the +* contributions from the current sub-system. +* If TRANS = 'T' RDSUM is not touched. +* NOTE: RDSUM only makes sense when STGSY2 is called by STGSYL. +* +* RDSCAL (input/output) REAL +* On entry, scaling factor used to prevent overflow in RDSUM. +* On exit, RDSCAL is updated w.r.t. the current contributions +* in RDSUM. +* If TRANS = 'T', RDSCAL is not touched. +* NOTE: RDSCAL only makes sense when STGSY2 is called by +* STGSYL. +* +* IPIV (input) INTEGER array, dimension (N). +* The pivot indices; for 1 <= i <= N, row i of the +* matrix has been interchanged with row IPIV(i). +* +* JPIV (input) INTEGER array, dimension (N). +* The pivot indices; for 1 <= j <= N, column j of the +* matrix has been interchanged with column JPIV(j). +* +* Further Details +* =============== +* +* Based on contributions by +* Bo Kagstrom and Peter Poromaa, Department of Computing Science, +* Umea University, S-901 87 Umea, Sweden. +* +* This routine is a further developed implementation of algorithm +* BSOLVE in [1] using complete pivoting in the LU factorization. +* +* [1] Bo Kagstrom and Lars Westin, +* Generalized Schur Methods with Condition Estimators for +* Solving the Generalized Sylvester Equation, IEEE Transactions +* on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751. +* +* [2] Peter Poromaa, +* On Efficient and Robust Estimators for the Separation +* between two Regular Matrix Pairs with Applications in +* Condition Estimation. Report IMINF-95.05, Departement of +* Computing Science, Umea University, S-901 87 Umea, Sweden, 1995. +* +* ===================================================================== +* +* .. Parameters .. + INTEGER MAXDIM + PARAMETER ( MAXDIM = 8 ) + REAL ZERO, ONE + PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) +* .. +* .. Local Scalars .. + INTEGER I, INFO, J, K + REAL BM, BP, PMONE, SMINU, SPLUS, TEMP +* .. +* .. Local Arrays .. + INTEGER IWORK( MAXDIM ) + REAL WORK( 4*MAXDIM ), XM( MAXDIM ), XP( MAXDIM ) +* .. +* .. External Subroutines .. + EXTERNAL SAXPY, SCOPY, SGECON, SGESC2, SLASSQ, SLASWP, + $ SSCAL +* .. +* .. External Functions .. + REAL SASUM, SDOT + EXTERNAL SASUM, SDOT +* .. +* .. Intrinsic Functions .. + INTRINSIC ABS, SQRT +* .. +* .. Executable Statements .. +* + IF( IJOB.NE.2 ) THEN +* +* Apply permutations IPIV to RHS +* + CALL SLASWP( 1, RHS, LDZ, 1, N-1, IPIV, 1 ) +* +* Solve for L-part choosing RHS either to +1 or -1. +* + PMONE = -ONE +* + DO 10 J = 1, N - 1 + BP = RHS( J ) + ONE + BM = RHS( J ) - ONE + SPLUS = ONE +* +* Look-ahead for L-part RHS(1:N-1) = + or -1, SPLUS and +* SMIN computed more efficiently than in BSOLVE [1]. +* + SPLUS = SPLUS + SDOT( N-J, Z( J+1, J ), 1, Z( J+1, J ), 1 ) + SMINU = SDOT( N-J, Z( J+1, J ), 1, RHS( J+1 ), 1 ) + SPLUS = SPLUS*RHS( J ) + IF( SPLUS.GT.SMINU ) THEN + RHS( J ) = BP + ELSE IF( SMINU.GT.SPLUS ) THEN + RHS( J ) = BM + ELSE +* +* In this case the updating sums are equal and we can +* choose RHS(J) +1 or -1. The first time this happens +* we choose -1, thereafter +1. This is a simple way to +* get good estimates of matrices like Byers well-known +* example (see [1]). (Not done in BSOLVE.) +* + RHS( J ) = RHS( J ) + PMONE + PMONE = ONE + END IF +* +* Compute the remaining r.h.s. +* + TEMP = -RHS( J ) + CALL SAXPY( N-J, TEMP, Z( J+1, J ), 1, RHS( J+1 ), 1 ) +* + 10 CONTINUE +* +* Solve for U-part, look-ahead for RHS(N) = +-1. This is not done +* in BSOLVE and will hopefully give us a better estimate because +* any ill-conditioning of the original matrix is transfered to U +* and not to L. U(N, N) is an approximation to sigma_min(LU). +* + CALL SCOPY( N-1, RHS, 1, XP, 1 ) + XP( N ) = RHS( N ) + ONE + RHS( N ) = RHS( N ) - ONE + SPLUS = ZERO + SMINU = ZERO + DO 30 I = N, 1, -1 + TEMP = ONE / Z( I, I ) + XP( I ) = XP( I )*TEMP + RHS( I ) = RHS( I )*TEMP + DO 20 K = I + 1, N + XP( I ) = XP( I ) - XP( K )*( Z( I, K )*TEMP ) + RHS( I ) = RHS( I ) - RHS( K )*( Z( I, K )*TEMP ) + 20 CONTINUE + SPLUS = SPLUS + ABS( XP( I ) ) + SMINU = SMINU + ABS( RHS( I ) ) + 30 CONTINUE + IF( SPLUS.GT.SMINU ) + $ CALL SCOPY( N, XP, 1, RHS, 1 ) +* +* Apply the permutations JPIV to the computed solution (RHS) +* + CALL SLASWP( 1, RHS, LDZ, 1, N-1, JPIV, -1 ) +* +* Compute the sum of squares +* + CALL SLASSQ( N, RHS, 1, RDSCAL, RDSUM ) +* + ELSE +* +* IJOB = 2, Compute approximate nullvector XM of Z +* + CALL SGECON( 'I', N, Z, LDZ, ONE, TEMP, WORK, IWORK, INFO ) + CALL SCOPY( N, WORK( N+1 ), 1, XM, 1 ) +* +* Compute RHS +* + CALL SLASWP( 1, XM, LDZ, 1, N-1, IPIV, -1 ) + TEMP = ONE / SQRT( SDOT( N, XM, 1, XM, 1 ) ) + CALL SSCAL( N, TEMP, XM, 1 ) + CALL SCOPY( N, XM, 1, XP, 1 ) + CALL SAXPY( N, ONE, RHS, 1, XP, 1 ) + CALL SAXPY( N, -ONE, XM, 1, RHS, 1 ) + CALL SGESC2( N, Z, LDZ, RHS, IPIV, JPIV, TEMP ) + CALL SGESC2( N, Z, LDZ, XP, IPIV, JPIV, TEMP ) + IF( SASUM( N, XP, 1 ).GT.SASUM( N, RHS, 1 ) ) + $ CALL SCOPY( N, XP, 1, RHS, 1 ) +* +* Compute the sum of squares +* + CALL SLASSQ( N, RHS, 1, RDSCAL, RDSUM ) +* + END IF +* + RETURN +* +* End of SLATDF +* + END |