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Mise à jour de lapack vers la version 3.3.0.
1: SUBROUTINE ZLASYF( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO ) 2: * 3: * -- LAPACK routine (version 3.2) -- 4: * -- LAPACK is a software package provided by Univ. of Tennessee, -- 5: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 6: * November 2006 7: * 8: * .. Scalar Arguments .. 9: CHARACTER UPLO 10: INTEGER INFO, KB, LDA, LDW, N, NB 11: * .. 12: * .. Array Arguments .. 13: INTEGER IPIV( * ) 14: COMPLEX*16 A( LDA, * ), W( LDW, * ) 15: * .. 16: * 17: * Purpose 18: * ======= 19: * 20: * ZLASYF computes a partial factorization of a complex symmetric matrix 21: * A using the Bunch-Kaufman diagonal pivoting method. The partial 22: * factorization has the form: 23: * 24: * A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or: 25: * ( 0 U22 ) ( 0 D ) ( U12' U22' ) 26: * 27: * A = ( L11 0 ) ( D 0 ) ( L11' L21' ) if UPLO = 'L' 28: * ( L21 I ) ( 0 A22 ) ( 0 I ) 29: * 30: * where the order of D is at most NB. The actual order is returned in 31: * the argument KB, and is either NB or NB-1, or N if N <= NB. 32: * Note that U' denotes the transpose of U. 33: * 34: * ZLASYF is an auxiliary routine called by ZSYTRF. It uses blocked code 35: * (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or 36: * A22 (if UPLO = 'L'). 37: * 38: * Arguments 39: * ========= 40: * 41: * UPLO (input) CHARACTER*1 42: * Specifies whether the upper or lower triangular part of the 43: * symmetric matrix A is stored: 44: * = 'U': Upper triangular 45: * = 'L': Lower triangular 46: * 47: * N (input) INTEGER 48: * The order of the matrix A. N >= 0. 49: * 50: * NB (input) INTEGER 51: * The maximum number of columns of the matrix A that should be 52: * factored. NB should be at least 2 to allow for 2-by-2 pivot 53: * blocks. 54: * 55: * KB (output) INTEGER 56: * The number of columns of A that were actually factored. 57: * KB is either NB-1 or NB, or N if N <= NB. 58: * 59: * A (input/output) COMPLEX*16 array, dimension (LDA,N) 60: * On entry, the symmetric matrix A. If UPLO = 'U', the leading 61: * n-by-n upper triangular part of A contains the upper 62: * triangular part of the matrix A, and the strictly lower 63: * triangular part of A is not referenced. If UPLO = 'L', the 64: * leading n-by-n lower triangular part of A contains the lower 65: * triangular part of the matrix A, and the strictly upper 66: * triangular part of A is not referenced. 67: * On exit, A contains details of the partial factorization. 68: * 69: * LDA (input) INTEGER 70: * The leading dimension of the array A. LDA >= max(1,N). 71: * 72: * IPIV (output) INTEGER array, dimension (N) 73: * Details of the interchanges and the block structure of D. 74: * If UPLO = 'U', only the last KB elements of IPIV are set; 75: * if UPLO = 'L', only the first KB elements are set. 76: * 77: * If IPIV(k) > 0, then rows and columns k and IPIV(k) were 78: * interchanged and D(k,k) is a 1-by-1 diagonal block. 79: * If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and 80: * columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) 81: * is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = 82: * IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were 83: * interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. 84: * 85: * W (workspace) COMPLEX*16 array, dimension (LDW,NB) 86: * 87: * LDW (input) INTEGER 88: * The leading dimension of the array W. LDW >= max(1,N). 89: * 90: * INFO (output) INTEGER 91: * = 0: successful exit 92: * > 0: if INFO = k, D(k,k) is exactly zero. The factorization 93: * has been completed, but the block diagonal matrix D is 94: * exactly singular. 95: * 96: * ===================================================================== 97: * 98: * .. Parameters .. 99: DOUBLE PRECISION ZERO, ONE 100: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 101: DOUBLE PRECISION EIGHT, SEVTEN 102: PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 ) 103: COMPLEX*16 CONE 104: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) ) 105: * .. 106: * .. Local Scalars .. 107: INTEGER IMAX, J, JB, JJ, JMAX, JP, K, KK, KKW, KP, 108: $ KSTEP, KW 109: DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, ROWMAX 110: COMPLEX*16 D11, D21, D22, R1, T, Z 111: * .. 112: * .. External Functions .. 113: LOGICAL LSAME 114: INTEGER IZAMAX 115: EXTERNAL LSAME, IZAMAX 116: * .. 117: * .. External Subroutines .. 118: EXTERNAL ZCOPY, ZGEMM, ZGEMV, ZSCAL, ZSWAP 119: * .. 120: * .. Intrinsic Functions .. 121: INTRINSIC ABS, DBLE, DIMAG, MAX, MIN, SQRT 122: * .. 123: * .. Statement Functions .. 124: DOUBLE PRECISION CABS1 125: * .. 126: * .. Statement Function definitions .. 127: CABS1( Z ) = ABS( DBLE( Z ) ) + ABS( DIMAG( Z ) ) 128: * .. 129: * .. Executable Statements .. 130: * 131: INFO = 0 132: * 133: * Initialize ALPHA for use in choosing pivot block size. 134: * 135: ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT 136: * 137: IF( LSAME( UPLO, 'U' ) ) THEN 138: * 139: * Factorize the trailing columns of A using the upper triangle 140: * of A and working backwards, and compute the matrix W = U12*D 141: * for use in updating A11 142: * 143: * K is the main loop index, decreasing from N in steps of 1 or 2 144: * 145: * KW is the column of W which corresponds to column K of A 146: * 147: K = N 148: 10 CONTINUE 149: KW = NB + K - N 150: * 151: * Exit from loop 152: * 153: IF( ( K.LE.N-NB+1 .AND. NB.LT.N ) .OR. K.LT.1 ) 154: $ GO TO 30 155: * 156: * Copy column K of A to column KW of W and update it 157: * 158: CALL ZCOPY( K, A( 1, K ), 1, W( 1, KW ), 1 ) 159: IF( K.LT.N ) 160: $ CALL ZGEMV( 'No transpose', K, N-K, -CONE, A( 1, K+1 ), LDA, 161: $ W( K, KW+1 ), LDW, CONE, W( 1, KW ), 1 ) 162: * 163: KSTEP = 1 164: * 165: * Determine rows and columns to be interchanged and whether 166: * a 1-by-1 or 2-by-2 pivot block will be used 167: * 168: ABSAKK = CABS1( W( K, KW ) ) 169: * 170: * IMAX is the row-index of the largest off-diagonal element in 171: * column K, and COLMAX is its absolute value 172: * 173: IF( K.GT.1 ) THEN 174: IMAX = IZAMAX( K-1, W( 1, KW ), 1 ) 175: COLMAX = CABS1( W( IMAX, KW ) ) 176: ELSE 177: COLMAX = ZERO 178: END IF 179: * 180: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN 181: * 182: * Column K is zero: set INFO and continue 183: * 184: IF( INFO.EQ.0 ) 185: $ INFO = K 186: KP = K 187: ELSE 188: IF( ABSAKK.GE.ALPHA*COLMAX ) THEN 189: * 190: * no interchange, use 1-by-1 pivot block 191: * 192: KP = K 193: ELSE 194: * 195: * Copy column IMAX to column KW-1 of W and update it 196: * 197: CALL ZCOPY( IMAX, A( 1, IMAX ), 1, W( 1, KW-1 ), 1 ) 198: CALL ZCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA, 199: $ W( IMAX+1, KW-1 ), 1 ) 200: IF( K.LT.N ) 201: $ CALL ZGEMV( 'No transpose', K, N-K, -CONE, 202: $ A( 1, K+1 ), LDA, W( IMAX, KW+1 ), LDW, 203: $ CONE, W( 1, KW-1 ), 1 ) 204: * 205: * JMAX is the column-index of the largest off-diagonal 206: * element in row IMAX, and ROWMAX is its absolute value 207: * 208: JMAX = IMAX + IZAMAX( K-IMAX, W( IMAX+1, KW-1 ), 1 ) 209: ROWMAX = CABS1( W( JMAX, KW-1 ) ) 210: IF( IMAX.GT.1 ) THEN 211: JMAX = IZAMAX( IMAX-1, W( 1, KW-1 ), 1 ) 212: ROWMAX = MAX( ROWMAX, CABS1( W( JMAX, KW-1 ) ) ) 213: END IF 214: * 215: IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN 216: * 217: * no interchange, use 1-by-1 pivot block 218: * 219: KP = K 220: ELSE IF( CABS1( W( IMAX, KW-1 ) ).GE.ALPHA*ROWMAX ) THEN 221: * 222: * interchange rows and columns K and IMAX, use 1-by-1 223: * pivot block 224: * 225: KP = IMAX 226: * 227: * copy column KW-1 of W to column KW 228: * 229: CALL ZCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 ) 230: ELSE 231: * 232: * interchange rows and columns K-1 and IMAX, use 2-by-2 233: * pivot block 234: * 235: KP = IMAX 236: KSTEP = 2 237: END IF 238: END IF 239: * 240: KK = K - KSTEP + 1 241: KKW = NB + KK - N 242: * 243: * Updated column KP is already stored in column KKW of W 244: * 245: IF( KP.NE.KK ) THEN 246: * 247: * Copy non-updated column KK to column KP 248: * 249: A( KP, K ) = A( KK, K ) 250: CALL ZCOPY( K-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ), 251: $ LDA ) 252: CALL ZCOPY( KP, A( 1, KK ), 1, A( 1, KP ), 1 ) 253: * 254: * Interchange rows KK and KP in last KK columns of A and W 255: * 256: CALL ZSWAP( N-KK+1, A( KK, KK ), LDA, A( KP, KK ), LDA ) 257: CALL ZSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ), 258: $ LDW ) 259: END IF 260: * 261: IF( KSTEP.EQ.1 ) THEN 262: * 263: * 1-by-1 pivot block D(k): column KW of W now holds 264: * 265: * W(k) = U(k)*D(k) 266: * 267: * where U(k) is the k-th column of U 268: * 269: * Store U(k) in column k of A 270: * 271: CALL ZCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 ) 272: R1 = CONE / A( K, K ) 273: CALL ZSCAL( K-1, R1, A( 1, K ), 1 ) 274: ELSE 275: * 276: * 2-by-2 pivot block D(k): columns KW and KW-1 of W now 277: * hold 278: * 279: * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k) 280: * 281: * where U(k) and U(k-1) are the k-th and (k-1)-th columns 282: * of U 283: * 284: IF( K.GT.2 ) THEN 285: * 286: * Store U(k) and U(k-1) in columns k and k-1 of A 287: * 288: D21 = W( K-1, KW ) 289: D11 = W( K, KW ) / D21 290: D22 = W( K-1, KW-1 ) / D21 291: T = CONE / ( D11*D22-CONE ) 292: D21 = T / D21 293: DO 20 J = 1, K - 2 294: A( J, K-1 ) = D21*( D11*W( J, KW-1 )-W( J, KW ) ) 295: A( J, K ) = D21*( D22*W( J, KW )-W( J, KW-1 ) ) 296: 20 CONTINUE 297: END IF 298: * 299: * Copy D(k) to A 300: * 301: A( K-1, K-1 ) = W( K-1, KW-1 ) 302: A( K-1, K ) = W( K-1, KW ) 303: A( K, K ) = W( K, KW ) 304: END IF 305: END IF 306: * 307: * Store details of the interchanges in IPIV 308: * 309: IF( KSTEP.EQ.1 ) THEN 310: IPIV( K ) = KP 311: ELSE 312: IPIV( K ) = -KP 313: IPIV( K-1 ) = -KP 314: END IF 315: * 316: * Decrease K and return to the start of the main loop 317: * 318: K = K - KSTEP 319: GO TO 10 320: * 321: 30 CONTINUE 322: * 323: * Update the upper triangle of A11 (= A(1:k,1:k)) as 324: * 325: * A11 := A11 - U12*D*U12' = A11 - U12*W' 326: * 327: * computing blocks of NB columns at a time 328: * 329: DO 50 J = ( ( K-1 ) / NB )*NB + 1, 1, -NB 330: JB = MIN( NB, K-J+1 ) 331: * 332: * Update the upper triangle of the diagonal block 333: * 334: DO 40 JJ = J, J + JB - 1 335: CALL ZGEMV( 'No transpose', JJ-J+1, N-K, -CONE, 336: $ A( J, K+1 ), LDA, W( JJ, KW+1 ), LDW, CONE, 337: $ A( J, JJ ), 1 ) 338: 40 CONTINUE 339: * 340: * Update the rectangular superdiagonal block 341: * 342: CALL ZGEMM( 'No transpose', 'Transpose', J-1, JB, N-K, 343: $ -CONE, A( 1, K+1 ), LDA, W( J, KW+1 ), LDW, 344: $ CONE, A( 1, J ), LDA ) 345: 50 CONTINUE 346: * 347: * Put U12 in standard form by partially undoing the interchanges 348: * in columns k+1:n 349: * 350: J = K + 1 351: 60 CONTINUE 352: JJ = J 353: JP = IPIV( J ) 354: IF( JP.LT.0 ) THEN 355: JP = -JP 356: J = J + 1 357: END IF 358: J = J + 1 359: IF( JP.NE.JJ .AND. J.LE.N ) 360: $ CALL ZSWAP( N-J+1, A( JP, J ), LDA, A( JJ, J ), LDA ) 361: IF( J.LE.N ) 362: $ GO TO 60 363: * 364: * Set KB to the number of columns factorized 365: * 366: KB = N - K 367: * 368: ELSE 369: * 370: * Factorize the leading columns of A using the lower triangle 371: * of A and working forwards, and compute the matrix W = L21*D 372: * for use in updating A22 373: * 374: * K is the main loop index, increasing from 1 in steps of 1 or 2 375: * 376: K = 1 377: 70 CONTINUE 378: * 379: * Exit from loop 380: * 381: IF( ( K.GE.NB .AND. NB.LT.N ) .OR. K.GT.N ) 382: $ GO TO 90 383: * 384: * Copy column K of A to column K of W and update it 385: * 386: CALL ZCOPY( N-K+1, A( K, K ), 1, W( K, K ), 1 ) 387: CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE, A( K, 1 ), LDA, 388: $ W( K, 1 ), LDW, CONE, W( K, K ), 1 ) 389: * 390: KSTEP = 1 391: * 392: * Determine rows and columns to be interchanged and whether 393: * a 1-by-1 or 2-by-2 pivot block will be used 394: * 395: ABSAKK = CABS1( W( K, K ) ) 396: * 397: * IMAX is the row-index of the largest off-diagonal element in 398: * column K, and COLMAX is its absolute value 399: * 400: IF( K.LT.N ) THEN 401: IMAX = K + IZAMAX( N-K, W( K+1, K ), 1 ) 402: COLMAX = CABS1( W( IMAX, K ) ) 403: ELSE 404: COLMAX = ZERO 405: END IF 406: * 407: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN 408: * 409: * Column K is zero: set INFO and continue 410: * 411: IF( INFO.EQ.0 ) 412: $ INFO = K 413: KP = K 414: ELSE 415: IF( ABSAKK.GE.ALPHA*COLMAX ) THEN 416: * 417: * no interchange, use 1-by-1 pivot block 418: * 419: KP = K 420: ELSE 421: * 422: * Copy column IMAX to column K+1 of W and update it 423: * 424: CALL ZCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1 ) 425: CALL ZCOPY( N-IMAX+1, A( IMAX, IMAX ), 1, W( IMAX, K+1 ), 426: $ 1 ) 427: CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE, A( K, 1 ), 428: $ LDA, W( IMAX, 1 ), LDW, CONE, W( K, K+1 ), 429: $ 1 ) 430: * 431: * JMAX is the column-index of the largest off-diagonal 432: * element in row IMAX, and ROWMAX is its absolute value 433: * 434: JMAX = K - 1 + IZAMAX( IMAX-K, W( K, K+1 ), 1 ) 435: ROWMAX = CABS1( W( JMAX, K+1 ) ) 436: IF( IMAX.LT.N ) THEN 437: JMAX = IMAX + IZAMAX( N-IMAX, W( IMAX+1, K+1 ), 1 ) 438: ROWMAX = MAX( ROWMAX, CABS1( W( JMAX, K+1 ) ) ) 439: END IF 440: * 441: IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN 442: * 443: * no interchange, use 1-by-1 pivot block 444: * 445: KP = K 446: ELSE IF( CABS1( W( IMAX, K+1 ) ).GE.ALPHA*ROWMAX ) THEN 447: * 448: * interchange rows and columns K and IMAX, use 1-by-1 449: * pivot block 450: * 451: KP = IMAX 452: * 453: * copy column K+1 of W to column K 454: * 455: CALL ZCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 ) 456: ELSE 457: * 458: * interchange rows and columns K+1 and IMAX, use 2-by-2 459: * pivot block 460: * 461: KP = IMAX 462: KSTEP = 2 463: END IF 464: END IF 465: * 466: KK = K + KSTEP - 1 467: * 468: * Updated column KP is already stored in column KK of W 469: * 470: IF( KP.NE.KK ) THEN 471: * 472: * Copy non-updated column KK to column KP 473: * 474: A( KP, K ) = A( KK, K ) 475: CALL ZCOPY( KP-K-1, A( K+1, KK ), 1, A( KP, K+1 ), LDA ) 476: CALL ZCOPY( N-KP+1, A( KP, KK ), 1, A( KP, KP ), 1 ) 477: * 478: * Interchange rows KK and KP in first KK columns of A and W 479: * 480: CALL ZSWAP( KK, A( KK, 1 ), LDA, A( KP, 1 ), LDA ) 481: CALL ZSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW ) 482: END IF 483: * 484: IF( KSTEP.EQ.1 ) THEN 485: * 486: * 1-by-1 pivot block D(k): column k of W now holds 487: * 488: * W(k) = L(k)*D(k) 489: * 490: * where L(k) is the k-th column of L 491: * 492: * Store L(k) in column k of A 493: * 494: CALL ZCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 ) 495: IF( K.LT.N ) THEN 496: R1 = CONE / A( K, K ) 497: CALL ZSCAL( N-K, R1, A( K+1, K ), 1 ) 498: END IF 499: ELSE 500: * 501: * 2-by-2 pivot block D(k): columns k and k+1 of W now hold 502: * 503: * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k) 504: * 505: * where L(k) and L(k+1) are the k-th and (k+1)-th columns 506: * of L 507: * 508: IF( K.LT.N-1 ) THEN 509: * 510: * Store L(k) and L(k+1) in columns k and k+1 of A 511: * 512: D21 = W( K+1, K ) 513: D11 = W( K+1, K+1 ) / D21 514: D22 = W( K, K ) / D21 515: T = CONE / ( D11*D22-CONE ) 516: D21 = T / D21 517: DO 80 J = K + 2, N 518: A( J, K ) = D21*( D11*W( J, K )-W( J, K+1 ) ) 519: A( J, K+1 ) = D21*( D22*W( J, K+1 )-W( J, K ) ) 520: 80 CONTINUE 521: END IF 522: * 523: * Copy D(k) to A 524: * 525: A( K, K ) = W( K, K ) 526: A( K+1, K ) = W( K+1, K ) 527: A( K+1, K+1 ) = W( K+1, K+1 ) 528: END IF 529: END IF 530: * 531: * Store details of the interchanges in IPIV 532: * 533: IF( KSTEP.EQ.1 ) THEN 534: IPIV( K ) = KP 535: ELSE 536: IPIV( K ) = -KP 537: IPIV( K+1 ) = -KP 538: END IF 539: * 540: * Increase K and return to the start of the main loop 541: * 542: K = K + KSTEP 543: GO TO 70 544: * 545: 90 CONTINUE 546: * 547: * Update the lower triangle of A22 (= A(k:n,k:n)) as 548: * 549: * A22 := A22 - L21*D*L21' = A22 - L21*W' 550: * 551: * computing blocks of NB columns at a time 552: * 553: DO 110 J = K, N, NB 554: JB = MIN( NB, N-J+1 ) 555: * 556: * Update the lower triangle of the diagonal block 557: * 558: DO 100 JJ = J, J + JB - 1 559: CALL ZGEMV( 'No transpose', J+JB-JJ, K-1, -CONE, 560: $ A( JJ, 1 ), LDA, W( JJ, 1 ), LDW, CONE, 561: $ A( JJ, JJ ), 1 ) 562: 100 CONTINUE 563: * 564: * Update the rectangular subdiagonal block 565: * 566: IF( J+JB.LE.N ) 567: $ CALL ZGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB, 568: $ K-1, -CONE, A( J+JB, 1 ), LDA, W( J, 1 ), 569: $ LDW, CONE, A( J+JB, J ), LDA ) 570: 110 CONTINUE 571: * 572: * Put L21 in standard form by partially undoing the interchanges 573: * in columns 1:k-1 574: * 575: J = K - 1 576: 120 CONTINUE 577: JJ = J 578: JP = IPIV( J ) 579: IF( JP.LT.0 ) THEN 580: JP = -JP 581: J = J - 1 582: END IF 583: J = J - 1 584: IF( JP.NE.JJ .AND. J.GE.1 ) 585: $ CALL ZSWAP( J, A( JP, 1 ), LDA, A( JJ, 1 ), LDA ) 586: IF( J.GE.1 ) 587: $ GO TO 120 588: * 589: * Set KB to the number of columns factorized 590: * 591: KB = K - 1 592: * 593: END IF 594: RETURN 595: * 596: * End of ZLASYF 597: * 598: END