Annotation of rpl/lapack/lapack/zlasyf_rk.f, revision 1.1
1.1 ! bertrand 1: *> \brief \b ZLASYF_RK computes a partial factorization of a complex 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 ZLASYF_RK + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlasyf_rk.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlasyf_rk.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlasyf_rk.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE ZLASYF_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: * COMPLEX*16 A( LDA, * ), E( * ), W( LDW, * )
! 31: * ..
! 32: *
! 33: *
! 34: *> \par Purpose:
! 35: * =============
! 36: *>
! 37: *> \verbatim
! 38: *> ZLASYF_RK computes a partial factorization of a complex 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: *> ZLASYF_RK is an auxiliary routine called by ZSYTRF_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 COMPLEX*16 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 COMPLEX*16 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 COMPLEX*16 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 complex16SYcomputational
! 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 ZLASYF_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: COMPLEX*16 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: COMPLEX*16 CONE, CZERO
! 287: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ),
! 288: $ CZERO = ( 0.0D+0, 0.0D+0 ) )
! 289: * ..
! 290: * .. Local Scalars ..
! 291: LOGICAL DONE
! 292: INTEGER IMAX, ITEMP, J, JB, JJ, JMAX, K, KK, KW, KKW,
! 293: $ KP, KSTEP, P, II
! 294: DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, ROWMAX, SFMIN, DTEMP
! 295: COMPLEX*16 D11, D12, D21, D22, R1, T, Z
! 296: * ..
! 297: * .. External Functions ..
! 298: LOGICAL LSAME
! 299: INTEGER IZAMAX
! 300: DOUBLE PRECISION DLAMCH
! 301: EXTERNAL LSAME, IZAMAX, DLAMCH
! 302: * ..
! 303: * .. External Subroutines ..
! 304: EXTERNAL ZCOPY, ZGEMM, ZGEMV, ZSCAL, ZSWAP
! 305: * ..
! 306: * .. Intrinsic Functions ..
! 307: INTRINSIC ABS, DBLE, DIMAG, MAX, MIN, SQRT
! 308: * ..
! 309: * .. Statement Functions ..
! 310: DOUBLE PRECISION CABS1
! 311: * ..
! 312: * .. Statement Function definitions ..
! 313: CABS1( Z ) = ABS( DBLE( Z ) ) + ABS( DIMAG( Z ) )
! 314: * ..
! 315: * .. Executable Statements ..
! 316: *
! 317: INFO = 0
! 318: *
! 319: * Initialize ALPHA for use in choosing pivot block size.
! 320: *
! 321: ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
! 322: *
! 323: * Compute machine safe minimum
! 324: *
! 325: SFMIN = DLAMCH( 'S' )
! 326: *
! 327: IF( LSAME( UPLO, 'U' ) ) THEN
! 328: *
! 329: * Factorize the trailing columns of A using the upper triangle
! 330: * of A and working backwards, and compute the matrix W = U12*D
! 331: * for use in updating A11
! 332: *
! 333: * Initilize the first entry of array E, where superdiagonal
! 334: * elements of D are stored
! 335: *
! 336: E( 1 ) = CZERO
! 337: *
! 338: * K is the main loop index, decreasing from N in steps of 1 or 2
! 339: *
! 340: K = N
! 341: 10 CONTINUE
! 342: *
! 343: * KW is the column of W which corresponds to column K of A
! 344: *
! 345: KW = NB + K - N
! 346: *
! 347: * Exit from loop
! 348: *
! 349: IF( ( K.LE.N-NB+1 .AND. NB.LT.N ) .OR. K.LT.1 )
! 350: $ GO TO 30
! 351: *
! 352: KSTEP = 1
! 353: P = K
! 354: *
! 355: * Copy column K of A to column KW of W and update it
! 356: *
! 357: CALL ZCOPY( K, A( 1, K ), 1, W( 1, KW ), 1 )
! 358: IF( K.LT.N )
! 359: $ CALL ZGEMV( 'No transpose', K, N-K, -CONE, A( 1, K+1 ),
! 360: $ LDA, W( K, KW+1 ), LDW, CONE, W( 1, KW ), 1 )
! 361: *
! 362: * Determine rows and columns to be interchanged and whether
! 363: * a 1-by-1 or 2-by-2 pivot block will be used
! 364: *
! 365: ABSAKK = CABS1( W( K, KW ) )
! 366: *
! 367: * IMAX is the row-index of the largest off-diagonal element in
! 368: * column K, and COLMAX is its absolute value.
! 369: * Determine both COLMAX and IMAX.
! 370: *
! 371: IF( K.GT.1 ) THEN
! 372: IMAX = IZAMAX( K-1, W( 1, KW ), 1 )
! 373: COLMAX = CABS1( W( IMAX, KW ) )
! 374: ELSE
! 375: COLMAX = ZERO
! 376: END IF
! 377: *
! 378: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
! 379: *
! 380: * Column K is zero or underflow: set INFO and continue
! 381: *
! 382: IF( INFO.EQ.0 )
! 383: $ INFO = K
! 384: KP = K
! 385: CALL ZCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
! 386: *
! 387: * Set E( K ) to zero
! 388: *
! 389: IF( K.GT.1 )
! 390: $ E( K ) = CZERO
! 391: *
! 392: ELSE
! 393: *
! 394: * ============================================================
! 395: *
! 396: * Test for interchange
! 397: *
! 398: * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
! 399: * (used to handle NaN and Inf)
! 400: *
! 401: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
! 402: *
! 403: * no interchange, use 1-by-1 pivot block
! 404: *
! 405: KP = K
! 406: *
! 407: ELSE
! 408: *
! 409: DONE = .FALSE.
! 410: *
! 411: * Loop until pivot found
! 412: *
! 413: 12 CONTINUE
! 414: *
! 415: * Begin pivot search loop body
! 416: *
! 417: *
! 418: * Copy column IMAX to column KW-1 of W and update it
! 419: *
! 420: CALL ZCOPY( IMAX, A( 1, IMAX ), 1, W( 1, KW-1 ), 1 )
! 421: CALL ZCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA,
! 422: $ W( IMAX+1, KW-1 ), 1 )
! 423: *
! 424: IF( K.LT.N )
! 425: $ CALL ZGEMV( 'No transpose', K, N-K, -CONE,
! 426: $ A( 1, K+1 ), LDA, W( IMAX, KW+1 ), LDW,
! 427: $ CONE, W( 1, KW-1 ), 1 )
! 428: *
! 429: * JMAX is the column-index of the largest off-diagonal
! 430: * element in row IMAX, and ROWMAX is its absolute value.
! 431: * Determine both ROWMAX and JMAX.
! 432: *
! 433: IF( IMAX.NE.K ) THEN
! 434: JMAX = IMAX + IZAMAX( K-IMAX, W( IMAX+1, KW-1 ),
! 435: $ 1 )
! 436: ROWMAX = CABS1( W( JMAX, KW-1 ) )
! 437: ELSE
! 438: ROWMAX = ZERO
! 439: END IF
! 440: *
! 441: IF( IMAX.GT.1 ) THEN
! 442: ITEMP = IZAMAX( IMAX-1, W( 1, KW-1 ), 1 )
! 443: DTEMP = CABS1( W( ITEMP, KW-1 ) )
! 444: IF( DTEMP.GT.ROWMAX ) THEN
! 445: ROWMAX = DTEMP
! 446: JMAX = ITEMP
! 447: END IF
! 448: END IF
! 449: *
! 450: * Equivalent to testing for
! 451: * CABS1( W( IMAX, KW-1 ) ).GE.ALPHA*ROWMAX
! 452: * (used to handle NaN and Inf)
! 453: *
! 454: IF( .NOT.(CABS1( W( IMAX, KW-1 ) ).LT.ALPHA*ROWMAX ) )
! 455: $ THEN
! 456: *
! 457: * interchange rows and columns K and IMAX,
! 458: * use 1-by-1 pivot block
! 459: *
! 460: KP = IMAX
! 461: *
! 462: * copy column KW-1 of W to column KW of W
! 463: *
! 464: CALL ZCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
! 465: *
! 466: DONE = .TRUE.
! 467: *
! 468: * Equivalent to testing for ROWMAX.EQ.COLMAX,
! 469: * (used to handle NaN and Inf)
! 470: *
! 471: ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
! 472: $ THEN
! 473: *
! 474: * interchange rows and columns K-1 and IMAX,
! 475: * use 2-by-2 pivot block
! 476: *
! 477: KP = IMAX
! 478: KSTEP = 2
! 479: DONE = .TRUE.
! 480: ELSE
! 481: *
! 482: * Pivot not found: set params and repeat
! 483: *
! 484: P = IMAX
! 485: COLMAX = ROWMAX
! 486: IMAX = JMAX
! 487: *
! 488: * Copy updated JMAXth (next IMAXth) column to Kth of W
! 489: *
! 490: CALL ZCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
! 491: *
! 492: END IF
! 493: *
! 494: * End pivot search loop body
! 495: *
! 496: IF( .NOT. DONE ) GOTO 12
! 497: *
! 498: END IF
! 499: *
! 500: * ============================================================
! 501: *
! 502: KK = K - KSTEP + 1
! 503: *
! 504: * KKW is the column of W which corresponds to column KK of A
! 505: *
! 506: KKW = NB + KK - N
! 507: *
! 508: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
! 509: *
! 510: * Copy non-updated column K to column P
! 511: *
! 512: CALL ZCOPY( K-P, A( P+1, K ), 1, A( P, P+1 ), LDA )
! 513: CALL ZCOPY( P, A( 1, K ), 1, A( 1, P ), 1 )
! 514: *
! 515: * Interchange rows K and P in last N-K+1 columns of A
! 516: * and last N-K+2 columns of W
! 517: *
! 518: CALL ZSWAP( N-K+1, A( K, K ), LDA, A( P, K ), LDA )
! 519: CALL ZSWAP( N-KK+1, W( K, KKW ), LDW, W( P, KKW ), LDW )
! 520: END IF
! 521: *
! 522: * Updated column KP is already stored in column KKW of W
! 523: *
! 524: IF( KP.NE.KK ) THEN
! 525: *
! 526: * Copy non-updated column KK to column KP
! 527: *
! 528: A( KP, K ) = A( KK, K )
! 529: CALL ZCOPY( K-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ),
! 530: $ LDA )
! 531: CALL ZCOPY( KP, A( 1, KK ), 1, A( 1, KP ), 1 )
! 532: *
! 533: * Interchange rows KK and KP in last N-KK+1 columns
! 534: * of A and W
! 535: *
! 536: CALL ZSWAP( N-KK+1, A( KK, KK ), LDA, A( KP, KK ), LDA )
! 537: CALL ZSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ),
! 538: $ LDW )
! 539: END IF
! 540: *
! 541: IF( KSTEP.EQ.1 ) THEN
! 542: *
! 543: * 1-by-1 pivot block D(k): column KW of W now holds
! 544: *
! 545: * W(k) = U(k)*D(k)
! 546: *
! 547: * where U(k) is the k-th column of U
! 548: *
! 549: * Store U(k) in column k of A
! 550: *
! 551: CALL ZCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
! 552: IF( K.GT.1 ) THEN
! 553: IF( CABS1( A( K, K ) ).GE.SFMIN ) THEN
! 554: R1 = CONE / A( K, K )
! 555: CALL ZSCAL( K-1, R1, A( 1, K ), 1 )
! 556: ELSE IF( A( K, K ).NE.CZERO ) THEN
! 557: DO 14 II = 1, K - 1
! 558: A( II, K ) = A( II, K ) / A( K, K )
! 559: 14 CONTINUE
! 560: END IF
! 561: *
! 562: * Store the superdiagonal element of D in array E
! 563: *
! 564: E( K ) = CZERO
! 565: *
! 566: END IF
! 567: *
! 568: ELSE
! 569: *
! 570: * 2-by-2 pivot block D(k): columns KW and KW-1 of W now
! 571: * hold
! 572: *
! 573: * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
! 574: *
! 575: * where U(k) and U(k-1) are the k-th and (k-1)-th columns
! 576: * of U
! 577: *
! 578: IF( K.GT.2 ) THEN
! 579: *
! 580: * Store U(k) and U(k-1) in columns k and k-1 of A
! 581: *
! 582: D12 = W( K-1, KW )
! 583: D11 = W( K, KW ) / D12
! 584: D22 = W( K-1, KW-1 ) / D12
! 585: T = CONE / ( D11*D22-CONE )
! 586: DO 20 J = 1, K - 2
! 587: A( J, K-1 ) = T*( (D11*W( J, KW-1 )-W( J, KW ) ) /
! 588: $ D12 )
! 589: A( J, K ) = T*( ( D22*W( J, KW )-W( J, KW-1 ) ) /
! 590: $ D12 )
! 591: 20 CONTINUE
! 592: END IF
! 593: *
! 594: * Copy diagonal elements of D(K) to A,
! 595: * copy superdiagonal element of D(K) to E(K) and
! 596: * ZERO out superdiagonal entry of A
! 597: *
! 598: A( K-1, K-1 ) = W( K-1, KW-1 )
! 599: A( K-1, K ) = CZERO
! 600: A( K, K ) = W( K, KW )
! 601: E( K ) = W( K-1, KW )
! 602: E( K-1 ) = CZERO
! 603: *
! 604: END IF
! 605: *
! 606: * End column K is nonsingular
! 607: *
! 608: END IF
! 609: *
! 610: * Store details of the interchanges in IPIV
! 611: *
! 612: IF( KSTEP.EQ.1 ) THEN
! 613: IPIV( K ) = KP
! 614: ELSE
! 615: IPIV( K ) = -P
! 616: IPIV( K-1 ) = -KP
! 617: END IF
! 618: *
! 619: * Decrease K and return to the start of the main loop
! 620: *
! 621: K = K - KSTEP
! 622: GO TO 10
! 623: *
! 624: 30 CONTINUE
! 625: *
! 626: * Update the upper triangle of A11 (= A(1:k,1:k)) as
! 627: *
! 628: * A11 := A11 - U12*D*U12**T = A11 - U12*W**T
! 629: *
! 630: * computing blocks of NB columns at a time
! 631: *
! 632: DO 50 J = ( ( K-1 ) / NB )*NB + 1, 1, -NB
! 633: JB = MIN( NB, K-J+1 )
! 634: *
! 635: * Update the upper triangle of the diagonal block
! 636: *
! 637: DO 40 JJ = J, J + JB - 1
! 638: CALL ZGEMV( 'No transpose', JJ-J+1, N-K, -CONE,
! 639: $ A( J, K+1 ), LDA, W( JJ, KW+1 ), LDW, CONE,
! 640: $ A( J, JJ ), 1 )
! 641: 40 CONTINUE
! 642: *
! 643: * Update the rectangular superdiagonal block
! 644: *
! 645: IF( J.GE.2 )
! 646: $ CALL ZGEMM( 'No transpose', 'Transpose', J-1, JB,
! 647: $ N-K, -CONE, A( 1, K+1 ), LDA, W( J, KW+1 ),
! 648: $ LDW, CONE, A( 1, J ), LDA )
! 649: 50 CONTINUE
! 650: *
! 651: * Set KB to the number of columns factorized
! 652: *
! 653: KB = N - K
! 654: *
! 655: ELSE
! 656: *
! 657: * Factorize the leading columns of A using the lower triangle
! 658: * of A and working forwards, and compute the matrix W = L21*D
! 659: * for use in updating A22
! 660: *
! 661: * Initilize the unused last entry of the subdiagonal array E.
! 662: *
! 663: E( N ) = CZERO
! 664: *
! 665: * K is the main loop index, increasing from 1 in steps of 1 or 2
! 666: *
! 667: K = 1
! 668: 70 CONTINUE
! 669: *
! 670: * Exit from loop
! 671: *
! 672: IF( ( K.GE.NB .AND. NB.LT.N ) .OR. K.GT.N )
! 673: $ GO TO 90
! 674: *
! 675: KSTEP = 1
! 676: P = K
! 677: *
! 678: * Copy column K of A to column K of W and update it
! 679: *
! 680: CALL ZCOPY( N-K+1, A( K, K ), 1, W( K, K ), 1 )
! 681: IF( K.GT.1 )
! 682: $ CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE, A( K, 1 ),
! 683: $ LDA, W( K, 1 ), LDW, CONE, W( K, K ), 1 )
! 684: *
! 685: * Determine rows and columns to be interchanged and whether
! 686: * a 1-by-1 or 2-by-2 pivot block will be used
! 687: *
! 688: ABSAKK = CABS1( W( K, K ) )
! 689: *
! 690: * IMAX is the row-index of the largest off-diagonal element in
! 691: * column K, and COLMAX is its absolute value.
! 692: * Determine both COLMAX and IMAX.
! 693: *
! 694: IF( K.LT.N ) THEN
! 695: IMAX = K + IZAMAX( N-K, W( K+1, K ), 1 )
! 696: COLMAX = CABS1( W( IMAX, K ) )
! 697: ELSE
! 698: COLMAX = ZERO
! 699: END IF
! 700: *
! 701: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
! 702: *
! 703: * Column K is zero or underflow: set INFO and continue
! 704: *
! 705: IF( INFO.EQ.0 )
! 706: $ INFO = K
! 707: KP = K
! 708: CALL ZCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
! 709: *
! 710: * Set E( K ) to zero
! 711: *
! 712: IF( K.LT.N )
! 713: $ E( K ) = CZERO
! 714: *
! 715: ELSE
! 716: *
! 717: * ============================================================
! 718: *
! 719: * Test for interchange
! 720: *
! 721: * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
! 722: * (used to handle NaN and Inf)
! 723: *
! 724: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
! 725: *
! 726: * no interchange, use 1-by-1 pivot block
! 727: *
! 728: KP = K
! 729: *
! 730: ELSE
! 731: *
! 732: DONE = .FALSE.
! 733: *
! 734: * Loop until pivot found
! 735: *
! 736: 72 CONTINUE
! 737: *
! 738: * Begin pivot search loop body
! 739: *
! 740: *
! 741: * Copy column IMAX to column K+1 of W and update it
! 742: *
! 743: CALL ZCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1)
! 744: CALL ZCOPY( N-IMAX+1, A( IMAX, IMAX ), 1,
! 745: $ W( IMAX, K+1 ), 1 )
! 746: IF( K.GT.1 )
! 747: $ CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE,
! 748: $ A( K, 1 ), LDA, W( IMAX, 1 ), LDW,
! 749: $ CONE, W( K, K+1 ), 1 )
! 750: *
! 751: * JMAX is the column-index of the largest off-diagonal
! 752: * element in row IMAX, and ROWMAX is its absolute value.
! 753: * Determine both ROWMAX and JMAX.
! 754: *
! 755: IF( IMAX.NE.K ) THEN
! 756: JMAX = K - 1 + IZAMAX( IMAX-K, W( K, K+1 ), 1 )
! 757: ROWMAX = CABS1( W( JMAX, K+1 ) )
! 758: ELSE
! 759: ROWMAX = ZERO
! 760: END IF
! 761: *
! 762: IF( IMAX.LT.N ) THEN
! 763: ITEMP = IMAX + IZAMAX( N-IMAX, W( IMAX+1, K+1 ), 1)
! 764: DTEMP = CABS1( W( ITEMP, K+1 ) )
! 765: IF( DTEMP.GT.ROWMAX ) THEN
! 766: ROWMAX = DTEMP
! 767: JMAX = ITEMP
! 768: END IF
! 769: END IF
! 770: *
! 771: * Equivalent to testing for
! 772: * CABS1( W( IMAX, K+1 ) ).GE.ALPHA*ROWMAX
! 773: * (used to handle NaN and Inf)
! 774: *
! 775: IF( .NOT.( CABS1( W( IMAX, K+1 ) ).LT.ALPHA*ROWMAX ) )
! 776: $ THEN
! 777: *
! 778: * interchange rows and columns K and IMAX,
! 779: * use 1-by-1 pivot block
! 780: *
! 781: KP = IMAX
! 782: *
! 783: * copy column K+1 of W to column K of W
! 784: *
! 785: CALL ZCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
! 786: *
! 787: DONE = .TRUE.
! 788: *
! 789: * Equivalent to testing for ROWMAX.EQ.COLMAX,
! 790: * (used to handle NaN and Inf)
! 791: *
! 792: ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
! 793: $ THEN
! 794: *
! 795: * interchange rows and columns K+1 and IMAX,
! 796: * use 2-by-2 pivot block
! 797: *
! 798: KP = IMAX
! 799: KSTEP = 2
! 800: DONE = .TRUE.
! 801: ELSE
! 802: *
! 803: * Pivot not found: set params and repeat
! 804: *
! 805: P = IMAX
! 806: COLMAX = ROWMAX
! 807: IMAX = JMAX
! 808: *
! 809: * Copy updated JMAXth (next IMAXth) column to Kth of W
! 810: *
! 811: CALL ZCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
! 812: *
! 813: END IF
! 814: *
! 815: * End pivot search loop body
! 816: *
! 817: IF( .NOT. DONE ) GOTO 72
! 818: *
! 819: END IF
! 820: *
! 821: * ============================================================
! 822: *
! 823: KK = K + KSTEP - 1
! 824: *
! 825: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
! 826: *
! 827: * Copy non-updated column K to column P
! 828: *
! 829: CALL ZCOPY( P-K, A( K, K ), 1, A( P, K ), LDA )
! 830: CALL ZCOPY( N-P+1, A( P, K ), 1, A( P, P ), 1 )
! 831: *
! 832: * Interchange rows K and P in first K columns of A
! 833: * and first K+1 columns of W
! 834: *
! 835: CALL ZSWAP( K, A( K, 1 ), LDA, A( P, 1 ), LDA )
! 836: CALL ZSWAP( KK, W( K, 1 ), LDW, W( P, 1 ), LDW )
! 837: END IF
! 838: *
! 839: * Updated column KP is already stored in column KK of W
! 840: *
! 841: IF( KP.NE.KK ) THEN
! 842: *
! 843: * Copy non-updated column KK to column KP
! 844: *
! 845: A( KP, K ) = A( KK, K )
! 846: CALL ZCOPY( KP-K-1, A( K+1, KK ), 1, A( KP, K+1 ), LDA )
! 847: CALL ZCOPY( N-KP+1, A( KP, KK ), 1, A( KP, KP ), 1 )
! 848: *
! 849: * Interchange rows KK and KP in first KK columns of A and W
! 850: *
! 851: CALL ZSWAP( KK, A( KK, 1 ), LDA, A( KP, 1 ), LDA )
! 852: CALL ZSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW )
! 853: END IF
! 854: *
! 855: IF( KSTEP.EQ.1 ) THEN
! 856: *
! 857: * 1-by-1 pivot block D(k): column k of W now holds
! 858: *
! 859: * W(k) = L(k)*D(k)
! 860: *
! 861: * where L(k) is the k-th column of L
! 862: *
! 863: * Store L(k) in column k of A
! 864: *
! 865: CALL ZCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
! 866: IF( K.LT.N ) THEN
! 867: IF( CABS1( A( K, K ) ).GE.SFMIN ) THEN
! 868: R1 = CONE / A( K, K )
! 869: CALL ZSCAL( N-K, R1, A( K+1, K ), 1 )
! 870: ELSE IF( A( K, K ).NE.CZERO ) THEN
! 871: DO 74 II = K + 1, N
! 872: A( II, K ) = A( II, K ) / A( K, K )
! 873: 74 CONTINUE
! 874: END IF
! 875: *
! 876: * Store the subdiagonal element of D in array E
! 877: *
! 878: E( K ) = CZERO
! 879: *
! 880: END IF
! 881: *
! 882: ELSE
! 883: *
! 884: * 2-by-2 pivot block D(k): columns k and k+1 of W now hold
! 885: *
! 886: * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
! 887: *
! 888: * where L(k) and L(k+1) are the k-th and (k+1)-th columns
! 889: * of L
! 890: *
! 891: IF( K.LT.N-1 ) THEN
! 892: *
! 893: * Store L(k) and L(k+1) in columns k and k+1 of A
! 894: *
! 895: D21 = W( K+1, K )
! 896: D11 = W( K+1, K+1 ) / D21
! 897: D22 = W( K, K ) / D21
! 898: T = CONE / ( D11*D22-CONE )
! 899: DO 80 J = K + 2, N
! 900: A( J, K ) = T*( ( D11*W( J, K )-W( J, K+1 ) ) /
! 901: $ D21 )
! 902: A( J, K+1 ) = T*( ( D22*W( J, K+1 )-W( J, K ) ) /
! 903: $ D21 )
! 904: 80 CONTINUE
! 905: END IF
! 906: *
! 907: * Copy diagonal elements of D(K) to A,
! 908: * copy subdiagonal element of D(K) to E(K) and
! 909: * ZERO out subdiagonal entry of A
! 910: *
! 911: A( K, K ) = W( K, K )
! 912: A( K+1, K ) = CZERO
! 913: A( K+1, K+1 ) = W( K+1, K+1 )
! 914: E( K ) = W( K+1, K )
! 915: E( K+1 ) = CZERO
! 916: *
! 917: END IF
! 918: *
! 919: * End column K is nonsingular
! 920: *
! 921: END IF
! 922: *
! 923: * Store details of the interchanges in IPIV
! 924: *
! 925: IF( KSTEP.EQ.1 ) THEN
! 926: IPIV( K ) = KP
! 927: ELSE
! 928: IPIV( K ) = -P
! 929: IPIV( K+1 ) = -KP
! 930: END IF
! 931: *
! 932: * Increase K and return to the start of the main loop
! 933: *
! 934: K = K + KSTEP
! 935: GO TO 70
! 936: *
! 937: 90 CONTINUE
! 938: *
! 939: * Update the lower triangle of A22 (= A(k:n,k:n)) as
! 940: *
! 941: * A22 := A22 - L21*D*L21**T = A22 - L21*W**T
! 942: *
! 943: * computing blocks of NB columns at a time
! 944: *
! 945: DO 110 J = K, N, NB
! 946: JB = MIN( NB, N-J+1 )
! 947: *
! 948: * Update the lower triangle of the diagonal block
! 949: *
! 950: DO 100 JJ = J, J + JB - 1
! 951: CALL ZGEMV( 'No transpose', J+JB-JJ, K-1, -CONE,
! 952: $ A( JJ, 1 ), LDA, W( JJ, 1 ), LDW, CONE,
! 953: $ A( JJ, JJ ), 1 )
! 954: 100 CONTINUE
! 955: *
! 956: * Update the rectangular subdiagonal block
! 957: *
! 958: IF( J+JB.LE.N )
! 959: $ CALL ZGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB,
! 960: $ K-1, -CONE, A( J+JB, 1 ), LDA, W( J, 1 ),
! 961: $ LDW, CONE, A( J+JB, J ), LDA )
! 962: 110 CONTINUE
! 963: *
! 964: * Set KB to the number of columns factorized
! 965: *
! 966: KB = K - 1
! 967: *
! 968: END IF
! 969: *
! 970: RETURN
! 971: *
! 972: * End of ZLASYF_RK
! 973: *
! 974: END
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