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Mise à jour de lapack vers la version 3.3.0.
1: SUBROUTINE DLAR1V( N, B1, BN, LAMBDA, D, L, LD, LLD, 2: $ PIVMIN, GAPTOL, Z, WANTNC, NEGCNT, ZTZ, MINGMA, 3: $ R, ISUPPZ, NRMINV, RESID, RQCORR, WORK ) 4: * 5: * -- LAPACK auxiliary routine (version 3.2) -- 6: * -- LAPACK is a software package provided by Univ. of Tennessee, -- 7: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 8: * November 2006 9: * 10: * .. Scalar Arguments .. 11: LOGICAL WANTNC 12: INTEGER B1, BN, N, NEGCNT, R 13: DOUBLE PRECISION GAPTOL, LAMBDA, MINGMA, NRMINV, PIVMIN, RESID, 14: $ RQCORR, ZTZ 15: * .. 16: * .. Array Arguments .. 17: INTEGER ISUPPZ( * ) 18: DOUBLE PRECISION D( * ), L( * ), LD( * ), LLD( * ), 19: $ WORK( * ) 20: DOUBLE PRECISION Z( * ) 21: * .. 22: * 23: * Purpose 24: * ======= 25: * 26: * DLAR1V computes the (scaled) r-th column of the inverse of 27: * the sumbmatrix in rows B1 through BN of the tridiagonal matrix 28: * L D L^T - sigma I. When sigma is close to an eigenvalue, the 29: * computed vector is an accurate eigenvector. Usually, r corresponds 30: * to the index where the eigenvector is largest in magnitude. 31: * The following steps accomplish this computation : 32: * (a) Stationary qd transform, L D L^T - sigma I = L(+) D(+) L(+)^T, 33: * (b) Progressive qd transform, L D L^T - sigma I = U(-) D(-) U(-)^T, 34: * (c) Computation of the diagonal elements of the inverse of 35: * L D L^T - sigma I by combining the above transforms, and choosing 36: * r as the index where the diagonal of the inverse is (one of the) 37: * largest in magnitude. 38: * (d) Computation of the (scaled) r-th column of the inverse using the 39: * twisted factorization obtained by combining the top part of the 40: * the stationary and the bottom part of the progressive transform. 41: * 42: * Arguments 43: * ========= 44: * 45: * N (input) INTEGER 46: * The order of the matrix L D L^T. 47: * 48: * B1 (input) INTEGER 49: * First index of the submatrix of L D L^T. 50: * 51: * BN (input) INTEGER 52: * Last index of the submatrix of L D L^T. 53: * 54: * LAMBDA (input) DOUBLE PRECISION 55: * The shift. In order to compute an accurate eigenvector, 56: * LAMBDA should be a good approximation to an eigenvalue 57: * of L D L^T. 58: * 59: * L (input) DOUBLE PRECISION array, dimension (N-1) 60: * The (n-1) subdiagonal elements of the unit bidiagonal matrix 61: * L, in elements 1 to N-1. 62: * 63: * D (input) DOUBLE PRECISION array, dimension (N) 64: * The n diagonal elements of the diagonal matrix D. 65: * 66: * LD (input) DOUBLE PRECISION array, dimension (N-1) 67: * The n-1 elements L(i)*D(i). 68: * 69: * LLD (input) DOUBLE PRECISION array, dimension (N-1) 70: * The n-1 elements L(i)*L(i)*D(i). 71: * 72: * PIVMIN (input) DOUBLE PRECISION 73: * The minimum pivot in the Sturm sequence. 74: * 75: * GAPTOL (input) DOUBLE PRECISION 76: * Tolerance that indicates when eigenvector entries are negligible 77: * w.r.t. their contribution to the residual. 78: * 79: * Z (input/output) DOUBLE PRECISION array, dimension (N) 80: * On input, all entries of Z must be set to 0. 81: * On output, Z contains the (scaled) r-th column of the 82: * inverse. The scaling is such that Z(R) equals 1. 83: * 84: * WANTNC (input) LOGICAL 85: * Specifies whether NEGCNT has to be computed. 86: * 87: * NEGCNT (output) INTEGER 88: * If WANTNC is .TRUE. then NEGCNT = the number of pivots < pivmin 89: * in the matrix factorization L D L^T, and NEGCNT = -1 otherwise. 90: * 91: * ZTZ (output) DOUBLE PRECISION 92: * The square of the 2-norm of Z. 93: * 94: * MINGMA (output) DOUBLE PRECISION 95: * The reciprocal of the largest (in magnitude) diagonal 96: * element of the inverse of L D L^T - sigma I. 97: * 98: * R (input/output) INTEGER 99: * The twist index for the twisted factorization used to 100: * compute Z. 101: * On input, 0 <= R <= N. If R is input as 0, R is set to 102: * the index where (L D L^T - sigma I)^{-1} is largest 103: * in magnitude. If 1 <= R <= N, R is unchanged. 104: * On output, R contains the twist index used to compute Z. 105: * Ideally, R designates the position of the maximum entry in the 106: * eigenvector. 107: * 108: * ISUPPZ (output) INTEGER array, dimension (2) 109: * The support of the vector in Z, i.e., the vector Z is 110: * nonzero only in elements ISUPPZ(1) through ISUPPZ( 2 ). 111: * 112: * NRMINV (output) DOUBLE PRECISION 113: * NRMINV = 1/SQRT( ZTZ ) 114: * 115: * RESID (output) DOUBLE PRECISION 116: * The residual of the FP vector. 117: * RESID = ABS( MINGMA )/SQRT( ZTZ ) 118: * 119: * RQCORR (output) DOUBLE PRECISION 120: * The Rayleigh Quotient correction to LAMBDA. 121: * RQCORR = MINGMA*TMP 122: * 123: * WORK (workspace) DOUBLE PRECISION array, dimension (4*N) 124: * 125: * Further Details 126: * =============== 127: * 128: * Based on contributions by 129: * Beresford Parlett, University of California, Berkeley, USA 130: * Jim Demmel, University of California, Berkeley, USA 131: * Inderjit Dhillon, University of Texas, Austin, USA 132: * Osni Marques, LBNL/NERSC, USA 133: * Christof Voemel, University of California, Berkeley, USA 134: * 135: * ===================================================================== 136: * 137: * .. Parameters .. 138: DOUBLE PRECISION ZERO, ONE 139: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) 140: 141: * .. 142: * .. Local Scalars .. 143: LOGICAL SAWNAN1, SAWNAN2 144: INTEGER I, INDLPL, INDP, INDS, INDUMN, NEG1, NEG2, R1, 145: $ R2 146: DOUBLE PRECISION DMINUS, DPLUS, EPS, S, TMP 147: * .. 148: * .. External Functions .. 149: LOGICAL DISNAN 150: DOUBLE PRECISION DLAMCH 151: EXTERNAL DISNAN, DLAMCH 152: * .. 153: * .. Intrinsic Functions .. 154: INTRINSIC ABS 155: * .. 156: * .. Executable Statements .. 157: * 158: EPS = DLAMCH( 'Precision' ) 159: 160: 161: IF( R.EQ.0 ) THEN 162: R1 = B1 163: R2 = BN 164: ELSE 165: R1 = R 166: R2 = R 167: END IF 168: 169: * Storage for LPLUS 170: INDLPL = 0 171: * Storage for UMINUS 172: INDUMN = N 173: INDS = 2*N + 1 174: INDP = 3*N + 1 175: 176: IF( B1.EQ.1 ) THEN 177: WORK( INDS ) = ZERO 178: ELSE 179: WORK( INDS+B1-1 ) = LLD( B1-1 ) 180: END IF 181: 182: * 183: * Compute the stationary transform (using the differential form) 184: * until the index R2. 185: * 186: SAWNAN1 = .FALSE. 187: NEG1 = 0 188: S = WORK( INDS+B1-1 ) - LAMBDA 189: DO 50 I = B1, R1 - 1 190: DPLUS = D( I ) + S 191: WORK( INDLPL+I ) = LD( I ) / DPLUS 192: IF(DPLUS.LT.ZERO) NEG1 = NEG1 + 1 193: WORK( INDS+I ) = S*WORK( INDLPL+I )*L( I ) 194: S = WORK( INDS+I ) - LAMBDA 195: 50 CONTINUE 196: SAWNAN1 = DISNAN( S ) 197: IF( SAWNAN1 ) GOTO 60 198: DO 51 I = R1, R2 - 1 199: DPLUS = D( I ) + S 200: WORK( INDLPL+I ) = LD( I ) / DPLUS 201: WORK( INDS+I ) = S*WORK( INDLPL+I )*L( I ) 202: S = WORK( INDS+I ) - LAMBDA 203: 51 CONTINUE 204: SAWNAN1 = DISNAN( S ) 205: * 206: 60 CONTINUE 207: IF( SAWNAN1 ) THEN 208: * Runs a slower version of the above loop if a NaN is detected 209: NEG1 = 0 210: S = WORK( INDS+B1-1 ) - LAMBDA 211: DO 70 I = B1, R1 - 1 212: DPLUS = D( I ) + S 213: IF(ABS(DPLUS).LT.PIVMIN) DPLUS = -PIVMIN 214: WORK( INDLPL+I ) = LD( I ) / DPLUS 215: IF(DPLUS.LT.ZERO) NEG1 = NEG1 + 1 216: WORK( INDS+I ) = S*WORK( INDLPL+I )*L( I ) 217: IF( WORK( INDLPL+I ).EQ.ZERO ) 218: $ WORK( INDS+I ) = LLD( I ) 219: S = WORK( INDS+I ) - LAMBDA 220: 70 CONTINUE 221: DO 71 I = R1, R2 - 1 222: DPLUS = D( I ) + S 223: IF(ABS(DPLUS).LT.PIVMIN) DPLUS = -PIVMIN 224: WORK( INDLPL+I ) = LD( I ) / DPLUS 225: WORK( INDS+I ) = S*WORK( INDLPL+I )*L( I ) 226: IF( WORK( INDLPL+I ).EQ.ZERO ) 227: $ WORK( INDS+I ) = LLD( I ) 228: S = WORK( INDS+I ) - LAMBDA 229: 71 CONTINUE 230: END IF 231: * 232: * Compute the progressive transform (using the differential form) 233: * until the index R1 234: * 235: SAWNAN2 = .FALSE. 236: NEG2 = 0 237: WORK( INDP+BN-1 ) = D( BN ) - LAMBDA 238: DO 80 I = BN - 1, R1, -1 239: DMINUS = LLD( I ) + WORK( INDP+I ) 240: TMP = D( I ) / DMINUS 241: IF(DMINUS.LT.ZERO) NEG2 = NEG2 + 1 242: WORK( INDUMN+I ) = L( I )*TMP 243: WORK( INDP+I-1 ) = WORK( INDP+I )*TMP - LAMBDA 244: 80 CONTINUE 245: TMP = WORK( INDP+R1-1 ) 246: SAWNAN2 = DISNAN( TMP ) 247: 248: IF( SAWNAN2 ) THEN 249: * Runs a slower version of the above loop if a NaN is detected 250: NEG2 = 0 251: DO 100 I = BN-1, R1, -1 252: DMINUS = LLD( I ) + WORK( INDP+I ) 253: IF(ABS(DMINUS).LT.PIVMIN) DMINUS = -PIVMIN 254: TMP = D( I ) / DMINUS 255: IF(DMINUS.LT.ZERO) NEG2 = NEG2 + 1 256: WORK( INDUMN+I ) = L( I )*TMP 257: WORK( INDP+I-1 ) = WORK( INDP+I )*TMP - LAMBDA 258: IF( TMP.EQ.ZERO ) 259: $ WORK( INDP+I-1 ) = D( I ) - LAMBDA 260: 100 CONTINUE 261: END IF 262: * 263: * Find the index (from R1 to R2) of the largest (in magnitude) 264: * diagonal element of the inverse 265: * 266: MINGMA = WORK( INDS+R1-1 ) + WORK( INDP+R1-1 ) 267: IF( MINGMA.LT.ZERO ) NEG1 = NEG1 + 1 268: IF( WANTNC ) THEN 269: NEGCNT = NEG1 + NEG2 270: ELSE 271: NEGCNT = -1 272: ENDIF 273: IF( ABS(MINGMA).EQ.ZERO ) 274: $ MINGMA = EPS*WORK( INDS+R1-1 ) 275: R = R1 276: DO 110 I = R1, R2 - 1 277: TMP = WORK( INDS+I ) + WORK( INDP+I ) 278: IF( TMP.EQ.ZERO ) 279: $ TMP = EPS*WORK( INDS+I ) 280: IF( ABS( TMP ).LE.ABS( MINGMA ) ) THEN 281: MINGMA = TMP 282: R = I + 1 283: END IF 284: 110 CONTINUE 285: * 286: * Compute the FP vector: solve N^T v = e_r 287: * 288: ISUPPZ( 1 ) = B1 289: ISUPPZ( 2 ) = BN 290: Z( R ) = ONE 291: ZTZ = ONE 292: * 293: * Compute the FP vector upwards from R 294: * 295: IF( .NOT.SAWNAN1 .AND. .NOT.SAWNAN2 ) THEN 296: DO 210 I = R-1, B1, -1 297: Z( I ) = -( WORK( INDLPL+I )*Z( I+1 ) ) 298: IF( (ABS(Z(I))+ABS(Z(I+1)))* ABS(LD(I)).LT.GAPTOL ) 299: $ THEN 300: Z( I ) = ZERO 301: ISUPPZ( 1 ) = I + 1 302: GOTO 220 303: ENDIF 304: ZTZ = ZTZ + Z( I )*Z( I ) 305: 210 CONTINUE 306: 220 CONTINUE 307: ELSE 308: * Run slower loop if NaN occurred. 309: DO 230 I = R - 1, B1, -1 310: IF( Z( I+1 ).EQ.ZERO ) THEN 311: Z( I ) = -( LD( I+1 ) / LD( I ) )*Z( I+2 ) 312: ELSE 313: Z( I ) = -( WORK( INDLPL+I )*Z( I+1 ) ) 314: END IF 315: IF( (ABS(Z(I))+ABS(Z(I+1)))* ABS(LD(I)).LT.GAPTOL ) 316: $ THEN 317: Z( I ) = ZERO 318: ISUPPZ( 1 ) = I + 1 319: GO TO 240 320: END IF 321: ZTZ = ZTZ + Z( I )*Z( I ) 322: 230 CONTINUE 323: 240 CONTINUE 324: ENDIF 325: 326: * Compute the FP vector downwards from R in blocks of size BLKSIZ 327: IF( .NOT.SAWNAN1 .AND. .NOT.SAWNAN2 ) THEN 328: DO 250 I = R, BN-1 329: Z( I+1 ) = -( WORK( INDUMN+I )*Z( I ) ) 330: IF( (ABS(Z(I))+ABS(Z(I+1)))* ABS(LD(I)).LT.GAPTOL ) 331: $ THEN 332: Z( I+1 ) = ZERO 333: ISUPPZ( 2 ) = I 334: GO TO 260 335: END IF 336: ZTZ = ZTZ + Z( I+1 )*Z( I+1 ) 337: 250 CONTINUE 338: 260 CONTINUE 339: ELSE 340: * Run slower loop if NaN occurred. 341: DO 270 I = R, BN - 1 342: IF( Z( I ).EQ.ZERO ) THEN 343: Z( I+1 ) = -( LD( I-1 ) / LD( I ) )*Z( I-1 ) 344: ELSE 345: Z( I+1 ) = -( WORK( INDUMN+I )*Z( I ) ) 346: END IF 347: IF( (ABS(Z(I))+ABS(Z(I+1)))* ABS(LD(I)).LT.GAPTOL ) 348: $ THEN 349: Z( I+1 ) = ZERO 350: ISUPPZ( 2 ) = I 351: GO TO 280 352: END IF 353: ZTZ = ZTZ + Z( I+1 )*Z( I+1 ) 354: 270 CONTINUE 355: 280 CONTINUE 356: END IF 357: * 358: * Compute quantities for convergence test 359: * 360: TMP = ONE / ZTZ 361: NRMINV = SQRT( TMP ) 362: RESID = ABS( MINGMA )*NRMINV 363: RQCORR = MINGMA*TMP 364: * 365: * 366: RETURN 367: * 368: * End of DLAR1V 369: * 370: END