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Mise à jour de lapack vers la version 3.3.0.
1: SUBROUTINE DBBCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P, Q, 2: $ THETA, PHI, U1, LDU1, U2, LDU2, V1T, LDV1T, 3: $ V2T, LDV2T, B11D, B11E, B12D, B12E, B21D, B21E, 4: $ B22D, B22E, WORK, LWORK, INFO ) 5: IMPLICIT NONE 6: * 7: * -- LAPACK routine (version 3.3.0) -- 8: * 9: * -- Contributed by Brian Sutton of the Randolph-Macon College -- 10: * -- November 2010 11: * 12: * -- LAPACK is a software package provided by Univ. of Tennessee, -- 13: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 14: * 15: * .. Scalar Arguments .. 16: CHARACTER JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS 17: INTEGER INFO, LDU1, LDU2, LDV1T, LDV2T, LWORK, M, P, Q 18: * .. 19: * .. Array Arguments .. 20: DOUBLE PRECISION B11D( * ), B11E( * ), B12D( * ), B12E( * ), 21: $ B21D( * ), B21E( * ), B22D( * ), B22E( * ), 22: $ PHI( * ), THETA( * ), WORK( * ) 23: DOUBLE PRECISION U1( LDU1, * ), U2( LDU2, * ), V1T( LDV1T, * ), 24: $ V2T( LDV2T, * ) 25: * .. 26: * 27: * Purpose 28: * ======= 29: * 30: * DBBCSD computes the CS decomposition of an orthogonal matrix in 31: * bidiagonal-block form, 32: * 33: * 34: * [ B11 | B12 0 0 ] 35: * [ 0 | 0 -I 0 ] 36: * X = [----------------] 37: * [ B21 | B22 0 0 ] 38: * [ 0 | 0 0 I ] 39: * 40: * [ C | -S 0 0 ] 41: * [ U1 | ] [ 0 | 0 -I 0 ] [ V1 | ]**T 42: * = [---------] [---------------] [---------] . 43: * [ | U2 ] [ S | C 0 0 ] [ | V2 ] 44: * [ 0 | 0 0 I ] 45: * 46: * X is M-by-M, its top-left block is P-by-Q, and Q must be no larger 47: * than P, M-P, or M-Q. (If Q is not the smallest index, then X must be 48: * transposed and/or permuted. This can be done in constant time using 49: * the TRANS and SIGNS options. See DORCSD for details.) 50: * 51: * The bidiagonal matrices B11, B12, B21, and B22 are represented 52: * implicitly by angles THETA(1:Q) and PHI(1:Q-1). 53: * 54: * The orthogonal matrices U1, U2, V1T, and V2T are input/output. 55: * The input matrices are pre- or post-multiplied by the appropriate 56: * singular vector matrices. 57: * 58: * Arguments 59: * ========= 60: * 61: * JOBU1 (input) CHARACTER 62: * = 'Y': U1 is updated; 63: * otherwise: U1 is not updated. 64: * 65: * JOBU2 (input) CHARACTER 66: * = 'Y': U2 is updated; 67: * otherwise: U2 is not updated. 68: * 69: * JOBV1T (input) CHARACTER 70: * = 'Y': V1T is updated; 71: * otherwise: V1T is not updated. 72: * 73: * JOBV2T (input) CHARACTER 74: * = 'Y': V2T is updated; 75: * otherwise: V2T is not updated. 76: * 77: * TRANS (input) CHARACTER 78: * = 'T': X, U1, U2, V1T, and V2T are stored in row-major 79: * order; 80: * otherwise: X, U1, U2, V1T, and V2T are stored in column- 81: * major order. 82: * 83: * M (input) INTEGER 84: * The number of rows and columns in X, the orthogonal matrix in 85: * bidiagonal-block form. 86: * 87: * P (input) INTEGER 88: * The number of rows in the top-left block of X. 0 <= P <= M. 89: * 90: * Q (input) INTEGER 91: * The number of columns in the top-left block of X. 92: * 0 <= Q <= MIN(P,M-P,M-Q). 93: * 94: * THETA (input/output) DOUBLE PRECISION array, dimension (Q) 95: * On entry, the angles THETA(1),...,THETA(Q) that, along with 96: * PHI(1), ...,PHI(Q-1), define the matrix in bidiagonal-block 97: * form. On exit, the angles whose cosines and sines define the 98: * diagonal blocks in the CS decomposition. 99: * 100: * PHI (input/workspace) DOUBLE PRECISION array, dimension (Q-1) 101: * The angles PHI(1),...,PHI(Q-1) that, along with THETA(1),..., 102: * THETA(Q), define the matrix in bidiagonal-block form. 103: * 104: * U1 (input/output) DOUBLE PRECISION array, dimension (LDU1,P) 105: * On entry, an LDU1-by-P matrix. On exit, U1 is postmultiplied 106: * by the left singular vector matrix common to [ B11 ; 0 ] and 107: * [ B12 0 0 ; 0 -I 0 0 ]. 108: * 109: * LDU1 (input) INTEGER 110: * The leading dimension of the array U1. 111: * 112: * U2 (input/output) DOUBLE PRECISION array, dimension (LDU2,M-P) 113: * On entry, an LDU2-by-(M-P) matrix. On exit, U2 is 114: * postmultiplied by the left singular vector matrix common to 115: * [ B21 ; 0 ] and [ B22 0 0 ; 0 0 I ]. 116: * 117: * LDU2 (input) INTEGER 118: * The leading dimension of the array U2. 119: * 120: * V1T (input/output) DOUBLE PRECISION array, dimension (LDV1T,Q) 121: * On entry, a LDV1T-by-Q matrix. On exit, V1T is premultiplied 122: * by the transpose of the right singular vector 123: * matrix common to [ B11 ; 0 ] and [ B21 ; 0 ]. 124: * 125: * LDV1T (input) INTEGER 126: * The leading dimension of the array V1T. 127: * 128: * V2T (input/output) DOUBLE PRECISION array, dimenison (LDV2T,M-Q) 129: * On entry, a LDV2T-by-(M-Q) matrix. On exit, V2T is 130: * premultiplied by the transpose of the right 131: * singular vector matrix common to [ B12 0 0 ; 0 -I 0 ] and 132: * [ B22 0 0 ; 0 0 I ]. 133: * 134: * LDV2T (input) INTEGER 135: * The leading dimension of the array V2T. 136: * 137: * B11D (output) DOUBLE PRECISION array, dimension (Q) 138: * When DBBCSD converges, B11D contains the cosines of THETA(1), 139: * ..., THETA(Q). If DBBCSD fails to converge, then B11D 140: * contains the diagonal of the partially reduced top-left 141: * block. 142: * 143: * B11E (output) DOUBLE PRECISION array, dimension (Q-1) 144: * When DBBCSD converges, B11E contains zeros. If DBBCSD fails 145: * to converge, then B11E contains the superdiagonal of the 146: * partially reduced top-left block. 147: * 148: * B12D (output) DOUBLE PRECISION array, dimension (Q) 149: * When DBBCSD converges, B12D contains the negative sines of 150: * THETA(1), ..., THETA(Q). If DBBCSD fails to converge, then 151: * B12D contains the diagonal of the partially reduced top-right 152: * block. 153: * 154: * B12E (output) DOUBLE PRECISION array, dimension (Q-1) 155: * When DBBCSD converges, B12E contains zeros. If DBBCSD fails 156: * to converge, then B12E contains the subdiagonal of the 157: * partially reduced top-right block. 158: * 159: * WORK (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) 160: * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 161: * 162: * LWORK (input) INTEGER 163: * The dimension of the array WORK. LWORK >= MAX(1,8*Q). 164: * 165: * If LWORK = -1, then a workspace query is assumed; the 166: * routine only calculates the optimal size of the WORK array, 167: * returns this value as the first entry of the work array, and 168: * no error message related to LWORK is issued by XERBLA. 169: * 170: * INFO (output) INTEGER 171: * = 0: successful exit. 172: * < 0: if INFO = -i, the i-th argument had an illegal value. 173: * > 0: if DBBCSD did not converge, INFO specifies the number 174: * of nonzero entries in PHI, and B11D, B11E, etc., 175: * contain the partially reduced matrix. 176: * 177: * Reference 178: * ========= 179: * 180: * [1] Brian D. Sutton. Computing the complete CS decomposition. Numer. 181: * Algorithms, 50(1):33-65, 2009. 182: * 183: * Internal Parameters 184: * =================== 185: * 186: * TOLMUL DOUBLE PRECISION, default = MAX(10,MIN(100,EPS**(-1/8))) 187: * TOLMUL controls the convergence criterion of the QR loop. 188: * Angles THETA(i), PHI(i) are rounded to 0 or PI/2 when they 189: * are within TOLMUL*EPS of either bound. 190: * 191: * =================================================================== 192: * 193: * .. Parameters .. 194: INTEGER MAXITR 195: PARAMETER ( MAXITR = 6 ) 196: DOUBLE PRECISION HUNDRED, MEIGHTH, ONE, PIOVER2, TEN, ZERO 197: PARAMETER ( HUNDRED = 100.0D0, MEIGHTH = -0.125D0, 198: $ ONE = 1.0D0, PIOVER2 = 1.57079632679489662D0, 199: $ TEN = 10.0D0, ZERO = 0.0D0 ) 200: DOUBLE PRECISION NEGONECOMPLEX 201: PARAMETER ( NEGONECOMPLEX = -1.0D0 ) 202: * .. 203: * .. Local Scalars .. 204: LOGICAL COLMAJOR, LQUERY, RESTART11, RESTART12, 205: $ RESTART21, RESTART22, WANTU1, WANTU2, WANTV1T, 206: $ WANTV2T 207: INTEGER I, IMIN, IMAX, ITER, IU1CS, IU1SN, IU2CS, 208: $ IU2SN, IV1TCS, IV1TSN, IV2TCS, IV2TSN, J, 209: $ LWORKMIN, LWORKOPT, MAXIT, MINI 210: DOUBLE PRECISION B11BULGE, B12BULGE, B21BULGE, B22BULGE, DUMMY, 211: $ EPS, MU, NU, R, SIGMA11, SIGMA21, 212: $ TEMP, THETAMAX, THETAMIN, THRESH, TOL, TOLMUL, 213: $ UNFL, X1, X2, Y1, Y2 214: * 215: * .. External Subroutines .. 216: EXTERNAL DLASR, DSCAL, DSWAP, DLARTGP, DLARTGS, DLAS2, 217: $ XERBLA 218: * .. 219: * .. External Functions .. 220: DOUBLE PRECISION DLAMCH 221: LOGICAL LSAME 222: EXTERNAL LSAME, DLAMCH 223: * .. 224: * .. Intrinsic Functions .. 225: INTRINSIC ABS, ATAN2, COS, MAX, MIN, SIN, SQRT 226: * .. 227: * .. Executable Statements .. 228: * 229: * Test input arguments 230: * 231: INFO = 0 232: LQUERY = LWORK .EQ. -1 233: WANTU1 = LSAME( JOBU1, 'Y' ) 234: WANTU2 = LSAME( JOBU2, 'Y' ) 235: WANTV1T = LSAME( JOBV1T, 'Y' ) 236: WANTV2T = LSAME( JOBV2T, 'Y' ) 237: COLMAJOR = .NOT. LSAME( TRANS, 'T' ) 238: * 239: IF( M .LT. 0 ) THEN 240: INFO = -6 241: ELSE IF( P .LT. 0 .OR. P .GT. M ) THEN 242: INFO = -7 243: ELSE IF( Q .LT. 0 .OR. Q .GT. M ) THEN 244: INFO = -8 245: ELSE IF( Q .GT. P .OR. Q .GT. M-P .OR. Q .GT. M-Q ) THEN 246: INFO = -8 247: ELSE IF( WANTU1 .AND. LDU1 .LT. P ) THEN 248: INFO = -12 249: ELSE IF( WANTU2 .AND. LDU2 .LT. M-P ) THEN 250: INFO = -14 251: ELSE IF( WANTV1T .AND. LDV1T .LT. Q ) THEN 252: INFO = -16 253: ELSE IF( WANTV2T .AND. LDV2T .LT. M-Q ) THEN 254: INFO = -18 255: END IF 256: * 257: * Quick return if Q = 0 258: * 259: IF( INFO .EQ. 0 .AND. Q .EQ. 0 ) THEN 260: LWORKMIN = 1 261: WORK(1) = LWORKMIN 262: RETURN 263: END IF 264: * 265: * Compute workspace 266: * 267: IF( INFO .EQ. 0 ) THEN 268: IU1CS = 1 269: IU1SN = IU1CS + Q 270: IU2CS = IU1SN + Q 271: IU2SN = IU2CS + Q 272: IV1TCS = IU2SN + Q 273: IV1TSN = IV1TCS + Q 274: IV2TCS = IV1TSN + Q 275: IV2TSN = IV2TCS + Q 276: LWORKOPT = IV2TSN + Q - 1 277: LWORKMIN = LWORKOPT 278: WORK(1) = LWORKOPT 279: IF( LWORK .LT. LWORKMIN .AND. .NOT. LQUERY ) THEN 280: INFO = -28 281: END IF 282: END IF 283: * 284: IF( INFO .NE. 0 ) THEN 285: CALL XERBLA( 'DBBCSD', -INFO ) 286: RETURN 287: ELSE IF( LQUERY ) THEN 288: RETURN 289: END IF 290: * 291: * Get machine constants 292: * 293: EPS = DLAMCH( 'Epsilon' ) 294: UNFL = DLAMCH( 'Safe minimum' ) 295: TOLMUL = MAX( TEN, MIN( HUNDRED, EPS**MEIGHTH ) ) 296: TOL = TOLMUL*EPS 297: THRESH = MAX( TOL, MAXITR*Q*Q*UNFL ) 298: * 299: * Test for negligible sines or cosines 300: * 301: DO I = 1, Q 302: IF( THETA(I) .LT. THRESH ) THEN 303: THETA(I) = ZERO 304: ELSE IF( THETA(I) .GT. PIOVER2-THRESH ) THEN 305: THETA(I) = PIOVER2 306: END IF 307: END DO 308: DO I = 1, Q-1 309: IF( PHI(I) .LT. THRESH ) THEN 310: PHI(I) = ZERO 311: ELSE IF( PHI(I) .GT. PIOVER2-THRESH ) THEN 312: PHI(I) = PIOVER2 313: END IF 314: END DO 315: * 316: * Initial deflation 317: * 318: IMAX = Q 319: DO WHILE( ( IMAX .GT. 1 ) .AND. ( PHI(IMAX-1) .EQ. ZERO ) ) 320: IMAX = IMAX - 1 321: END DO 322: IMIN = IMAX - 1 323: IF ( IMIN .GT. 1 ) THEN 324: DO WHILE( PHI(IMIN-1) .NE. ZERO ) 325: IMIN = IMIN - 1 326: IF ( IMIN .LE. 1 ) EXIT 327: END DO 328: END IF 329: * 330: * Initialize iteration counter 331: * 332: MAXIT = MAXITR*Q*Q 333: ITER = 0 334: * 335: * Begin main iteration loop 336: * 337: DO WHILE( IMAX .GT. 1 ) 338: * 339: * Compute the matrix entries 340: * 341: B11D(IMIN) = COS( THETA(IMIN) ) 342: B21D(IMIN) = -SIN( THETA(IMIN) ) 343: DO I = IMIN, IMAX - 1 344: B11E(I) = -SIN( THETA(I) ) * SIN( PHI(I) ) 345: B11D(I+1) = COS( THETA(I+1) ) * COS( PHI(I) ) 346: B12D(I) = SIN( THETA(I) ) * COS( PHI(I) ) 347: B12E(I) = COS( THETA(I+1) ) * SIN( PHI(I) ) 348: B21E(I) = -COS( THETA(I) ) * SIN( PHI(I) ) 349: B21D(I+1) = -SIN( THETA(I+1) ) * COS( PHI(I) ) 350: B22D(I) = COS( THETA(I) ) * COS( PHI(I) ) 351: B22E(I) = -SIN( THETA(I+1) ) * SIN( PHI(I) ) 352: END DO 353: B12D(IMAX) = SIN( THETA(IMAX) ) 354: B22D(IMAX) = COS( THETA(IMAX) ) 355: * 356: * Abort if not converging; otherwise, increment ITER 357: * 358: IF( ITER .GT. MAXIT ) THEN 359: INFO = 0 360: DO I = 1, Q 361: IF( PHI(I) .NE. ZERO ) 362: $ INFO = INFO + 1 363: END DO 364: RETURN 365: END IF 366: * 367: ITER = ITER + IMAX - IMIN 368: * 369: * Compute shifts 370: * 371: THETAMAX = THETA(IMIN) 372: THETAMIN = THETA(IMIN) 373: DO I = IMIN+1, IMAX 374: IF( THETA(I) > THETAMAX ) 375: $ THETAMAX = THETA(I) 376: IF( THETA(I) < THETAMIN ) 377: $ THETAMIN = THETA(I) 378: END DO 379: * 380: IF( THETAMAX .GT. PIOVER2 - THRESH ) THEN 381: * 382: * Zero on diagonals of B11 and B22; induce deflation with a 383: * zero shift 384: * 385: MU = ZERO 386: NU = ONE 387: * 388: ELSE IF( THETAMIN .LT. THRESH ) THEN 389: * 390: * Zero on diagonals of B12 and B22; induce deflation with a 391: * zero shift 392: * 393: MU = ONE 394: NU = ZERO 395: * 396: ELSE 397: * 398: * Compute shifts for B11 and B21 and use the lesser 399: * 400: CALL DLAS2( B11D(IMAX-1), B11E(IMAX-1), B11D(IMAX), SIGMA11, 401: $ DUMMY ) 402: CALL DLAS2( B21D(IMAX-1), B21E(IMAX-1), B21D(IMAX), SIGMA21, 403: $ DUMMY ) 404: * 405: IF( SIGMA11 .LE. SIGMA21 ) THEN 406: MU = SIGMA11 407: NU = SQRT( ONE - MU**2 ) 408: IF( MU .LT. THRESH ) THEN 409: MU = ZERO 410: NU = ONE 411: END IF 412: ELSE 413: NU = SIGMA21 414: MU = SQRT( 1.0 - NU**2 ) 415: IF( NU .LT. THRESH ) THEN 416: MU = ONE 417: NU = ZERO 418: END IF 419: END IF 420: END IF 421: * 422: * Rotate to produce bulges in B11 and B21 423: * 424: IF( MU .LE. NU ) THEN 425: CALL DLARTGS( B11D(IMIN), B11E(IMIN), MU, 426: $ WORK(IV1TCS+IMIN-1), WORK(IV1TSN+IMIN-1) ) 427: ELSE 428: CALL DLARTGS( B21D(IMIN), B21E(IMIN), NU, 429: $ WORK(IV1TCS+IMIN-1), WORK(IV1TSN+IMIN-1) ) 430: END IF 431: * 432: TEMP = WORK(IV1TCS+IMIN-1)*B11D(IMIN) + 433: $ WORK(IV1TSN+IMIN-1)*B11E(IMIN) 434: B11E(IMIN) = WORK(IV1TCS+IMIN-1)*B11E(IMIN) - 435: $ WORK(IV1TSN+IMIN-1)*B11D(IMIN) 436: B11D(IMIN) = TEMP 437: B11BULGE = WORK(IV1TSN+IMIN-1)*B11D(IMIN+1) 438: B11D(IMIN+1) = WORK(IV1TCS+IMIN-1)*B11D(IMIN+1) 439: TEMP = WORK(IV1TCS+IMIN-1)*B21D(IMIN) + 440: $ WORK(IV1TSN+IMIN-1)*B21E(IMIN) 441: B21E(IMIN) = WORK(IV1TCS+IMIN-1)*B21E(IMIN) - 442: $ WORK(IV1TSN+IMIN-1)*B21D(IMIN) 443: B21D(IMIN) = TEMP 444: B21BULGE = WORK(IV1TSN+IMIN-1)*B21D(IMIN+1) 445: B21D(IMIN+1) = WORK(IV1TCS+IMIN-1)*B21D(IMIN+1) 446: * 447: * Compute THETA(IMIN) 448: * 449: THETA( IMIN ) = ATAN2( SQRT( B21D(IMIN)**2+B21BULGE**2 ), 450: $ SQRT( B11D(IMIN)**2+B11BULGE**2 ) ) 451: * 452: * Chase the bulges in B11(IMIN+1,IMIN) and B21(IMIN+1,IMIN) 453: * 454: IF( B11D(IMIN)**2+B11BULGE**2 .GT. THRESH**2 ) THEN 455: CALL DLARTGP( B11BULGE, B11D(IMIN), WORK(IU1SN+IMIN-1), 456: $ WORK(IU1CS+IMIN-1), R ) 457: ELSE IF( MU .LE. NU ) THEN 458: CALL DLARTGS( B11E( IMIN ), B11D( IMIN + 1 ), MU, 459: $ WORK(IU1CS+IMIN-1), WORK(IU1SN+IMIN-1) ) 460: ELSE 461: CALL DLARTGS( B12D( IMIN ), B12E( IMIN ), NU, 462: $ WORK(IU1CS+IMIN-1), WORK(IU1SN+IMIN-1) ) 463: END IF 464: IF( B21D(IMIN)**2+B21BULGE**2 .GT. THRESH**2 ) THEN 465: CALL DLARTGP( B21BULGE, B21D(IMIN), WORK(IU2SN+IMIN-1), 466: $ WORK(IU2CS+IMIN-1), R ) 467: ELSE IF( NU .LT. MU ) THEN 468: CALL DLARTGS( B21E( IMIN ), B21D( IMIN + 1 ), NU, 469: $ WORK(IU2CS+IMIN-1), WORK(IU2SN+IMIN-1) ) 470: ELSE 471: CALL DLARTGS( B22D(IMIN), B22E(IMIN), MU, 472: $ WORK(IU2CS+IMIN-1), WORK(IU2SN+IMIN-1) ) 473: END IF 474: WORK(IU2CS+IMIN-1) = -WORK(IU2CS+IMIN-1) 475: WORK(IU2SN+IMIN-1) = -WORK(IU2SN+IMIN-1) 476: * 477: TEMP = WORK(IU1CS+IMIN-1)*B11E(IMIN) + 478: $ WORK(IU1SN+IMIN-1)*B11D(IMIN+1) 479: B11D(IMIN+1) = WORK(IU1CS+IMIN-1)*B11D(IMIN+1) - 480: $ WORK(IU1SN+IMIN-1)*B11E(IMIN) 481: B11E(IMIN) = TEMP 482: IF( IMAX .GT. IMIN+1 ) THEN 483: B11BULGE = WORK(IU1SN+IMIN-1)*B11E(IMIN+1) 484: B11E(IMIN+1) = WORK(IU1CS+IMIN-1)*B11E(IMIN+1) 485: END IF 486: TEMP = WORK(IU1CS+IMIN-1)*B12D(IMIN) + 487: $ WORK(IU1SN+IMIN-1)*B12E(IMIN) 488: B12E(IMIN) = WORK(IU1CS+IMIN-1)*B12E(IMIN) - 489: $ WORK(IU1SN+IMIN-1)*B12D(IMIN) 490: B12D(IMIN) = TEMP 491: B12BULGE = WORK(IU1SN+IMIN-1)*B12D(IMIN+1) 492: B12D(IMIN+1) = WORK(IU1CS+IMIN-1)*B12D(IMIN+1) 493: TEMP = WORK(IU2CS+IMIN-1)*B21E(IMIN) + 494: $ WORK(IU2SN+IMIN-1)*B21D(IMIN+1) 495: B21D(IMIN+1) = WORK(IU2CS+IMIN-1)*B21D(IMIN+1) - 496: $ WORK(IU2SN+IMIN-1)*B21E(IMIN) 497: B21E(IMIN) = TEMP 498: IF( IMAX .GT. IMIN+1 ) THEN 499: B21BULGE = WORK(IU2SN+IMIN-1)*B21E(IMIN+1) 500: B21E(IMIN+1) = WORK(IU2CS+IMIN-1)*B21E(IMIN+1) 501: END IF 502: TEMP = WORK(IU2CS+IMIN-1)*B22D(IMIN) + 503: $ WORK(IU2SN+IMIN-1)*B22E(IMIN) 504: B22E(IMIN) = WORK(IU2CS+IMIN-1)*B22E(IMIN) - 505: $ WORK(IU2SN+IMIN-1)*B22D(IMIN) 506: B22D(IMIN) = TEMP 507: B22BULGE = WORK(IU2SN+IMIN-1)*B22D(IMIN+1) 508: B22D(IMIN+1) = WORK(IU2CS+IMIN-1)*B22D(IMIN+1) 509: * 510: * Inner loop: chase bulges from B11(IMIN,IMIN+2), 511: * B12(IMIN,IMIN+1), B21(IMIN,IMIN+2), and B22(IMIN,IMIN+1) to 512: * bottom-right 513: * 514: DO I = IMIN+1, IMAX-1 515: * 516: * Compute PHI(I-1) 517: * 518: X1 = SIN(THETA(I-1))*B11E(I-1) + COS(THETA(I-1))*B21E(I-1) 519: X2 = SIN(THETA(I-1))*B11BULGE + COS(THETA(I-1))*B21BULGE 520: Y1 = SIN(THETA(I-1))*B12D(I-1) + COS(THETA(I-1))*B22D(I-1) 521: Y2 = SIN(THETA(I-1))*B12BULGE + COS(THETA(I-1))*B22BULGE 522: * 523: PHI(I-1) = ATAN2( SQRT(X1**2+X2**2), SQRT(Y1**2+Y2**2) ) 524: * 525: * Determine if there are bulges to chase or if a new direct 526: * summand has been reached 527: * 528: RESTART11 = B11E(I-1)**2 + B11BULGE**2 .LE. THRESH**2 529: RESTART21 = B21E(I-1)**2 + B21BULGE**2 .LE. THRESH**2 530: RESTART12 = B12D(I-1)**2 + B12BULGE**2 .LE. THRESH**2 531: RESTART22 = B22D(I-1)**2 + B22BULGE**2 .LE. THRESH**2 532: * 533: * If possible, chase bulges from B11(I-1,I+1), B12(I-1,I), 534: * B21(I-1,I+1), and B22(I-1,I). If necessary, restart bulge- 535: * chasing by applying the original shift again. 536: * 537: IF( .NOT. RESTART11 .AND. .NOT. RESTART21 ) THEN 538: CALL DLARTGP( X2, X1, WORK(IV1TSN+I-1), WORK(IV1TCS+I-1), 539: $ R ) 540: ELSE IF( .NOT. RESTART11 .AND. RESTART21 ) THEN 541: CALL DLARTGP( B11BULGE, B11E(I-1), WORK(IV1TSN+I-1), 542: $ WORK(IV1TCS+I-1), R ) 543: ELSE IF( RESTART11 .AND. .NOT. RESTART21 ) THEN 544: CALL DLARTGP( B21BULGE, B21E(I-1), WORK(IV1TSN+I-1), 545: $ WORK(IV1TCS+I-1), R ) 546: ELSE IF( MU .LE. NU ) THEN 547: CALL DLARTGS( B11D(I), B11E(I), MU, WORK(IV1TCS+I-1), 548: $ WORK(IV1TSN+I-1) ) 549: ELSE 550: CALL DLARTGS( B21D(I), B21E(I), NU, WORK(IV1TCS+I-1), 551: $ WORK(IV1TSN+I-1) ) 552: END IF 553: WORK(IV1TCS+I-1) = -WORK(IV1TCS+I-1) 554: WORK(IV1TSN+I-1) = -WORK(IV1TSN+I-1) 555: IF( .NOT. RESTART12 .AND. .NOT. RESTART22 ) THEN 556: CALL DLARTGP( Y2, Y1, WORK(IV2TSN+I-1-1), 557: $ WORK(IV2TCS+I-1-1), R ) 558: ELSE IF( .NOT. RESTART12 .AND. RESTART22 ) THEN 559: CALL DLARTGP( B12BULGE, B12D(I-1), WORK(IV2TSN+I-1-1), 560: $ WORK(IV2TCS+I-1-1), R ) 561: ELSE IF( RESTART12 .AND. .NOT. RESTART22 ) THEN 562: CALL DLARTGP( B22BULGE, B22D(I-1), WORK(IV2TSN+I-1-1), 563: $ WORK(IV2TCS+I-1-1), R ) 564: ELSE IF( NU .LT. MU ) THEN 565: CALL DLARTGS( B12E(I-1), B12D(I), NU, WORK(IV2TCS+I-1-1), 566: $ WORK(IV2TSN+I-1-1) ) 567: ELSE 568: CALL DLARTGS( B22E(I-1), B22D(I), MU, WORK(IV2TCS+I-1-1), 569: $ WORK(IV2TSN+I-1-1) ) 570: END IF 571: * 572: TEMP = WORK(IV1TCS+I-1)*B11D(I) + WORK(IV1TSN+I-1)*B11E(I) 573: B11E(I) = WORK(IV1TCS+I-1)*B11E(I) - 574: $ WORK(IV1TSN+I-1)*B11D(I) 575: B11D(I) = TEMP 576: B11BULGE = WORK(IV1TSN+I-1)*B11D(I+1) 577: B11D(I+1) = WORK(IV1TCS+I-1)*B11D(I+1) 578: TEMP = WORK(IV1TCS+I-1)*B21D(I) + WORK(IV1TSN+I-1)*B21E(I) 579: B21E(I) = WORK(IV1TCS+I-1)*B21E(I) - 580: $ WORK(IV1TSN+I-1)*B21D(I) 581: B21D(I) = TEMP 582: B21BULGE = WORK(IV1TSN+I-1)*B21D(I+1) 583: B21D(I+1) = WORK(IV1TCS+I-1)*B21D(I+1) 584: TEMP = WORK(IV2TCS+I-1-1)*B12E(I-1) + 585: $ WORK(IV2TSN+I-1-1)*B12D(I) 586: B12D(I) = WORK(IV2TCS+I-1-1)*B12D(I) - 587: $ WORK(IV2TSN+I-1-1)*B12E(I-1) 588: B12E(I-1) = TEMP 589: B12BULGE = WORK(IV2TSN+I-1-1)*B12E(I) 590: B12E(I) = WORK(IV2TCS+I-1-1)*B12E(I) 591: TEMP = WORK(IV2TCS+I-1-1)*B22E(I-1) + 592: $ WORK(IV2TSN+I-1-1)*B22D(I) 593: B22D(I) = WORK(IV2TCS+I-1-1)*B22D(I) - 594: $ WORK(IV2TSN+I-1-1)*B22E(I-1) 595: B22E(I-1) = TEMP 596: B22BULGE = WORK(IV2TSN+I-1-1)*B22E(I) 597: B22E(I) = WORK(IV2TCS+I-1-1)*B22E(I) 598: * 599: * Compute THETA(I) 600: * 601: X1 = COS(PHI(I-1))*B11D(I) + SIN(PHI(I-1))*B12E(I-1) 602: X2 = COS(PHI(I-1))*B11BULGE + SIN(PHI(I-1))*B12BULGE 603: Y1 = COS(PHI(I-1))*B21D(I) + SIN(PHI(I-1))*B22E(I-1) 604: Y2 = COS(PHI(I-1))*B21BULGE + SIN(PHI(I-1))*B22BULGE 605: * 606: THETA(I) = ATAN2( SQRT(Y1**2+Y2**2), SQRT(X1**2+X2**2) ) 607: * 608: * Determine if there are bulges to chase or if a new direct 609: * summand has been reached 610: * 611: RESTART11 = B11D(I)**2 + B11BULGE**2 .LE. THRESH**2 612: RESTART12 = B12E(I-1)**2 + B12BULGE**2 .LE. THRESH**2 613: RESTART21 = B21D(I)**2 + B21BULGE**2 .LE. THRESH**2 614: RESTART22 = B22E(I-1)**2 + B22BULGE**2 .LE. THRESH**2 615: * 616: * If possible, chase bulges from B11(I+1,I), B12(I+1,I-1), 617: * B21(I+1,I), and B22(I+1,I-1). If necessary, restart bulge- 618: * chasing by applying the original shift again. 619: * 620: IF( .NOT. RESTART11 .AND. .NOT. RESTART12 ) THEN 621: CALL DLARTGP( X2, X1, WORK(IU1SN+I-1), WORK(IU1CS+I-1), 622: $ R ) 623: ELSE IF( .NOT. RESTART11 .AND. RESTART12 ) THEN 624: CALL DLARTGP( B11BULGE, B11D(I), WORK(IU1SN+I-1), 625: $ WORK(IU1CS+I-1), R ) 626: ELSE IF( RESTART11 .AND. .NOT. RESTART12 ) THEN 627: CALL DLARTGP( B12BULGE, B12E(I-1), WORK(IU1SN+I-1), 628: $ WORK(IU1CS+I-1), R ) 629: ELSE IF( MU .LE. NU ) THEN 630: CALL DLARTGS( B11E(I), B11D(I+1), MU, WORK(IU1CS+I-1), 631: $ WORK(IU1SN+I-1) ) 632: ELSE 633: CALL DLARTGS( B12D(I), B12E(I), NU, WORK(IU1CS+I-1), 634: $ WORK(IU1SN+I-1) ) 635: END IF 636: IF( .NOT. RESTART21 .AND. .NOT. RESTART22 ) THEN 637: CALL DLARTGP( Y2, Y1, WORK(IU2SN+I-1), WORK(IU2CS+I-1), 638: $ R ) 639: ELSE IF( .NOT. RESTART21 .AND. RESTART22 ) THEN 640: CALL DLARTGP( B21BULGE, B21D(I), WORK(IU2SN+I-1), 641: $ WORK(IU2CS+I-1), R ) 642: ELSE IF( RESTART21 .AND. .NOT. RESTART22 ) THEN 643: CALL DLARTGP( B22BULGE, B22E(I-1), WORK(IU2SN+I-1), 644: $ WORK(IU2CS+I-1), R ) 645: ELSE IF( NU .LT. MU ) THEN 646: CALL DLARTGS( B21E(I), B21E(I+1), NU, WORK(IU2CS+I-1), 647: $ WORK(IU2SN+I-1) ) 648: ELSE 649: CALL DLARTGS( B22D(I), B22E(I), MU, WORK(IU2CS+I-1), 650: $ WORK(IU2SN+I-1) ) 651: END IF 652: WORK(IU2CS+I-1) = -WORK(IU2CS+I-1) 653: WORK(IU2SN+I-1) = -WORK(IU2SN+I-1) 654: * 655: TEMP = WORK(IU1CS+I-1)*B11E(I) + WORK(IU1SN+I-1)*B11D(I+1) 656: B11D(I+1) = WORK(IU1CS+I-1)*B11D(I+1) - 657: $ WORK(IU1SN+I-1)*B11E(I) 658: B11E(I) = TEMP 659: IF( I .LT. IMAX - 1 ) THEN 660: B11BULGE = WORK(IU1SN+I-1)*B11E(I+1) 661: B11E(I+1) = WORK(IU1CS+I-1)*B11E(I+1) 662: END IF 663: TEMP = WORK(IU2CS+I-1)*B21E(I) + WORK(IU2SN+I-1)*B21D(I+1) 664: B21D(I+1) = WORK(IU2CS+I-1)*B21D(I+1) - 665: $ WORK(IU2SN+I-1)*B21E(I) 666: B21E(I) = TEMP 667: IF( I .LT. IMAX - 1 ) THEN 668: B21BULGE = WORK(IU2SN+I-1)*B21E(I+1) 669: B21E(I+1) = WORK(IU2CS+I-1)*B21E(I+1) 670: END IF 671: TEMP = WORK(IU1CS+I-1)*B12D(I) + WORK(IU1SN+I-1)*B12E(I) 672: B12E(I) = WORK(IU1CS+I-1)*B12E(I) - WORK(IU1SN+I-1)*B12D(I) 673: B12D(I) = TEMP 674: B12BULGE = WORK(IU1SN+I-1)*B12D(I+1) 675: B12D(I+1) = WORK(IU1CS+I-1)*B12D(I+1) 676: TEMP = WORK(IU2CS+I-1)*B22D(I) + WORK(IU2SN+I-1)*B22E(I) 677: B22E(I) = WORK(IU2CS+I-1)*B22E(I) - WORK(IU2SN+I-1)*B22D(I) 678: B22D(I) = TEMP 679: B22BULGE = WORK(IU2SN+I-1)*B22D(I+1) 680: B22D(I+1) = WORK(IU2CS+I-1)*B22D(I+1) 681: * 682: END DO 683: * 684: * Compute PHI(IMAX-1) 685: * 686: X1 = SIN(THETA(IMAX-1))*B11E(IMAX-1) + 687: $ COS(THETA(IMAX-1))*B21E(IMAX-1) 688: Y1 = SIN(THETA(IMAX-1))*B12D(IMAX-1) + 689: $ COS(THETA(IMAX-1))*B22D(IMAX-1) 690: Y2 = SIN(THETA(IMAX-1))*B12BULGE + COS(THETA(IMAX-1))*B22BULGE 691: * 692: PHI(IMAX-1) = ATAN2( ABS(X1), SQRT(Y1**2+Y2**2) ) 693: * 694: * Chase bulges from B12(IMAX-1,IMAX) and B22(IMAX-1,IMAX) 695: * 696: RESTART12 = B12D(IMAX-1)**2 + B12BULGE**2 .LE. THRESH**2 697: RESTART22 = B22D(IMAX-1)**2 + B22BULGE**2 .LE. THRESH**2 698: * 699: IF( .NOT. RESTART12 .AND. .NOT. RESTART22 ) THEN 700: CALL DLARTGP( Y2, Y1, WORK(IV2TSN+IMAX-1-1), 701: $ WORK(IV2TCS+IMAX-1-1), R ) 702: ELSE IF( .NOT. RESTART12 .AND. RESTART22 ) THEN 703: CALL DLARTGP( B12BULGE, B12D(IMAX-1), WORK(IV2TSN+IMAX-1-1), 704: $ WORK(IV2TCS+IMAX-1-1), R ) 705: ELSE IF( RESTART12 .AND. .NOT. RESTART22 ) THEN 706: CALL DLARTGP( B22BULGE, B22D(IMAX-1), WORK(IV2TSN+IMAX-1-1), 707: $ WORK(IV2TCS+IMAX-1-1), R ) 708: ELSE IF( NU .LT. MU ) THEN 709: CALL DLARTGS( B12E(IMAX-1), B12D(IMAX), NU, 710: $ WORK(IV2TCS+IMAX-1-1), WORK(IV2TSN+IMAX-1-1) ) 711: ELSE 712: CALL DLARTGS( B22E(IMAX-1), B22D(IMAX), MU, 713: $ WORK(IV2TCS+IMAX-1-1), WORK(IV2TSN+IMAX-1-1) ) 714: END IF 715: * 716: TEMP = WORK(IV2TCS+IMAX-1-1)*B12E(IMAX-1) + 717: $ WORK(IV2TSN+IMAX-1-1)*B12D(IMAX) 718: B12D(IMAX) = WORK(IV2TCS+IMAX-1-1)*B12D(IMAX) - 719: $ WORK(IV2TSN+IMAX-1-1)*B12E(IMAX-1) 720: B12E(IMAX-1) = TEMP 721: TEMP = WORK(IV2TCS+IMAX-1-1)*B22E(IMAX-1) + 722: $ WORK(IV2TSN+IMAX-1-1)*B22D(IMAX) 723: B22D(IMAX) = WORK(IV2TCS+IMAX-1-1)*B22D(IMAX) - 724: $ WORK(IV2TSN+IMAX-1-1)*B22E(IMAX-1) 725: B22E(IMAX-1) = TEMP 726: * 727: * Update singular vectors 728: * 729: IF( WANTU1 ) THEN 730: IF( COLMAJOR ) THEN 731: CALL DLASR( 'R', 'V', 'F', P, IMAX-IMIN+1, 732: $ WORK(IU1CS+IMIN-1), WORK(IU1SN+IMIN-1), 733: $ U1(1,IMIN), LDU1 ) 734: ELSE 735: CALL DLASR( 'L', 'V', 'F', IMAX-IMIN+1, P, 736: $ WORK(IU1CS+IMIN-1), WORK(IU1SN+IMIN-1), 737: $ U1(IMIN,1), LDU1 ) 738: END IF 739: END IF 740: IF( WANTU2 ) THEN 741: IF( COLMAJOR ) THEN 742: CALL DLASR( 'R', 'V', 'F', M-P, IMAX-IMIN+1, 743: $ WORK(IU2CS+IMIN-1), WORK(IU2SN+IMIN-1), 744: $ U2(1,IMIN), LDU2 ) 745: ELSE 746: CALL DLASR( 'L', 'V', 'F', IMAX-IMIN+1, M-P, 747: $ WORK(IU2CS+IMIN-1), WORK(IU2SN+IMIN-1), 748: $ U2(IMIN,1), LDU2 ) 749: END IF 750: END IF 751: IF( WANTV1T ) THEN 752: IF( COLMAJOR ) THEN 753: CALL DLASR( 'L', 'V', 'F', IMAX-IMIN+1, Q, 754: $ WORK(IV1TCS+IMIN-1), WORK(IV1TSN+IMIN-1), 755: $ V1T(IMIN,1), LDV1T ) 756: ELSE 757: CALL DLASR( 'R', 'V', 'F', Q, IMAX-IMIN+1, 758: $ WORK(IV1TCS+IMIN-1), WORK(IV1TSN+IMIN-1), 759: $ V1T(1,IMIN), LDV1T ) 760: END IF 761: END IF 762: IF( WANTV2T ) THEN 763: IF( COLMAJOR ) THEN 764: CALL DLASR( 'L', 'V', 'F', IMAX-IMIN+1, M-Q, 765: $ WORK(IV2TCS+IMIN-1), WORK(IV2TSN+IMIN-1), 766: $ V2T(IMIN,1), LDV2T ) 767: ELSE 768: CALL DLASR( 'R', 'V', 'F', M-Q, IMAX-IMIN+1, 769: $ WORK(IV2TCS+IMIN-1), WORK(IV2TSN+IMIN-1), 770: $ V2T(1,IMIN), LDV2T ) 771: END IF 772: END IF 773: * 774: * Fix signs on B11(IMAX-1,IMAX) and B21(IMAX-1,IMAX) 775: * 776: IF( B11E(IMAX-1)+B21E(IMAX-1) .GT. 0 ) THEN 777: B11D(IMAX) = -B11D(IMAX) 778: B21D(IMAX) = -B21D(IMAX) 779: IF( WANTV1T ) THEN 780: IF( COLMAJOR ) THEN 781: CALL DSCAL( Q, NEGONECOMPLEX, V1T(IMAX,1), LDV1T ) 782: ELSE 783: CALL DSCAL( Q, NEGONECOMPLEX, V1T(1,IMAX), 1 ) 784: END IF 785: END IF 786: END IF 787: * 788: * Compute THETA(IMAX) 789: * 790: X1 = COS(PHI(IMAX-1))*B11D(IMAX) + 791: $ SIN(PHI(IMAX-1))*B12E(IMAX-1) 792: Y1 = COS(PHI(IMAX-1))*B21D(IMAX) + 793: $ SIN(PHI(IMAX-1))*B22E(IMAX-1) 794: * 795: THETA(IMAX) = ATAN2( ABS(Y1), ABS(X1) ) 796: * 797: * Fix signs on B11(IMAX,IMAX), B12(IMAX,IMAX-1), B21(IMAX,IMAX), 798: * and B22(IMAX,IMAX-1) 799: * 800: IF( B11D(IMAX)+B12E(IMAX-1) .LT. 0 ) THEN 801: B12D(IMAX) = -B12D(IMAX) 802: IF( WANTU1 ) THEN 803: IF( COLMAJOR ) THEN 804: CALL DSCAL( P, NEGONECOMPLEX, U1(1,IMAX), 1 ) 805: ELSE 806: CALL DSCAL( P, NEGONECOMPLEX, U1(IMAX,1), LDU1 ) 807: END IF 808: END IF 809: END IF 810: IF( B21D(IMAX)+B22E(IMAX-1) .GT. 0 ) THEN 811: B22D(IMAX) = -B22D(IMAX) 812: IF( WANTU2 ) THEN 813: IF( COLMAJOR ) THEN 814: CALL DSCAL( M-P, NEGONECOMPLEX, U2(1,IMAX), 1 ) 815: ELSE 816: CALL DSCAL( M-P, NEGONECOMPLEX, U2(IMAX,1), LDU2 ) 817: END IF 818: END IF 819: END IF 820: * 821: * Fix signs on B12(IMAX,IMAX) and B22(IMAX,IMAX) 822: * 823: IF( B12D(IMAX)+B22D(IMAX) .LT. 0 ) THEN 824: IF( WANTV2T ) THEN 825: IF( COLMAJOR ) THEN 826: CALL DSCAL( M-Q, NEGONECOMPLEX, V2T(IMAX,1), LDV2T ) 827: ELSE 828: CALL DSCAL( M-Q, NEGONECOMPLEX, V2T(1,IMAX), 1 ) 829: END IF 830: END IF 831: END IF 832: * 833: * Test for negligible sines or cosines 834: * 835: DO I = IMIN, IMAX 836: IF( THETA(I) .LT. THRESH ) THEN 837: THETA(I) = ZERO 838: ELSE IF( THETA(I) .GT. PIOVER2-THRESH ) THEN 839: THETA(I) = PIOVER2 840: END IF 841: END DO 842: DO I = IMIN, IMAX-1 843: IF( PHI(I) .LT. THRESH ) THEN 844: PHI(I) = ZERO 845: ELSE IF( PHI(I) .GT. PIOVER2-THRESH ) THEN 846: PHI(I) = PIOVER2 847: END IF 848: END DO 849: * 850: * Deflate 851: * 852: IF (IMAX .GT. 1) THEN 853: DO WHILE( PHI(IMAX-1) .EQ. ZERO ) 854: IMAX = IMAX - 1 855: IF (IMAX .LE. 1) EXIT 856: END DO 857: END IF 858: IF( IMIN .GT. IMAX - 1 ) 859: $ IMIN = IMAX - 1 860: IF (IMIN .GT. 1) THEN 861: DO WHILE (PHI(IMIN-1) .NE. ZERO) 862: IMIN = IMIN - 1 863: IF (IMIN .LE. 1) EXIT 864: END DO 865: END IF 866: * 867: * Repeat main iteration loop 868: * 869: END DO 870: * 871: * Postprocessing: order THETA from least to greatest 872: * 873: DO I = 1, Q 874: * 875: MINI = I 876: THETAMIN = THETA(I) 877: DO J = I+1, Q 878: IF( THETA(J) .LT. THETAMIN ) THEN 879: MINI = J 880: THETAMIN = THETA(J) 881: END IF 882: END DO 883: * 884: IF( MINI .NE. I ) THEN 885: THETA(MINI) = THETA(I) 886: THETA(I) = THETAMIN 887: IF( COLMAJOR ) THEN 888: IF( WANTU1 ) 889: $ CALL DSWAP( P, U1(1,I), 1, U1(1,MINI), 1 ) 890: IF( WANTU2 ) 891: $ CALL DSWAP( M-P, U2(1,I), 1, U2(1,MINI), 1 ) 892: IF( WANTV1T ) 893: $ CALL DSWAP( Q, V1T(I,1), LDV1T, V1T(MINI,1), LDV1T ) 894: IF( WANTV2T ) 895: $ CALL DSWAP( M-Q, V2T(I,1), LDV2T, V2T(MINI,1), 896: $ LDV2T ) 897: ELSE 898: IF( WANTU1 ) 899: $ CALL DSWAP( P, U1(I,1), LDU1, U1(MINI,1), LDU1 ) 900: IF( WANTU2 ) 901: $ CALL DSWAP( M-P, U2(I,1), LDU2, U2(MINI,1), LDU2 ) 902: IF( WANTV1T ) 903: $ CALL DSWAP( Q, V1T(1,I), 1, V1T(1,MINI), 1 ) 904: IF( WANTV2T ) 905: $ CALL DSWAP( M-Q, V2T(1,I), 1, V2T(1,MINI), 1 ) 906: END IF 907: END IF 908: * 909: END DO 910: * 911: RETURN 912: * 913: * End of DBBCSD 914: * 915: END 916: