Annotation of rpl/lapack/lapack/dlasyf_rk.f, revision 1.1
1.1 ! bertrand 1: *> \brief \b DLASYF_RK computes a partial factorization of a real symmetric indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method.
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download DLASYF_RK + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasyf_rk.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasyf_rk.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasyf_rk.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE DLASYF_RK( UPLO, N, NB, KB, A, LDA, E, IPIV, W, LDW,
! 22: * INFO )
! 23: *
! 24: * .. Scalar Arguments ..
! 25: * CHARACTER UPLO
! 26: * INTEGER INFO, KB, LDA, LDW, N, NB
! 27: * ..
! 28: * .. Array Arguments ..
! 29: * INTEGER IPIV( * )
! 30: * DOUBLE PRECISION A( LDA, * ), E( * ), W( LDW, * )
! 31: * ..
! 32: *
! 33: *
! 34: *> \par Purpose:
! 35: * =============
! 36: *>
! 37: *> \verbatim
! 38: *> DLASYF_RK computes a partial factorization of a real symmetric
! 39: *> matrix A using the bounded Bunch-Kaufman (rook) diagonal
! 40: *> pivoting method. The partial factorization has the form:
! 41: *>
! 42: *> A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or:
! 43: *> ( 0 U22 ) ( 0 D ) ( U12**T U22**T )
! 44: *>
! 45: *> A = ( L11 0 ) ( D 0 ) ( L11**T L21**T ) if UPLO = 'L',
! 46: *> ( L21 I ) ( 0 A22 ) ( 0 I )
! 47: *>
! 48: *> where the order of D is at most NB. The actual order is returned in
! 49: *> the argument KB, and is either NB or NB-1, or N if N <= NB.
! 50: *>
! 51: *> DLASYF_RK is an auxiliary routine called by DSYTRF_RK. It uses
! 52: *> blocked code (calling Level 3 BLAS) to update the submatrix
! 53: *> A11 (if UPLO = 'U') or A22 (if UPLO = 'L').
! 54: *> \endverbatim
! 55: *
! 56: * Arguments:
! 57: * ==========
! 58: *
! 59: *> \param[in] UPLO
! 60: *> \verbatim
! 61: *> UPLO is CHARACTER*1
! 62: *> Specifies whether the upper or lower triangular part of the
! 63: *> symmetric matrix A is stored:
! 64: *> = 'U': Upper triangular
! 65: *> = 'L': Lower triangular
! 66: *> \endverbatim
! 67: *>
! 68: *> \param[in] N
! 69: *> \verbatim
! 70: *> N is INTEGER
! 71: *> The order of the matrix A. N >= 0.
! 72: *> \endverbatim
! 73: *>
! 74: *> \param[in] NB
! 75: *> \verbatim
! 76: *> NB is INTEGER
! 77: *> The maximum number of columns of the matrix A that should be
! 78: *> factored. NB should be at least 2 to allow for 2-by-2 pivot
! 79: *> blocks.
! 80: *> \endverbatim
! 81: *>
! 82: *> \param[out] KB
! 83: *> \verbatim
! 84: *> KB is INTEGER
! 85: *> The number of columns of A that were actually factored.
! 86: *> KB is either NB-1 or NB, or N if N <= NB.
! 87: *> \endverbatim
! 88: *>
! 89: *> \param[in,out] A
! 90: *> \verbatim
! 91: *> A is DOUBLE PRECISION array, dimension (LDA,N)
! 92: *> On entry, the symmetric matrix A.
! 93: *> If UPLO = 'U': the leading N-by-N upper triangular part
! 94: *> of A contains the upper triangular part of the matrix A,
! 95: *> and the strictly lower triangular part of A is not
! 96: *> referenced.
! 97: *>
! 98: *> If UPLO = 'L': the leading N-by-N lower triangular part
! 99: *> of A contains the lower triangular part of the matrix A,
! 100: *> and the strictly upper triangular part of A is not
! 101: *> referenced.
! 102: *>
! 103: *> On exit, contains:
! 104: *> a) ONLY diagonal elements of the symmetric block diagonal
! 105: *> matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
! 106: *> (superdiagonal (or subdiagonal) elements of D
! 107: *> are stored on exit in array E), and
! 108: *> b) If UPLO = 'U': factor U in the superdiagonal part of A.
! 109: *> If UPLO = 'L': factor L in the subdiagonal part of A.
! 110: *> \endverbatim
! 111: *>
! 112: *> \param[in] LDA
! 113: *> \verbatim
! 114: *> LDA is INTEGER
! 115: *> The leading dimension of the array A. LDA >= max(1,N).
! 116: *> \endverbatim
! 117: *>
! 118: *> \param[out] E
! 119: *> \verbatim
! 120: *> E is DOUBLE PRECISION array, dimension (N)
! 121: *> On exit, contains the superdiagonal (or subdiagonal)
! 122: *> elements of the symmetric block diagonal matrix D
! 123: *> with 1-by-1 or 2-by-2 diagonal blocks, where
! 124: *> If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
! 125: *> If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
! 126: *>
! 127: *> NOTE: For 1-by-1 diagonal block D(k), where
! 128: *> 1 <= k <= N, the element E(k) is set to 0 in both
! 129: *> UPLO = 'U' or UPLO = 'L' cases.
! 130: *> \endverbatim
! 131: *>
! 132: *> \param[out] IPIV
! 133: *> \verbatim
! 134: *> IPIV is INTEGER array, dimension (N)
! 135: *> IPIV describes the permutation matrix P in the factorization
! 136: *> of matrix A as follows. The absolute value of IPIV(k)
! 137: *> represents the index of row and column that were
! 138: *> interchanged with the k-th row and column. The value of UPLO
! 139: *> describes the order in which the interchanges were applied.
! 140: *> Also, the sign of IPIV represents the block structure of
! 141: *> the symmetric block diagonal matrix D with 1-by-1 or 2-by-2
! 142: *> diagonal blocks which correspond to 1 or 2 interchanges
! 143: *> at each factorization step.
! 144: *>
! 145: *> If UPLO = 'U',
! 146: *> ( in factorization order, k decreases from N to 1 ):
! 147: *> a) A single positive entry IPIV(k) > 0 means:
! 148: *> D(k,k) is a 1-by-1 diagonal block.
! 149: *> If IPIV(k) != k, rows and columns k and IPIV(k) were
! 150: *> interchanged in the submatrix A(1:N,N-KB+1:N);
! 151: *> If IPIV(k) = k, no interchange occurred.
! 152: *>
! 153: *>
! 154: *> b) A pair of consecutive negative entries
! 155: *> IPIV(k) < 0 and IPIV(k-1) < 0 means:
! 156: *> D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
! 157: *> (NOTE: negative entries in IPIV appear ONLY in pairs).
! 158: *> 1) If -IPIV(k) != k, rows and columns
! 159: *> k and -IPIV(k) were interchanged
! 160: *> in the matrix A(1:N,N-KB+1:N).
! 161: *> If -IPIV(k) = k, no interchange occurred.
! 162: *> 2) If -IPIV(k-1) != k-1, rows and columns
! 163: *> k-1 and -IPIV(k-1) were interchanged
! 164: *> in the submatrix A(1:N,N-KB+1:N).
! 165: *> If -IPIV(k-1) = k-1, no interchange occurred.
! 166: *>
! 167: *> c) In both cases a) and b) is always ABS( IPIV(k) ) <= k.
! 168: *>
! 169: *> d) NOTE: Any entry IPIV(k) is always NONZERO on output.
! 170: *>
! 171: *> If UPLO = 'L',
! 172: *> ( in factorization order, k increases from 1 to N ):
! 173: *> a) A single positive entry IPIV(k) > 0 means:
! 174: *> D(k,k) is a 1-by-1 diagonal block.
! 175: *> If IPIV(k) != k, rows and columns k and IPIV(k) were
! 176: *> interchanged in the submatrix A(1:N,1:KB).
! 177: *> If IPIV(k) = k, no interchange occurred.
! 178: *>
! 179: *> b) A pair of consecutive negative entries
! 180: *> IPIV(k) < 0 and IPIV(k+1) < 0 means:
! 181: *> D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
! 182: *> (NOTE: negative entries in IPIV appear ONLY in pairs).
! 183: *> 1) If -IPIV(k) != k, rows and columns
! 184: *> k and -IPIV(k) were interchanged
! 185: *> in the submatrix A(1:N,1:KB).
! 186: *> If -IPIV(k) = k, no interchange occurred.
! 187: *> 2) If -IPIV(k+1) != k+1, rows and columns
! 188: *> k-1 and -IPIV(k-1) were interchanged
! 189: *> in the submatrix A(1:N,1:KB).
! 190: *> If -IPIV(k+1) = k+1, no interchange occurred.
! 191: *>
! 192: *> c) In both cases a) and b) is always ABS( IPIV(k) ) >= k.
! 193: *>
! 194: *> d) NOTE: Any entry IPIV(k) is always NONZERO on output.
! 195: *> \endverbatim
! 196: *>
! 197: *> \param[out] W
! 198: *> \verbatim
! 199: *> W is DOUBLE PRECISION array, dimension (LDW,NB)
! 200: *> \endverbatim
! 201: *>
! 202: *> \param[in] LDW
! 203: *> \verbatim
! 204: *> LDW is INTEGER
! 205: *> The leading dimension of the array W. LDW >= max(1,N).
! 206: *> \endverbatim
! 207: *>
! 208: *> \param[out] INFO
! 209: *> \verbatim
! 210: *> INFO is INTEGER
! 211: *> = 0: successful exit
! 212: *>
! 213: *> < 0: If INFO = -k, the k-th argument had an illegal value
! 214: *>
! 215: *> > 0: If INFO = k, the matrix A is singular, because:
! 216: *> If UPLO = 'U': column k in the upper
! 217: *> triangular part of A contains all zeros.
! 218: *> If UPLO = 'L': column k in the lower
! 219: *> triangular part of A contains all zeros.
! 220: *>
! 221: *> Therefore D(k,k) is exactly zero, and superdiagonal
! 222: *> elements of column k of U (or subdiagonal elements of
! 223: *> column k of L ) are all zeros. The factorization has
! 224: *> been completed, but the block diagonal matrix D is
! 225: *> exactly singular, and division by zero will occur if
! 226: *> it is used to solve a system of equations.
! 227: *>
! 228: *> NOTE: INFO only stores the first occurrence of
! 229: *> a singularity, any subsequent occurrence of singularity
! 230: *> is not stored in INFO even though the factorization
! 231: *> always completes.
! 232: *> \endverbatim
! 233: *
! 234: * Authors:
! 235: * ========
! 236: *
! 237: *> \author Univ. of Tennessee
! 238: *> \author Univ. of California Berkeley
! 239: *> \author Univ. of Colorado Denver
! 240: *> \author NAG Ltd.
! 241: *
! 242: *> \date December 2016
! 243: *
! 244: *> \ingroup doubleSYcomputational
! 245: *
! 246: *> \par Contributors:
! 247: * ==================
! 248: *>
! 249: *> \verbatim
! 250: *>
! 251: *> December 2016, Igor Kozachenko,
! 252: *> Computer Science Division,
! 253: *> University of California, Berkeley
! 254: *>
! 255: *> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
! 256: *> School of Mathematics,
! 257: *> University of Manchester
! 258: *>
! 259: *> \endverbatim
! 260: *
! 261: * =====================================================================
! 262: SUBROUTINE DLASYF_RK( UPLO, N, NB, KB, A, LDA, E, IPIV, W, LDW,
! 263: $ INFO )
! 264: *
! 265: * -- LAPACK computational routine (version 3.7.0) --
! 266: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 267: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 268: * December 2016
! 269: *
! 270: * .. Scalar Arguments ..
! 271: CHARACTER UPLO
! 272: INTEGER INFO, KB, LDA, LDW, N, NB
! 273: * ..
! 274: * .. Array Arguments ..
! 275: INTEGER IPIV( * )
! 276: DOUBLE PRECISION A( LDA, * ), E( * ), W( LDW, * )
! 277: * ..
! 278: *
! 279: * =====================================================================
! 280: *
! 281: * .. Parameters ..
! 282: DOUBLE PRECISION ZERO, ONE
! 283: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
! 284: DOUBLE PRECISION EIGHT, SEVTEN
! 285: PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
! 286: * ..
! 287: * .. Local Scalars ..
! 288: LOGICAL DONE
! 289: INTEGER IMAX, ITEMP, J, JB, JJ, JMAX, K, KK, KW, KKW,
! 290: $ KP, KSTEP, P, II
! 291: DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22,
! 292: $ DTEMP, R1, ROWMAX, T, SFMIN
! 293: * ..
! 294: * .. External Functions ..
! 295: LOGICAL LSAME
! 296: INTEGER IDAMAX
! 297: DOUBLE PRECISION DLAMCH
! 298: EXTERNAL LSAME, IDAMAX, DLAMCH
! 299: * ..
! 300: * .. External Subroutines ..
! 301: EXTERNAL DCOPY, DGEMM, DGEMV, DSCAL, DSWAP
! 302: * ..
! 303: * .. Intrinsic Functions ..
! 304: INTRINSIC ABS, MAX, MIN, SQRT
! 305: * ..
! 306: * .. Executable Statements ..
! 307: *
! 308: INFO = 0
! 309: *
! 310: * Initialize ALPHA for use in choosing pivot block size.
! 311: *
! 312: ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
! 313: *
! 314: * Compute machine safe minimum
! 315: *
! 316: SFMIN = DLAMCH( 'S' )
! 317: *
! 318: IF( LSAME( UPLO, 'U' ) ) THEN
! 319: *
! 320: * Factorize the trailing columns of A using the upper triangle
! 321: * of A and working backwards, and compute the matrix W = U12*D
! 322: * for use in updating A11
! 323: *
! 324: * Initilize the first entry of array E, where superdiagonal
! 325: * elements of D are stored
! 326: *
! 327: E( 1 ) = ZERO
! 328: *
! 329: * K is the main loop index, decreasing from N in steps of 1 or 2
! 330: *
! 331: K = N
! 332: 10 CONTINUE
! 333: *
! 334: * KW is the column of W which corresponds to column K of A
! 335: *
! 336: KW = NB + K - N
! 337: *
! 338: * Exit from loop
! 339: *
! 340: IF( ( K.LE.N-NB+1 .AND. NB.LT.N ) .OR. K.LT.1 )
! 341: $ GO TO 30
! 342: *
! 343: KSTEP = 1
! 344: P = K
! 345: *
! 346: * Copy column K of A to column KW of W and update it
! 347: *
! 348: CALL DCOPY( K, A( 1, K ), 1, W( 1, KW ), 1 )
! 349: IF( K.LT.N )
! 350: $ CALL DGEMV( 'No transpose', K, N-K, -ONE, A( 1, K+1 ),
! 351: $ LDA, W( K, KW+1 ), LDW, ONE, W( 1, KW ), 1 )
! 352: *
! 353: * Determine rows and columns to be interchanged and whether
! 354: * a 1-by-1 or 2-by-2 pivot block will be used
! 355: *
! 356: ABSAKK = ABS( W( K, KW ) )
! 357: *
! 358: * IMAX is the row-index of the largest off-diagonal element in
! 359: * column K, and COLMAX is its absolute value.
! 360: * Determine both COLMAX and IMAX.
! 361: *
! 362: IF( K.GT.1 ) THEN
! 363: IMAX = IDAMAX( K-1, W( 1, KW ), 1 )
! 364: COLMAX = ABS( W( IMAX, KW ) )
! 365: ELSE
! 366: COLMAX = ZERO
! 367: END IF
! 368: *
! 369: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
! 370: *
! 371: * Column K is zero or underflow: set INFO and continue
! 372: *
! 373: IF( INFO.EQ.0 )
! 374: $ INFO = K
! 375: KP = K
! 376: CALL DCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
! 377: *
! 378: * Set E( K ) to zero
! 379: *
! 380: IF( K.GT.1 )
! 381: $ E( K ) = ZERO
! 382: *
! 383: ELSE
! 384: *
! 385: * ============================================================
! 386: *
! 387: * Test for interchange
! 388: *
! 389: * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
! 390: * (used to handle NaN and Inf)
! 391: *
! 392: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
! 393: *
! 394: * no interchange, use 1-by-1 pivot block
! 395: *
! 396: KP = K
! 397: *
! 398: ELSE
! 399: *
! 400: DONE = .FALSE.
! 401: *
! 402: * Loop until pivot found
! 403: *
! 404: 12 CONTINUE
! 405: *
! 406: * Begin pivot search loop body
! 407: *
! 408: *
! 409: * Copy column IMAX to column KW-1 of W and update it
! 410: *
! 411: CALL DCOPY( IMAX, A( 1, IMAX ), 1, W( 1, KW-1 ), 1 )
! 412: CALL DCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA,
! 413: $ W( IMAX+1, KW-1 ), 1 )
! 414: *
! 415: IF( K.LT.N )
! 416: $ CALL DGEMV( 'No transpose', K, N-K, -ONE,
! 417: $ A( 1, K+1 ), LDA, W( IMAX, KW+1 ), LDW,
! 418: $ ONE, W( 1, KW-1 ), 1 )
! 419: *
! 420: * JMAX is the column-index of the largest off-diagonal
! 421: * element in row IMAX, and ROWMAX is its absolute value.
! 422: * Determine both ROWMAX and JMAX.
! 423: *
! 424: IF( IMAX.NE.K ) THEN
! 425: JMAX = IMAX + IDAMAX( K-IMAX, W( IMAX+1, KW-1 ),
! 426: $ 1 )
! 427: ROWMAX = ABS( W( JMAX, KW-1 ) )
! 428: ELSE
! 429: ROWMAX = ZERO
! 430: END IF
! 431: *
! 432: IF( IMAX.GT.1 ) THEN
! 433: ITEMP = IDAMAX( IMAX-1, W( 1, KW-1 ), 1 )
! 434: DTEMP = ABS( W( ITEMP, KW-1 ) )
! 435: IF( DTEMP.GT.ROWMAX ) THEN
! 436: ROWMAX = DTEMP
! 437: JMAX = ITEMP
! 438: END IF
! 439: END IF
! 440: *
! 441: * Equivalent to testing for
! 442: * ABS( W( IMAX, KW-1 ) ).GE.ALPHA*ROWMAX
! 443: * (used to handle NaN and Inf)
! 444: *
! 445: IF( .NOT.(ABS( W( IMAX, KW-1 ) ).LT.ALPHA*ROWMAX ) )
! 446: $ THEN
! 447: *
! 448: * interchange rows and columns K and IMAX,
! 449: * use 1-by-1 pivot block
! 450: *
! 451: KP = IMAX
! 452: *
! 453: * copy column KW-1 of W to column KW of W
! 454: *
! 455: CALL DCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
! 456: *
! 457: DONE = .TRUE.
! 458: *
! 459: * Equivalent to testing for ROWMAX.EQ.COLMAX,
! 460: * (used to handle NaN and Inf)
! 461: *
! 462: ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
! 463: $ THEN
! 464: *
! 465: * interchange rows and columns K-1 and IMAX,
! 466: * use 2-by-2 pivot block
! 467: *
! 468: KP = IMAX
! 469: KSTEP = 2
! 470: DONE = .TRUE.
! 471: ELSE
! 472: *
! 473: * Pivot not found: set params and repeat
! 474: *
! 475: P = IMAX
! 476: COLMAX = ROWMAX
! 477: IMAX = JMAX
! 478: *
! 479: * Copy updated JMAXth (next IMAXth) column to Kth of W
! 480: *
! 481: CALL DCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
! 482: *
! 483: END IF
! 484: *
! 485: * End pivot search loop body
! 486: *
! 487: IF( .NOT. DONE ) GOTO 12
! 488: *
! 489: END IF
! 490: *
! 491: * ============================================================
! 492: *
! 493: KK = K - KSTEP + 1
! 494: *
! 495: * KKW is the column of W which corresponds to column KK of A
! 496: *
! 497: KKW = NB + KK - N
! 498: *
! 499: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
! 500: *
! 501: * Copy non-updated column K to column P
! 502: *
! 503: CALL DCOPY( K-P, A( P+1, K ), 1, A( P, P+1 ), LDA )
! 504: CALL DCOPY( P, A( 1, K ), 1, A( 1, P ), 1 )
! 505: *
! 506: * Interchange rows K and P in last N-K+1 columns of A
! 507: * and last N-K+2 columns of W
! 508: *
! 509: CALL DSWAP( N-K+1, A( K, K ), LDA, A( P, K ), LDA )
! 510: CALL DSWAP( N-KK+1, W( K, KKW ), LDW, W( P, KKW ), LDW )
! 511: END IF
! 512: *
! 513: * Updated column KP is already stored in column KKW of W
! 514: *
! 515: IF( KP.NE.KK ) THEN
! 516: *
! 517: * Copy non-updated column KK to column KP
! 518: *
! 519: A( KP, K ) = A( KK, K )
! 520: CALL DCOPY( K-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ),
! 521: $ LDA )
! 522: CALL DCOPY( KP, A( 1, KK ), 1, A( 1, KP ), 1 )
! 523: *
! 524: * Interchange rows KK and KP in last N-KK+1 columns
! 525: * of A and W
! 526: *
! 527: CALL DSWAP( N-KK+1, A( KK, KK ), LDA, A( KP, KK ), LDA )
! 528: CALL DSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ),
! 529: $ LDW )
! 530: END IF
! 531: *
! 532: IF( KSTEP.EQ.1 ) THEN
! 533: *
! 534: * 1-by-1 pivot block D(k): column KW of W now holds
! 535: *
! 536: * W(k) = U(k)*D(k)
! 537: *
! 538: * where U(k) is the k-th column of U
! 539: *
! 540: * Store U(k) in column k of A
! 541: *
! 542: CALL DCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
! 543: IF( K.GT.1 ) THEN
! 544: IF( ABS( A( K, K ) ).GE.SFMIN ) THEN
! 545: R1 = ONE / A( K, K )
! 546: CALL DSCAL( K-1, R1, A( 1, K ), 1 )
! 547: ELSE IF( A( K, K ).NE.ZERO ) THEN
! 548: DO 14 II = 1, K - 1
! 549: A( II, K ) = A( II, K ) / A( K, K )
! 550: 14 CONTINUE
! 551: END IF
! 552: *
! 553: * Store the superdiagonal element of D in array E
! 554: *
! 555: E( K ) = ZERO
! 556: *
! 557: END IF
! 558: *
! 559: ELSE
! 560: *
! 561: * 2-by-2 pivot block D(k): columns KW and KW-1 of W now
! 562: * hold
! 563: *
! 564: * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
! 565: *
! 566: * where U(k) and U(k-1) are the k-th and (k-1)-th columns
! 567: * of U
! 568: *
! 569: IF( K.GT.2 ) THEN
! 570: *
! 571: * Store U(k) and U(k-1) in columns k and k-1 of A
! 572: *
! 573: D12 = W( K-1, KW )
! 574: D11 = W( K, KW ) / D12
! 575: D22 = W( K-1, KW-1 ) / D12
! 576: T = ONE / ( D11*D22-ONE )
! 577: DO 20 J = 1, K - 2
! 578: A( J, K-1 ) = T*( (D11*W( J, KW-1 )-W( J, KW ) ) /
! 579: $ D12 )
! 580: A( J, K ) = T*( ( D22*W( J, KW )-W( J, KW-1 ) ) /
! 581: $ D12 )
! 582: 20 CONTINUE
! 583: END IF
! 584: *
! 585: * Copy diagonal elements of D(K) to A,
! 586: * copy superdiagonal element of D(K) to E(K) and
! 587: * ZERO out superdiagonal entry of A
! 588: *
! 589: A( K-1, K-1 ) = W( K-1, KW-1 )
! 590: A( K-1, K ) = ZERO
! 591: A( K, K ) = W( K, KW )
! 592: E( K ) = W( K-1, KW )
! 593: E( K-1 ) = ZERO
! 594: *
! 595: END IF
! 596: *
! 597: * End column K is nonsingular
! 598: *
! 599: END IF
! 600: *
! 601: * Store details of the interchanges in IPIV
! 602: *
! 603: IF( KSTEP.EQ.1 ) THEN
! 604: IPIV( K ) = KP
! 605: ELSE
! 606: IPIV( K ) = -P
! 607: IPIV( K-1 ) = -KP
! 608: END IF
! 609: *
! 610: * Decrease K and return to the start of the main loop
! 611: *
! 612: K = K - KSTEP
! 613: GO TO 10
! 614: *
! 615: 30 CONTINUE
! 616: *
! 617: * Update the upper triangle of A11 (= A(1:k,1:k)) as
! 618: *
! 619: * A11 := A11 - U12*D*U12**T = A11 - U12*W**T
! 620: *
! 621: * computing blocks of NB columns at a time
! 622: *
! 623: DO 50 J = ( ( K-1 ) / NB )*NB + 1, 1, -NB
! 624: JB = MIN( NB, K-J+1 )
! 625: *
! 626: * Update the upper triangle of the diagonal block
! 627: *
! 628: DO 40 JJ = J, J + JB - 1
! 629: CALL DGEMV( 'No transpose', JJ-J+1, N-K, -ONE,
! 630: $ A( J, K+1 ), LDA, W( JJ, KW+1 ), LDW, ONE,
! 631: $ A( J, JJ ), 1 )
! 632: 40 CONTINUE
! 633: *
! 634: * Update the rectangular superdiagonal block
! 635: *
! 636: IF( J.GE.2 )
! 637: $ CALL DGEMM( 'No transpose', 'Transpose', J-1, JB,
! 638: $ N-K, -ONE, A( 1, K+1 ), LDA, W( J, KW+1 ),
! 639: $ LDW, ONE, A( 1, J ), LDA )
! 640: 50 CONTINUE
! 641: *
! 642: * Set KB to the number of columns factorized
! 643: *
! 644: KB = N - K
! 645: *
! 646: ELSE
! 647: *
! 648: * Factorize the leading columns of A using the lower triangle
! 649: * of A and working forwards, and compute the matrix W = L21*D
! 650: * for use in updating A22
! 651: *
! 652: * Initilize the unused last entry of the subdiagonal array E.
! 653: *
! 654: E( N ) = ZERO
! 655: *
! 656: * K is the main loop index, increasing from 1 in steps of 1 or 2
! 657: *
! 658: K = 1
! 659: 70 CONTINUE
! 660: *
! 661: * Exit from loop
! 662: *
! 663: IF( ( K.GE.NB .AND. NB.LT.N ) .OR. K.GT.N )
! 664: $ GO TO 90
! 665: *
! 666: KSTEP = 1
! 667: P = K
! 668: *
! 669: * Copy column K of A to column K of W and update it
! 670: *
! 671: CALL DCOPY( N-K+1, A( K, K ), 1, W( K, K ), 1 )
! 672: IF( K.GT.1 )
! 673: $ CALL DGEMV( 'No transpose', N-K+1, K-1, -ONE, A( K, 1 ),
! 674: $ LDA, W( K, 1 ), LDW, ONE, W( K, K ), 1 )
! 675: *
! 676: * Determine rows and columns to be interchanged and whether
! 677: * a 1-by-1 or 2-by-2 pivot block will be used
! 678: *
! 679: ABSAKK = ABS( W( K, K ) )
! 680: *
! 681: * IMAX is the row-index of the largest off-diagonal element in
! 682: * column K, and COLMAX is its absolute value.
! 683: * Determine both COLMAX and IMAX.
! 684: *
! 685: IF( K.LT.N ) THEN
! 686: IMAX = K + IDAMAX( N-K, W( K+1, K ), 1 )
! 687: COLMAX = ABS( W( IMAX, K ) )
! 688: ELSE
! 689: COLMAX = ZERO
! 690: END IF
! 691: *
! 692: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
! 693: *
! 694: * Column K is zero or underflow: set INFO and continue
! 695: *
! 696: IF( INFO.EQ.0 )
! 697: $ INFO = K
! 698: KP = K
! 699: CALL DCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
! 700: *
! 701: * Set E( K ) to zero
! 702: *
! 703: IF( K.LT.N )
! 704: $ E( K ) = ZERO
! 705: *
! 706: ELSE
! 707: *
! 708: * ============================================================
! 709: *
! 710: * Test for interchange
! 711: *
! 712: * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
! 713: * (used to handle NaN and Inf)
! 714: *
! 715: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
! 716: *
! 717: * no interchange, use 1-by-1 pivot block
! 718: *
! 719: KP = K
! 720: *
! 721: ELSE
! 722: *
! 723: DONE = .FALSE.
! 724: *
! 725: * Loop until pivot found
! 726: *
! 727: 72 CONTINUE
! 728: *
! 729: * Begin pivot search loop body
! 730: *
! 731: *
! 732: * Copy column IMAX to column K+1 of W and update it
! 733: *
! 734: CALL DCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1)
! 735: CALL DCOPY( N-IMAX+1, A( IMAX, IMAX ), 1,
! 736: $ W( IMAX, K+1 ), 1 )
! 737: IF( K.GT.1 )
! 738: $ CALL DGEMV( 'No transpose', N-K+1, K-1, -ONE,
! 739: $ A( K, 1 ), LDA, W( IMAX, 1 ), LDW,
! 740: $ ONE, W( K, K+1 ), 1 )
! 741: *
! 742: * JMAX is the column-index of the largest off-diagonal
! 743: * element in row IMAX, and ROWMAX is its absolute value.
! 744: * Determine both ROWMAX and JMAX.
! 745: *
! 746: IF( IMAX.NE.K ) THEN
! 747: JMAX = K - 1 + IDAMAX( IMAX-K, W( K, K+1 ), 1 )
! 748: ROWMAX = ABS( W( JMAX, K+1 ) )
! 749: ELSE
! 750: ROWMAX = ZERO
! 751: END IF
! 752: *
! 753: IF( IMAX.LT.N ) THEN
! 754: ITEMP = IMAX + IDAMAX( N-IMAX, W( IMAX+1, K+1 ), 1)
! 755: DTEMP = ABS( W( ITEMP, K+1 ) )
! 756: IF( DTEMP.GT.ROWMAX ) THEN
! 757: ROWMAX = DTEMP
! 758: JMAX = ITEMP
! 759: END IF
! 760: END IF
! 761: *
! 762: * Equivalent to testing for
! 763: * ABS( W( IMAX, K+1 ) ).GE.ALPHA*ROWMAX
! 764: * (used to handle NaN and Inf)
! 765: *
! 766: IF( .NOT.( ABS( W( IMAX, K+1 ) ).LT.ALPHA*ROWMAX ) )
! 767: $ THEN
! 768: *
! 769: * interchange rows and columns K and IMAX,
! 770: * use 1-by-1 pivot block
! 771: *
! 772: KP = IMAX
! 773: *
! 774: * copy column K+1 of W to column K of W
! 775: *
! 776: CALL DCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
! 777: *
! 778: DONE = .TRUE.
! 779: *
! 780: * Equivalent to testing for ROWMAX.EQ.COLMAX,
! 781: * (used to handle NaN and Inf)
! 782: *
! 783: ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
! 784: $ THEN
! 785: *
! 786: * interchange rows and columns K+1 and IMAX,
! 787: * use 2-by-2 pivot block
! 788: *
! 789: KP = IMAX
! 790: KSTEP = 2
! 791: DONE = .TRUE.
! 792: ELSE
! 793: *
! 794: * Pivot not found: set params and repeat
! 795: *
! 796: P = IMAX
! 797: COLMAX = ROWMAX
! 798: IMAX = JMAX
! 799: *
! 800: * Copy updated JMAXth (next IMAXth) column to Kth of W
! 801: *
! 802: CALL DCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
! 803: *
! 804: END IF
! 805: *
! 806: * End pivot search loop body
! 807: *
! 808: IF( .NOT. DONE ) GOTO 72
! 809: *
! 810: END IF
! 811: *
! 812: * ============================================================
! 813: *
! 814: KK = K + KSTEP - 1
! 815: *
! 816: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
! 817: *
! 818: * Copy non-updated column K to column P
! 819: *
! 820: CALL DCOPY( P-K, A( K, K ), 1, A( P, K ), LDA )
! 821: CALL DCOPY( N-P+1, A( P, K ), 1, A( P, P ), 1 )
! 822: *
! 823: * Interchange rows K and P in first K columns of A
! 824: * and first K+1 columns of W
! 825: *
! 826: CALL DSWAP( K, A( K, 1 ), LDA, A( P, 1 ), LDA )
! 827: CALL DSWAP( KK, W( K, 1 ), LDW, W( P, 1 ), LDW )
! 828: END IF
! 829: *
! 830: * Updated column KP is already stored in column KK of W
! 831: *
! 832: IF( KP.NE.KK ) THEN
! 833: *
! 834: * Copy non-updated column KK to column KP
! 835: *
! 836: A( KP, K ) = A( KK, K )
! 837: CALL DCOPY( KP-K-1, A( K+1, KK ), 1, A( KP, K+1 ), LDA )
! 838: CALL DCOPY( N-KP+1, A( KP, KK ), 1, A( KP, KP ), 1 )
! 839: *
! 840: * Interchange rows KK and KP in first KK columns of A and W
! 841: *
! 842: CALL DSWAP( KK, A( KK, 1 ), LDA, A( KP, 1 ), LDA )
! 843: CALL DSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW )
! 844: END IF
! 845: *
! 846: IF( KSTEP.EQ.1 ) THEN
! 847: *
! 848: * 1-by-1 pivot block D(k): column k of W now holds
! 849: *
! 850: * W(k) = L(k)*D(k)
! 851: *
! 852: * where L(k) is the k-th column of L
! 853: *
! 854: * Store L(k) in column k of A
! 855: *
! 856: CALL DCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
! 857: IF( K.LT.N ) THEN
! 858: IF( ABS( A( K, K ) ).GE.SFMIN ) THEN
! 859: R1 = ONE / A( K, K )
! 860: CALL DSCAL( N-K, R1, A( K+1, K ), 1 )
! 861: ELSE IF( A( K, K ).NE.ZERO ) THEN
! 862: DO 74 II = K + 1, N
! 863: A( II, K ) = A( II, K ) / A( K, K )
! 864: 74 CONTINUE
! 865: END IF
! 866: *
! 867: * Store the subdiagonal element of D in array E
! 868: *
! 869: E( K ) = ZERO
! 870: *
! 871: END IF
! 872: *
! 873: ELSE
! 874: *
! 875: * 2-by-2 pivot block D(k): columns k and k+1 of W now hold
! 876: *
! 877: * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
! 878: *
! 879: * where L(k) and L(k+1) are the k-th and (k+1)-th columns
! 880: * of L
! 881: *
! 882: IF( K.LT.N-1 ) THEN
! 883: *
! 884: * Store L(k) and L(k+1) in columns k and k+1 of A
! 885: *
! 886: D21 = W( K+1, K )
! 887: D11 = W( K+1, K+1 ) / D21
! 888: D22 = W( K, K ) / D21
! 889: T = ONE / ( D11*D22-ONE )
! 890: DO 80 J = K + 2, N
! 891: A( J, K ) = T*( ( D11*W( J, K )-W( J, K+1 ) ) /
! 892: $ D21 )
! 893: A( J, K+1 ) = T*( ( D22*W( J, K+1 )-W( J, K ) ) /
! 894: $ D21 )
! 895: 80 CONTINUE
! 896: END IF
! 897: *
! 898: * Copy diagonal elements of D(K) to A,
! 899: * copy subdiagonal element of D(K) to E(K) and
! 900: * ZERO out subdiagonal entry of A
! 901: *
! 902: A( K, K ) = W( K, K )
! 903: A( K+1, K ) = ZERO
! 904: A( K+1, K+1 ) = W( K+1, K+1 )
! 905: E( K ) = W( K+1, K )
! 906: E( K+1 ) = ZERO
! 907: *
! 908: END IF
! 909: *
! 910: * End column K is nonsingular
! 911: *
! 912: END IF
! 913: *
! 914: * Store details of the interchanges in IPIV
! 915: *
! 916: IF( KSTEP.EQ.1 ) THEN
! 917: IPIV( K ) = KP
! 918: ELSE
! 919: IPIV( K ) = -P
! 920: IPIV( K+1 ) = -KP
! 921: END IF
! 922: *
! 923: * Increase K and return to the start of the main loop
! 924: *
! 925: K = K + KSTEP
! 926: GO TO 70
! 927: *
! 928: 90 CONTINUE
! 929: *
! 930: * Update the lower triangle of A22 (= A(k:n,k:n)) as
! 931: *
! 932: * A22 := A22 - L21*D*L21**T = A22 - L21*W**T
! 933: *
! 934: * computing blocks of NB columns at a time
! 935: *
! 936: DO 110 J = K, N, NB
! 937: JB = MIN( NB, N-J+1 )
! 938: *
! 939: * Update the lower triangle of the diagonal block
! 940: *
! 941: DO 100 JJ = J, J + JB - 1
! 942: CALL DGEMV( 'No transpose', J+JB-JJ, K-1, -ONE,
! 943: $ A( JJ, 1 ), LDA, W( JJ, 1 ), LDW, ONE,
! 944: $ A( JJ, JJ ), 1 )
! 945: 100 CONTINUE
! 946: *
! 947: * Update the rectangular subdiagonal block
! 948: *
! 949: IF( J+JB.LE.N )
! 950: $ CALL DGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB,
! 951: $ K-1, -ONE, A( J+JB, 1 ), LDA, W( J, 1 ),
! 952: $ LDW, ONE, A( J+JB, J ), LDA )
! 953: 110 CONTINUE
! 954: *
! 955: * Set KB to the number of columns factorized
! 956: *
! 957: KB = K - 1
! 958: *
! 959: END IF
! 960: *
! 961: RETURN
! 962: *
! 963: * End of DLASYF_RK
! 964: *
! 965: END
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