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Mise à jour de lapack vers la version 3.2.2.
1: SUBROUTINE DLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV, 2: $ JPIV ) 3: * 4: * -- LAPACK auxiliary routine (version 3.2.2) -- 5: * -- LAPACK is a software package provided by Univ. of Tennessee, -- 6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 7: * June 2010 8: * 9: * .. Scalar Arguments .. 10: INTEGER IJOB, LDZ, N 11: DOUBLE PRECISION RDSCAL, RDSUM 12: * .. 13: * .. Array Arguments .. 14: INTEGER IPIV( * ), JPIV( * ) 15: DOUBLE PRECISION RHS( * ), Z( LDZ, * ) 16: * .. 17: * 18: * Purpose 19: * ======= 20: * 21: * DLATDF uses the LU factorization of the n-by-n matrix Z computed by 22: * DGETC2 and computes a contribution to the reciprocal Dif-estimate 23: * by solving Z * x = b for x, and choosing the r.h.s. b such that 24: * the norm of x is as large as possible. On entry RHS = b holds the 25: * contribution from earlier solved sub-systems, and on return RHS = x. 26: * 27: * The factorization of Z returned by DGETC2 has the form Z = P*L*U*Q, 28: * where P and Q are permutation matrices. L is lower triangular with 29: * unit diagonal elements and U is upper triangular. 30: * 31: * Arguments 32: * ========= 33: * 34: * IJOB (input) INTEGER 35: * IJOB = 2: First compute an approximative null-vector e 36: * of Z using DGECON, e is normalized and solve for 37: * Zx = +-e - f with the sign giving the greater value 38: * of 2-norm(x). About 5 times as expensive as Default. 39: * IJOB .ne. 2: Local look ahead strategy where all entries of 40: * the r.h.s. b is choosen as either +1 or -1 (Default). 41: * 42: * N (input) INTEGER 43: * The number of columns of the matrix Z. 44: * 45: * Z (input) DOUBLE PRECISION array, dimension (LDZ, N) 46: * On entry, the LU part of the factorization of the n-by-n 47: * matrix Z computed by DGETC2: Z = P * L * U * Q 48: * 49: * LDZ (input) INTEGER 50: * The leading dimension of the array Z. LDA >= max(1, N). 51: * 52: * RHS (input/output) DOUBLE PRECISION array, dimension (N) 53: * On entry, RHS contains contributions from other subsystems. 54: * On exit, RHS contains the solution of the subsystem with 55: * entries acoording to the value of IJOB (see above). 56: * 57: * RDSUM (input/output) DOUBLE PRECISION 58: * On entry, the sum of squares of computed contributions to 59: * the Dif-estimate under computation by DTGSYL, where the 60: * scaling factor RDSCAL (see below) has been factored out. 61: * On exit, the corresponding sum of squares updated with the 62: * contributions from the current sub-system. 63: * If TRANS = 'T' RDSUM is not touched. 64: * NOTE: RDSUM only makes sense when DTGSY2 is called by STGSYL. 65: * 66: * RDSCAL (input/output) DOUBLE PRECISION 67: * On entry, scaling factor used to prevent overflow in RDSUM. 68: * On exit, RDSCAL is updated w.r.t. the current contributions 69: * in RDSUM. 70: * If TRANS = 'T', RDSCAL is not touched. 71: * NOTE: RDSCAL only makes sense when DTGSY2 is called by 72: * DTGSYL. 73: * 74: * IPIV (input) INTEGER array, dimension (N). 75: * The pivot indices; for 1 <= i <= N, row i of the 76: * matrix has been interchanged with row IPIV(i). 77: * 78: * JPIV (input) INTEGER array, dimension (N). 79: * The pivot indices; for 1 <= j <= N, column j of the 80: * matrix has been interchanged with column JPIV(j). 81: * 82: * Further Details 83: * =============== 84: * 85: * Based on contributions by 86: * Bo Kagstrom and Peter Poromaa, Department of Computing Science, 87: * Umea University, S-901 87 Umea, Sweden. 88: * 89: * This routine is a further developed implementation of algorithm 90: * BSOLVE in [1] using complete pivoting in the LU factorization. 91: * 92: * [1] Bo Kagstrom and Lars Westin, 93: * Generalized Schur Methods with Condition Estimators for 94: * Solving the Generalized Sylvester Equation, IEEE Transactions 95: * on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751. 96: * 97: * [2] Peter Poromaa, 98: * On Efficient and Robust Estimators for the Separation 99: * between two Regular Matrix Pairs with Applications in 100: * Condition Estimation. Report IMINF-95.05, Departement of 101: * Computing Science, Umea University, S-901 87 Umea, Sweden, 1995. 102: * 103: * ===================================================================== 104: * 105: * .. Parameters .. 106: INTEGER MAXDIM 107: PARAMETER ( MAXDIM = 8 ) 108: DOUBLE PRECISION ZERO, ONE 109: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 110: * .. 111: * .. Local Scalars .. 112: INTEGER I, INFO, J, K 113: DOUBLE PRECISION BM, BP, PMONE, SMINU, SPLUS, TEMP 114: * .. 115: * .. Local Arrays .. 116: INTEGER IWORK( MAXDIM ) 117: DOUBLE PRECISION WORK( 4*MAXDIM ), XM( MAXDIM ), XP( MAXDIM ) 118: * .. 119: * .. External Subroutines .. 120: EXTERNAL DAXPY, DCOPY, DGECON, DGESC2, DLASSQ, DLASWP, 121: $ DSCAL 122: * .. 123: * .. External Functions .. 124: DOUBLE PRECISION DASUM, DDOT 125: EXTERNAL DASUM, DDOT 126: * .. 127: * .. Intrinsic Functions .. 128: INTRINSIC ABS, SQRT 129: * .. 130: * .. Executable Statements .. 131: * 132: IF( IJOB.NE.2 ) THEN 133: * 134: * Apply permutations IPIV to RHS 135: * 136: CALL DLASWP( 1, RHS, LDZ, 1, N-1, IPIV, 1 ) 137: * 138: * Solve for L-part choosing RHS either to +1 or -1. 139: * 140: PMONE = -ONE 141: * 142: DO 10 J = 1, N - 1 143: BP = RHS( J ) + ONE 144: BM = RHS( J ) - ONE 145: SPLUS = ONE 146: * 147: * Look-ahead for L-part RHS(1:N-1) = + or -1, SPLUS and 148: * SMIN computed more efficiently than in BSOLVE [1]. 149: * 150: SPLUS = SPLUS + DDOT( N-J, Z( J+1, J ), 1, Z( J+1, J ), 1 ) 151: SMINU = DDOT( N-J, Z( J+1, J ), 1, RHS( J+1 ), 1 ) 152: SPLUS = SPLUS*RHS( J ) 153: IF( SPLUS.GT.SMINU ) THEN 154: RHS( J ) = BP 155: ELSE IF( SMINU.GT.SPLUS ) THEN 156: RHS( J ) = BM 157: ELSE 158: * 159: * In this case the updating sums are equal and we can 160: * choose RHS(J) +1 or -1. The first time this happens 161: * we choose -1, thereafter +1. This is a simple way to 162: * get good estimates of matrices like Byers well-known 163: * example (see [1]). (Not done in BSOLVE.) 164: * 165: RHS( J ) = RHS( J ) + PMONE 166: PMONE = ONE 167: END IF 168: * 169: * Compute the remaining r.h.s. 170: * 171: TEMP = -RHS( J ) 172: CALL DAXPY( N-J, TEMP, Z( J+1, J ), 1, RHS( J+1 ), 1 ) 173: * 174: 10 CONTINUE 175: * 176: * Solve for U-part, look-ahead for RHS(N) = +-1. This is not done 177: * in BSOLVE and will hopefully give us a better estimate because 178: * any ill-conditioning of the original matrix is transfered to U 179: * and not to L. U(N, N) is an approximation to sigma_min(LU). 180: * 181: CALL DCOPY( N-1, RHS, 1, XP, 1 ) 182: XP( N ) = RHS( N ) + ONE 183: RHS( N ) = RHS( N ) - ONE 184: SPLUS = ZERO 185: SMINU = ZERO 186: DO 30 I = N, 1, -1 187: TEMP = ONE / Z( I, I ) 188: XP( I ) = XP( I )*TEMP 189: RHS( I ) = RHS( I )*TEMP 190: DO 20 K = I + 1, N 191: XP( I ) = XP( I ) - XP( K )*( Z( I, K )*TEMP ) 192: RHS( I ) = RHS( I ) - RHS( K )*( Z( I, K )*TEMP ) 193: 20 CONTINUE 194: SPLUS = SPLUS + ABS( XP( I ) ) 195: SMINU = SMINU + ABS( RHS( I ) ) 196: 30 CONTINUE 197: IF( SPLUS.GT.SMINU ) 198: $ CALL DCOPY( N, XP, 1, RHS, 1 ) 199: * 200: * Apply the permutations JPIV to the computed solution (RHS) 201: * 202: CALL DLASWP( 1, RHS, LDZ, 1, N-1, JPIV, -1 ) 203: * 204: * Compute the sum of squares 205: * 206: CALL DLASSQ( N, RHS, 1, RDSCAL, RDSUM ) 207: * 208: ELSE 209: * 210: * IJOB = 2, Compute approximate nullvector XM of Z 211: * 212: CALL DGECON( 'I', N, Z, LDZ, ONE, TEMP, WORK, IWORK, INFO ) 213: CALL DCOPY( N, WORK( N+1 ), 1, XM, 1 ) 214: * 215: * Compute RHS 216: * 217: CALL DLASWP( 1, XM, LDZ, 1, N-1, IPIV, -1 ) 218: TEMP = ONE / SQRT( DDOT( N, XM, 1, XM, 1 ) ) 219: CALL DSCAL( N, TEMP, XM, 1 ) 220: CALL DCOPY( N, XM, 1, XP, 1 ) 221: CALL DAXPY( N, ONE, RHS, 1, XP, 1 ) 222: CALL DAXPY( N, -ONE, XM, 1, RHS, 1 ) 223: CALL DGESC2( N, Z, LDZ, RHS, IPIV, JPIV, TEMP ) 224: CALL DGESC2( N, Z, LDZ, XP, IPIV, JPIV, TEMP ) 225: IF( DASUM( N, XP, 1 ).GT.DASUM( N, RHS, 1 ) ) 226: $ CALL DCOPY( N, XP, 1, RHS, 1 ) 227: * 228: * Compute the sum of squares 229: * 230: CALL DLASSQ( N, RHS, 1, RDSCAL, RDSUM ) 231: * 232: END IF 233: * 234: RETURN 235: * 236: * End of DLATDF 237: * 238: END