Annotation of rpl/lapack/lapack/zhetf2_rook.f, revision 1.1
1.1 ! bertrand 1: *> \brief \b ZHETF2_ROOK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman ("rook") diagonal pivoting method (unblocked algorithm).
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download ZHETF2_ROOK + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zhetf2_rook.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zhetf2_rook.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zhetf2_rook.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE ZHETF2_ROOK( UPLO, N, A, LDA, IPIV, INFO )
! 22: *
! 23: * .. Scalar Arguments ..
! 24: * CHARACTER UPLO
! 25: * INTEGER INFO, LDA, N
! 26: * ..
! 27: * .. Array Arguments ..
! 28: * INTEGER IPIV( * )
! 29: * COMPLEX*16 A( LDA, * )
! 30: * ..
! 31: *
! 32: *
! 33: *> \par Purpose:
! 34: * =============
! 35: *>
! 36: *> \verbatim
! 37: *>
! 38: *> ZHETF2_ROOK computes the factorization of a complex Hermitian matrix A
! 39: *> using the bounded Bunch-Kaufman ("rook") diagonal pivoting method:
! 40: *>
! 41: *> A = U*D*U**H or A = L*D*L**H
! 42: *>
! 43: *> where U (or L) is a product of permutation and unit upper (lower)
! 44: *> triangular matrices, U**H is the conjugate transpose of U, and D is
! 45: *> Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
! 46: *>
! 47: *> This is the unblocked version of the algorithm, calling Level 2 BLAS.
! 48: *> \endverbatim
! 49: *
! 50: * Arguments:
! 51: * ==========
! 52: *
! 53: *> \param[in] UPLO
! 54: *> \verbatim
! 55: *> UPLO is CHARACTER*1
! 56: *> Specifies whether the upper or lower triangular part of the
! 57: *> Hermitian matrix A is stored:
! 58: *> = 'U': Upper triangular
! 59: *> = 'L': Lower triangular
! 60: *> \endverbatim
! 61: *>
! 62: *> \param[in] N
! 63: *> \verbatim
! 64: *> N is INTEGER
! 65: *> The order of the matrix A. N >= 0.
! 66: *> \endverbatim
! 67: *>
! 68: *> \param[in,out] A
! 69: *> \verbatim
! 70: *> A is COMPLEX*16 array, dimension (LDA,N)
! 71: *> On entry, the Hermitian matrix A. If UPLO = 'U', the leading
! 72: *> n-by-n upper triangular part of A contains the upper
! 73: *> triangular part of the matrix A, and the strictly lower
! 74: *> triangular part of A is not referenced. If UPLO = 'L', the
! 75: *> leading n-by-n lower triangular part of A contains the lower
! 76: *> triangular part of the matrix A, and the strictly upper
! 77: *> triangular part of A is not referenced.
! 78: *>
! 79: *> On exit, the block diagonal matrix D and the multipliers used
! 80: *> to obtain the factor U or L (see below for further details).
! 81: *> \endverbatim
! 82: *>
! 83: *> \param[in] LDA
! 84: *> \verbatim
! 85: *> LDA is INTEGER
! 86: *> The leading dimension of the array A. LDA >= max(1,N).
! 87: *> \endverbatim
! 88: *>
! 89: *> \param[out] IPIV
! 90: *> \verbatim
! 91: *> IPIV is INTEGER array, dimension (N)
! 92: *> Details of the interchanges and the block structure of D.
! 93: *>
! 94: *> If UPLO = 'U':
! 95: *> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
! 96: *> interchanged and D(k,k) is a 1-by-1 diagonal block.
! 97: *>
! 98: *> If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and
! 99: *> columns k and -IPIV(k) were interchanged and rows and
! 100: *> columns k-1 and -IPIV(k-1) were inerchaged,
! 101: *> D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
! 102: *>
! 103: *> If UPLO = 'L':
! 104: *> If IPIV(k) > 0, then rows and columns k and IPIV(k)
! 105: *> were interchanged and D(k,k) is a 1-by-1 diagonal block.
! 106: *>
! 107: *> If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and
! 108: *> columns k and -IPIV(k) were interchanged and rows and
! 109: *> columns k+1 and -IPIV(k+1) were inerchaged,
! 110: *> D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
! 111: *> \endverbatim
! 112: *>
! 113: *> \param[out] INFO
! 114: *> \verbatim
! 115: *> INFO is INTEGER
! 116: *> = 0: successful exit
! 117: *> < 0: if INFO = -k, the k-th argument had an illegal value
! 118: *> > 0: if INFO = k, D(k,k) is exactly zero. The factorization
! 119: *> has been completed, but the block diagonal matrix D is
! 120: *> exactly singular, and division by zero will occur if it
! 121: *> is used to solve a system of equations.
! 122: *> \endverbatim
! 123: *
! 124: * Authors:
! 125: * ========
! 126: *
! 127: *> \author Univ. of Tennessee
! 128: *> \author Univ. of California Berkeley
! 129: *> \author Univ. of Colorado Denver
! 130: *> \author NAG Ltd.
! 131: *
! 132: *> \date November 2013
! 133: *
! 134: *> \ingroup complex16HEcomputational
! 135: *
! 136: *> \par Further Details:
! 137: * =====================
! 138: *>
! 139: *> \verbatim
! 140: *>
! 141: *> If UPLO = 'U', then A = U*D*U**H, where
! 142: *> U = P(n)*U(n)* ... *P(k)U(k)* ...,
! 143: *> i.e., U is a product of terms P(k)*U(k), where k decreases from n to
! 144: *> 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
! 145: *> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
! 146: *> defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
! 147: *> that if the diagonal block D(k) is of order s (s = 1 or 2), then
! 148: *>
! 149: *> ( I v 0 ) k-s
! 150: *> U(k) = ( 0 I 0 ) s
! 151: *> ( 0 0 I ) n-k
! 152: *> k-s s n-k
! 153: *>
! 154: *> If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
! 155: *> If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
! 156: *> and A(k,k), and v overwrites A(1:k-2,k-1:k).
! 157: *>
! 158: *> If UPLO = 'L', then A = L*D*L**H, where
! 159: *> L = P(1)*L(1)* ... *P(k)*L(k)* ...,
! 160: *> i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
! 161: *> n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
! 162: *> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
! 163: *> defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
! 164: *> that if the diagonal block D(k) is of order s (s = 1 or 2), then
! 165: *>
! 166: *> ( I 0 0 ) k-1
! 167: *> L(k) = ( 0 I 0 ) s
! 168: *> ( 0 v I ) n-k-s+1
! 169: *> k-1 s n-k-s+1
! 170: *>
! 171: *> If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
! 172: *> If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
! 173: *> and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
! 174: *> \endverbatim
! 175: *
! 176: *> \par Contributors:
! 177: * ==================
! 178: *>
! 179: *> \verbatim
! 180: *>
! 181: *> November 2013, Igor Kozachenko,
! 182: *> Computer Science Division,
! 183: *> University of California, Berkeley
! 184: *>
! 185: *> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
! 186: *> School of Mathematics,
! 187: *> University of Manchester
! 188: *>
! 189: *> 01-01-96 - Based on modifications by
! 190: *> J. Lewis, Boeing Computer Services Company
! 191: *> A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
! 192: *> \endverbatim
! 193: *
! 194: * =====================================================================
! 195: SUBROUTINE ZHETF2_ROOK( UPLO, N, A, LDA, IPIV, INFO )
! 196: *
! 197: * -- LAPACK computational routine (version 3.5.0) --
! 198: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 199: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 200: * November 2013
! 201: *
! 202: * .. Scalar Arguments ..
! 203: CHARACTER UPLO
! 204: INTEGER INFO, LDA, N
! 205: * ..
! 206: * .. Array Arguments ..
! 207: INTEGER IPIV( * )
! 208: COMPLEX*16 A( LDA, * )
! 209: * ..
! 210: *
! 211: * ======================================================================
! 212: *
! 213: * .. Parameters ..
! 214: DOUBLE PRECISION ZERO, ONE
! 215: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
! 216: DOUBLE PRECISION EIGHT, SEVTEN
! 217: PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
! 218: * ..
! 219: * .. Local Scalars ..
! 220: LOGICAL DONE, UPPER
! 221: INTEGER I, II, IMAX, ITEMP, J, JMAX, K, KK, KP, KSTEP,
! 222: $ P
! 223: DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D, D11, D22, R1, DTEMP,
! 224: $ ROWMAX, TT, SFMIN
! 225: COMPLEX*16 D12, D21, T, WK, WKM1, WKP1, Z
! 226: * ..
! 227: * .. External Functions ..
! 228: *
! 229: LOGICAL LSAME
! 230: INTEGER IZAMAX
! 231: DOUBLE PRECISION DLAMCH, DLAPY2
! 232: EXTERNAL LSAME, IZAMAX, DLAMCH, DLAPY2
! 233: * ..
! 234: * .. External Subroutines ..
! 235: EXTERNAL XERBLA, ZDSCAL, ZHER, ZSWAP
! 236: * ..
! 237: * .. Intrinsic Functions ..
! 238: INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX, SQRT
! 239: * ..
! 240: * .. Statement Functions ..
! 241: DOUBLE PRECISION CABS1
! 242: * ..
! 243: * .. Statement Function definitions ..
! 244: CABS1( Z ) = ABS( DBLE( Z ) ) + ABS( DIMAG( Z ) )
! 245: * ..
! 246: * .. Executable Statements ..
! 247: *
! 248: * Test the input parameters.
! 249: *
! 250: INFO = 0
! 251: UPPER = LSAME( UPLO, 'U' )
! 252: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
! 253: INFO = -1
! 254: ELSE IF( N.LT.0 ) THEN
! 255: INFO = -2
! 256: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
! 257: INFO = -4
! 258: END IF
! 259: IF( INFO.NE.0 ) THEN
! 260: CALL XERBLA( 'ZHETF2_ROOK', -INFO )
! 261: RETURN
! 262: END IF
! 263: *
! 264: * Initialize ALPHA for use in choosing pivot block size.
! 265: *
! 266: ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
! 267: *
! 268: * Compute machine safe minimum
! 269: *
! 270: SFMIN = DLAMCH( 'S' )
! 271: *
! 272: IF( UPPER ) THEN
! 273: *
! 274: * Factorize A as U*D*U**H using the upper triangle of A
! 275: *
! 276: * K is the main loop index, decreasing from N to 1 in steps of
! 277: * 1 or 2
! 278: *
! 279: K = N
! 280: 10 CONTINUE
! 281: *
! 282: * If K < 1, exit from loop
! 283: *
! 284: IF( K.LT.1 )
! 285: $ GO TO 70
! 286: KSTEP = 1
! 287: P = K
! 288: *
! 289: * Determine rows and columns to be interchanged and whether
! 290: * a 1-by-1 or 2-by-2 pivot block will be used
! 291: *
! 292: ABSAKK = ABS( DBLE( A( K, K ) ) )
! 293: *
! 294: * IMAX is the row-index of the largest off-diagonal element in
! 295: * column K, and COLMAX is its absolute value.
! 296: * Determine both COLMAX and IMAX.
! 297: *
! 298: IF( K.GT.1 ) THEN
! 299: IMAX = IZAMAX( K-1, A( 1, K ), 1 )
! 300: COLMAX = CABS1( A( IMAX, K ) )
! 301: ELSE
! 302: COLMAX = ZERO
! 303: END IF
! 304: *
! 305: IF( ( MAX( ABSAKK, COLMAX ).EQ.ZERO ) ) THEN
! 306: *
! 307: * Column K is zero or underflow: set INFO and continue
! 308: *
! 309: IF( INFO.EQ.0 )
! 310: $ INFO = K
! 311: KP = K
! 312: A( K, K ) = DBLE( A( K, K ) )
! 313: ELSE
! 314: *
! 315: * ============================================================
! 316: *
! 317: * BEGIN pivot search
! 318: *
! 319: * Case(1)
! 320: * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
! 321: * (used to handle NaN and Inf)
! 322: *
! 323: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
! 324: *
! 325: * no interchange, use 1-by-1 pivot block
! 326: *
! 327: KP = K
! 328: *
! 329: ELSE
! 330: *
! 331: DONE = .FALSE.
! 332: *
! 333: * Loop until pivot found
! 334: *
! 335: 12 CONTINUE
! 336: *
! 337: * BEGIN pivot search loop body
! 338: *
! 339: *
! 340: * JMAX is the column-index of the largest off-diagonal
! 341: * element in row IMAX, and ROWMAX is its absolute value.
! 342: * Determine both ROWMAX and JMAX.
! 343: *
! 344: IF( IMAX.NE.K ) THEN
! 345: JMAX = IMAX + IZAMAX( K-IMAX, A( IMAX, IMAX+1 ),
! 346: $ LDA )
! 347: ROWMAX = CABS1( A( IMAX, JMAX ) )
! 348: ELSE
! 349: ROWMAX = ZERO
! 350: END IF
! 351: *
! 352: IF( IMAX.GT.1 ) THEN
! 353: ITEMP = IZAMAX( IMAX-1, A( 1, IMAX ), 1 )
! 354: DTEMP = CABS1( A( ITEMP, IMAX ) )
! 355: IF( DTEMP.GT.ROWMAX ) THEN
! 356: ROWMAX = DTEMP
! 357: JMAX = ITEMP
! 358: END IF
! 359: END IF
! 360: *
! 361: * Case(2)
! 362: * Equivalent to testing for
! 363: * ABS( REAL( W( IMAX,KW-1 ) ) ).GE.ALPHA*ROWMAX
! 364: * (used to handle NaN and Inf)
! 365: *
! 366: IF( .NOT.( ABS( DBLE( A( IMAX, IMAX ) ) )
! 367: $ .LT.ALPHA*ROWMAX ) ) THEN
! 368: *
! 369: * interchange rows and columns K and IMAX,
! 370: * use 1-by-1 pivot block
! 371: *
! 372: KP = IMAX
! 373: DONE = .TRUE.
! 374: *
! 375: * Case(3)
! 376: * Equivalent to testing for ROWMAX.EQ.COLMAX,
! 377: * (used to handle NaN and Inf)
! 378: *
! 379: ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
! 380: $ THEN
! 381: *
! 382: * interchange rows and columns K-1 and IMAX,
! 383: * use 2-by-2 pivot block
! 384: *
! 385: KP = IMAX
! 386: KSTEP = 2
! 387: DONE = .TRUE.
! 388: *
! 389: * Case(4)
! 390: ELSE
! 391: *
! 392: * Pivot not found: set params and repeat
! 393: *
! 394: P = IMAX
! 395: COLMAX = ROWMAX
! 396: IMAX = JMAX
! 397: END IF
! 398: *
! 399: * END pivot search loop body
! 400: *
! 401: IF( .NOT.DONE ) GOTO 12
! 402: *
! 403: END IF
! 404: *
! 405: * END pivot search
! 406: *
! 407: * ============================================================
! 408: *
! 409: * KK is the column of A where pivoting step stopped
! 410: *
! 411: KK = K - KSTEP + 1
! 412: *
! 413: * For only a 2x2 pivot, interchange rows and columns K and P
! 414: * in the leading submatrix A(1:k,1:k)
! 415: *
! 416: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
! 417: * (1) Swap columnar parts
! 418: IF( P.GT.1 )
! 419: $ CALL ZSWAP( P-1, A( 1, K ), 1, A( 1, P ), 1 )
! 420: * (2) Swap and conjugate middle parts
! 421: DO 14 J = P + 1, K - 1
! 422: T = DCONJG( A( J, K ) )
! 423: A( J, K ) = DCONJG( A( P, J ) )
! 424: A( P, J ) = T
! 425: 14 CONTINUE
! 426: * (3) Swap and conjugate corner elements at row-col interserction
! 427: A( P, K ) = DCONJG( A( P, K ) )
! 428: * (4) Swap diagonal elements at row-col intersection
! 429: R1 = DBLE( A( K, K ) )
! 430: A( K, K ) = DBLE( A( P, P ) )
! 431: A( P, P ) = R1
! 432: END IF
! 433: *
! 434: * For both 1x1 and 2x2 pivots, interchange rows and
! 435: * columns KK and KP in the leading submatrix A(1:k,1:k)
! 436: *
! 437: IF( KP.NE.KK ) THEN
! 438: * (1) Swap columnar parts
! 439: IF( KP.GT.1 )
! 440: $ CALL ZSWAP( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
! 441: * (2) Swap and conjugate middle parts
! 442: DO 15 J = KP + 1, KK - 1
! 443: T = DCONJG( A( J, KK ) )
! 444: A( J, KK ) = DCONJG( A( KP, J ) )
! 445: A( KP, J ) = T
! 446: 15 CONTINUE
! 447: * (3) Swap and conjugate corner elements at row-col interserction
! 448: A( KP, KK ) = DCONJG( A( KP, KK ) )
! 449: * (4) Swap diagonal elements at row-col intersection
! 450: R1 = DBLE( A( KK, KK ) )
! 451: A( KK, KK ) = DBLE( A( KP, KP ) )
! 452: A( KP, KP ) = R1
! 453: *
! 454: IF( KSTEP.EQ.2 ) THEN
! 455: * (*) Make sure that diagonal element of pivot is real
! 456: A( K, K ) = DBLE( A( K, K ) )
! 457: * (5) Swap row elements
! 458: T = A( K-1, K )
! 459: A( K-1, K ) = A( KP, K )
! 460: A( KP, K ) = T
! 461: END IF
! 462: ELSE
! 463: * (*) Make sure that diagonal element of pivot is real
! 464: A( K, K ) = DBLE( A( K, K ) )
! 465: IF( KSTEP.EQ.2 )
! 466: $ A( K-1, K-1 ) = DBLE( A( K-1, K-1 ) )
! 467: END IF
! 468: *
! 469: * Update the leading submatrix
! 470: *
! 471: IF( KSTEP.EQ.1 ) THEN
! 472: *
! 473: * 1-by-1 pivot block D(k): column k now holds
! 474: *
! 475: * W(k) = U(k)*D(k)
! 476: *
! 477: * where U(k) is the k-th column of U
! 478: *
! 479: IF( K.GT.1 ) THEN
! 480: *
! 481: * Perform a rank-1 update of A(1:k-1,1:k-1) and
! 482: * store U(k) in column k
! 483: *
! 484: IF( ABS( DBLE( A( K, K ) ) ).GE.SFMIN ) THEN
! 485: *
! 486: * Perform a rank-1 update of A(1:k-1,1:k-1) as
! 487: * A := A - U(k)*D(k)*U(k)**T
! 488: * = A - W(k)*1/D(k)*W(k)**T
! 489: *
! 490: D11 = ONE / DBLE( A( K, K ) )
! 491: CALL ZHER( UPLO, K-1, -D11, A( 1, K ), 1, A, LDA )
! 492: *
! 493: * Store U(k) in column k
! 494: *
! 495: CALL ZDSCAL( K-1, D11, A( 1, K ), 1 )
! 496: ELSE
! 497: *
! 498: * Store L(k) in column K
! 499: *
! 500: D11 = DBLE( A( K, K ) )
! 501: DO 16 II = 1, K - 1
! 502: A( II, K ) = A( II, K ) / D11
! 503: 16 CONTINUE
! 504: *
! 505: * Perform a rank-1 update of A(k+1:n,k+1:n) as
! 506: * A := A - U(k)*D(k)*U(k)**T
! 507: * = A - W(k)*(1/D(k))*W(k)**T
! 508: * = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T
! 509: *
! 510: CALL ZHER( UPLO, K-1, -D11, A( 1, K ), 1, A, LDA )
! 511: END IF
! 512: END IF
! 513: *
! 514: ELSE
! 515: *
! 516: * 2-by-2 pivot block D(k): columns k and k-1 now hold
! 517: *
! 518: * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
! 519: *
! 520: * where U(k) and U(k-1) are the k-th and (k-1)-th columns
! 521: * of U
! 522: *
! 523: * Perform a rank-2 update of A(1:k-2,1:k-2) as
! 524: *
! 525: * A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T
! 526: * = A - ( ( A(k-1)A(k) )*inv(D(k)) ) * ( A(k-1)A(k) )**T
! 527: *
! 528: * and store L(k) and L(k+1) in columns k and k+1
! 529: *
! 530: IF( K.GT.2 ) THEN
! 531: * D = |A12|
! 532: D = DLAPY2( DBLE( A( K-1, K ) ),
! 533: $ DIMAG( A( K-1, K ) ) )
! 534: D11 = A( K, K ) / D
! 535: D22 = A( K-1, K-1 ) / D
! 536: D12 = A( K-1, K ) / D
! 537: TT = ONE / ( D11*D22-ONE )
! 538: *
! 539: DO 30 J = K - 2, 1, -1
! 540: *
! 541: * Compute D21 * ( W(k)W(k+1) ) * inv(D(k)) for row J
! 542: *
! 543: WKM1 = TT*( D11*A( J, K-1 )-DCONJG( D12 )*
! 544: $ A( J, K ) )
! 545: WK = TT*( D22*A( J, K )-D12*A( J, K-1 ) )
! 546: *
! 547: * Perform a rank-2 update of A(1:k-2,1:k-2)
! 548: *
! 549: DO 20 I = J, 1, -1
! 550: A( I, J ) = A( I, J ) -
! 551: $ ( A( I, K ) / D )*DCONJG( WK ) -
! 552: $ ( A( I, K-1 ) / D )*DCONJG( WKM1 )
! 553: 20 CONTINUE
! 554: *
! 555: * Store U(k) and U(k-1) in cols k and k-1 for row J
! 556: *
! 557: A( J, K ) = WK / D
! 558: A( J, K-1 ) = WKM1 / D
! 559: * (*) Make sure that diagonal element of pivot is real
! 560: A( J, J ) = DCMPLX( DBLE( A( J, J ) ), ZERO )
! 561: *
! 562: 30 CONTINUE
! 563: *
! 564: END IF
! 565: *
! 566: END IF
! 567: *
! 568: END IF
! 569: *
! 570: * Store details of the interchanges in IPIV
! 571: *
! 572: IF( KSTEP.EQ.1 ) THEN
! 573: IPIV( K ) = KP
! 574: ELSE
! 575: IPIV( K ) = -P
! 576: IPIV( K-1 ) = -KP
! 577: END IF
! 578: *
! 579: * Decrease K and return to the start of the main loop
! 580: *
! 581: K = K - KSTEP
! 582: GO TO 10
! 583: *
! 584: ELSE
! 585: *
! 586: * Factorize A as L*D*L**H using the lower triangle of A
! 587: *
! 588: * K is the main loop index, increasing from 1 to N in steps of
! 589: * 1 or 2
! 590: *
! 591: K = 1
! 592: 40 CONTINUE
! 593: *
! 594: * If K > N, exit from loop
! 595: *
! 596: IF( K.GT.N )
! 597: $ GO TO 70
! 598: KSTEP = 1
! 599: P = K
! 600: *
! 601: * Determine rows and columns to be interchanged and whether
! 602: * a 1-by-1 or 2-by-2 pivot block will be used
! 603: *
! 604: ABSAKK = ABS( DBLE( A( K, K ) ) )
! 605: *
! 606: * IMAX is the row-index of the largest off-diagonal element in
! 607: * column K, and COLMAX is its absolute value.
! 608: * Determine both COLMAX and IMAX.
! 609: *
! 610: IF( K.LT.N ) THEN
! 611: IMAX = K + IZAMAX( N-K, A( K+1, K ), 1 )
! 612: COLMAX = CABS1( A( IMAX, K ) )
! 613: ELSE
! 614: COLMAX = ZERO
! 615: END IF
! 616: *
! 617: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
! 618: *
! 619: * Column K is zero or underflow: set INFO and continue
! 620: *
! 621: IF( INFO.EQ.0 )
! 622: $ INFO = K
! 623: KP = K
! 624: A( K, K ) = DBLE( A( K, K ) )
! 625: ELSE
! 626: *
! 627: * ============================================================
! 628: *
! 629: * BEGIN pivot search
! 630: *
! 631: * Case(1)
! 632: * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
! 633: * (used to handle NaN and Inf)
! 634: *
! 635: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
! 636: *
! 637: * no interchange, use 1-by-1 pivot block
! 638: *
! 639: KP = K
! 640: *
! 641: ELSE
! 642: *
! 643: DONE = .FALSE.
! 644: *
! 645: * Loop until pivot found
! 646: *
! 647: 42 CONTINUE
! 648: *
! 649: * BEGIN pivot search loop body
! 650: *
! 651: *
! 652: * JMAX is the column-index of the largest off-diagonal
! 653: * element in row IMAX, and ROWMAX is its absolute value.
! 654: * Determine both ROWMAX and JMAX.
! 655: *
! 656: IF( IMAX.NE.K ) THEN
! 657: JMAX = K - 1 + IZAMAX( IMAX-K, A( IMAX, K ), LDA )
! 658: ROWMAX = CABS1( A( IMAX, JMAX ) )
! 659: ELSE
! 660: ROWMAX = ZERO
! 661: END IF
! 662: *
! 663: IF( IMAX.LT.N ) THEN
! 664: ITEMP = IMAX + IZAMAX( N-IMAX, A( IMAX+1, IMAX ),
! 665: $ 1 )
! 666: DTEMP = CABS1( A( ITEMP, IMAX ) )
! 667: IF( DTEMP.GT.ROWMAX ) THEN
! 668: ROWMAX = DTEMP
! 669: JMAX = ITEMP
! 670: END IF
! 671: END IF
! 672: *
! 673: * Case(2)
! 674: * Equivalent to testing for
! 675: * ABS( REAL( W( IMAX,KW-1 ) ) ).GE.ALPHA*ROWMAX
! 676: * (used to handle NaN and Inf)
! 677: *
! 678: IF( .NOT.( ABS( DBLE( A( IMAX, IMAX ) ) )
! 679: $ .LT.ALPHA*ROWMAX ) ) THEN
! 680: *
! 681: * interchange rows and columns K and IMAX,
! 682: * use 1-by-1 pivot block
! 683: *
! 684: KP = IMAX
! 685: DONE = .TRUE.
! 686: *
! 687: * Case(3)
! 688: * Equivalent to testing for ROWMAX.EQ.COLMAX,
! 689: * (used to handle NaN and Inf)
! 690: *
! 691: ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
! 692: $ THEN
! 693: *
! 694: * interchange rows and columns K+1 and IMAX,
! 695: * use 2-by-2 pivot block
! 696: *
! 697: KP = IMAX
! 698: KSTEP = 2
! 699: DONE = .TRUE.
! 700: *
! 701: * Case(4)
! 702: ELSE
! 703: *
! 704: * Pivot not found: set params and repeat
! 705: *
! 706: P = IMAX
! 707: COLMAX = ROWMAX
! 708: IMAX = JMAX
! 709: END IF
! 710: *
! 711: *
! 712: * END pivot search loop body
! 713: *
! 714: IF( .NOT.DONE ) GOTO 42
! 715: *
! 716: END IF
! 717: *
! 718: * END pivot search
! 719: *
! 720: * ============================================================
! 721: *
! 722: * KK is the column of A where pivoting step stopped
! 723: *
! 724: KK = K + KSTEP - 1
! 725: *
! 726: * For only a 2x2 pivot, interchange rows and columns K and P
! 727: * in the trailing submatrix A(k:n,k:n)
! 728: *
! 729: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
! 730: * (1) Swap columnar parts
! 731: IF( P.LT.N )
! 732: $ CALL ZSWAP( N-P, A( P+1, K ), 1, A( P+1, P ), 1 )
! 733: * (2) Swap and conjugate middle parts
! 734: DO 44 J = K + 1, P - 1
! 735: T = DCONJG( A( J, K ) )
! 736: A( J, K ) = DCONJG( A( P, J ) )
! 737: A( P, J ) = T
! 738: 44 CONTINUE
! 739: * (3) Swap and conjugate corner elements at row-col interserction
! 740: A( P, K ) = DCONJG( A( P, K ) )
! 741: * (4) Swap diagonal elements at row-col intersection
! 742: R1 = DBLE( A( K, K ) )
! 743: A( K, K ) = DBLE( A( P, P ) )
! 744: A( P, P ) = R1
! 745: END IF
! 746: *
! 747: * For both 1x1 and 2x2 pivots, interchange rows and
! 748: * columns KK and KP in the trailing submatrix A(k:n,k:n)
! 749: *
! 750: IF( KP.NE.KK ) THEN
! 751: * (1) Swap columnar parts
! 752: IF( KP.LT.N )
! 753: $ CALL ZSWAP( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 )
! 754: * (2) Swap and conjugate middle parts
! 755: DO 45 J = KK + 1, KP - 1
! 756: T = DCONJG( A( J, KK ) )
! 757: A( J, KK ) = DCONJG( A( KP, J ) )
! 758: A( KP, J ) = T
! 759: 45 CONTINUE
! 760: * (3) Swap and conjugate corner elements at row-col interserction
! 761: A( KP, KK ) = DCONJG( A( KP, KK ) )
! 762: * (4) Swap diagonal elements at row-col intersection
! 763: R1 = DBLE( A( KK, KK ) )
! 764: A( KK, KK ) = DBLE( A( KP, KP ) )
! 765: A( KP, KP ) = R1
! 766: *
! 767: IF( KSTEP.EQ.2 ) THEN
! 768: * (*) Make sure that diagonal element of pivot is real
! 769: A( K, K ) = DBLE( A( K, K ) )
! 770: * (5) Swap row elements
! 771: T = A( K+1, K )
! 772: A( K+1, K ) = A( KP, K )
! 773: A( KP, K ) = T
! 774: END IF
! 775: ELSE
! 776: * (*) Make sure that diagonal element of pivot is real
! 777: A( K, K ) = DBLE( A( K, K ) )
! 778: IF( KSTEP.EQ.2 )
! 779: $ A( K+1, K+1 ) = DBLE( A( K+1, K+1 ) )
! 780: END IF
! 781: *
! 782: * Update the trailing submatrix
! 783: *
! 784: IF( KSTEP.EQ.1 ) THEN
! 785: *
! 786: * 1-by-1 pivot block D(k): column k of A now holds
! 787: *
! 788: * W(k) = L(k)*D(k),
! 789: *
! 790: * where L(k) is the k-th column of L
! 791: *
! 792: IF( K.LT.N ) THEN
! 793: *
! 794: * Perform a rank-1 update of A(k+1:n,k+1:n) and
! 795: * store L(k) in column k
! 796: *
! 797: * Handle division by a small number
! 798: *
! 799: IF( ABS( DBLE( A( K, K ) ) ).GE.SFMIN ) THEN
! 800: *
! 801: * Perform a rank-1 update of A(k+1:n,k+1:n) as
! 802: * A := A - L(k)*D(k)*L(k)**T
! 803: * = A - W(k)*(1/D(k))*W(k)**T
! 804: *
! 805: D11 = ONE / DBLE( A( K, K ) )
! 806: CALL ZHER( UPLO, N-K, -D11, A( K+1, K ), 1,
! 807: $ A( K+1, K+1 ), LDA )
! 808: *
! 809: * Store L(k) in column k
! 810: *
! 811: CALL ZDSCAL( N-K, D11, A( K+1, K ), 1 )
! 812: ELSE
! 813: *
! 814: * Store L(k) in column k
! 815: *
! 816: D11 = DBLE( A( K, K ) )
! 817: DO 46 II = K + 1, N
! 818: A( II, K ) = A( II, K ) / D11
! 819: 46 CONTINUE
! 820: *
! 821: * Perform a rank-1 update of A(k+1:n,k+1:n) as
! 822: * A := A - L(k)*D(k)*L(k)**T
! 823: * = A - W(k)*(1/D(k))*W(k)**T
! 824: * = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T
! 825: *
! 826: CALL ZHER( UPLO, N-K, -D11, A( K+1, K ), 1,
! 827: $ A( K+1, K+1 ), LDA )
! 828: END IF
! 829: END IF
! 830: *
! 831: ELSE
! 832: *
! 833: * 2-by-2 pivot block D(k): columns k and k+1 now hold
! 834: *
! 835: * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
! 836: *
! 837: * where L(k) and L(k+1) are the k-th and (k+1)-th columns
! 838: * of L
! 839: *
! 840: *
! 841: * Perform a rank-2 update of A(k+2:n,k+2:n) as
! 842: *
! 843: * A := A - ( L(k) L(k+1) ) * D(k) * ( L(k) L(k+1) )**T
! 844: * = A - ( ( A(k)A(k+1) )*inv(D(k) ) * ( A(k)A(k+1) )**T
! 845: *
! 846: * and store L(k) and L(k+1) in columns k and k+1
! 847: *
! 848: IF( K.LT.N-1 ) THEN
! 849: * D = |A21|
! 850: D = DLAPY2( DBLE( A( K+1, K ) ),
! 851: $ DIMAG( A( K+1, K ) ) )
! 852: D11 = DBLE( A( K+1, K+1 ) ) / D
! 853: D22 = DBLE( A( K, K ) ) / D
! 854: D21 = A( K+1, K ) / D
! 855: TT = ONE / ( D11*D22-ONE )
! 856: *
! 857: DO 60 J = K + 2, N
! 858: *
! 859: * Compute D21 * ( W(k)W(k+1) ) * inv(D(k)) for row J
! 860: *
! 861: WK = TT*( D11*A( J, K )-D21*A( J, K+1 ) )
! 862: WKP1 = TT*( D22*A( J, K+1 )-DCONJG( D21 )*
! 863: $ A( J, K ) )
! 864: *
! 865: * Perform a rank-2 update of A(k+2:n,k+2:n)
! 866: *
! 867: DO 50 I = J, N
! 868: A( I, J ) = A( I, J ) -
! 869: $ ( A( I, K ) / D )*DCONJG( WK ) -
! 870: $ ( A( I, K+1 ) / D )*DCONJG( WKP1 )
! 871: 50 CONTINUE
! 872: *
! 873: * Store L(k) and L(k+1) in cols k and k+1 for row J
! 874: *
! 875: A( J, K ) = WK / D
! 876: A( J, K+1 ) = WKP1 / D
! 877: * (*) Make sure that diagonal element of pivot is real
! 878: A( J, J ) = DCMPLX( DBLE( A( J, J ) ), ZERO )
! 879: *
! 880: 60 CONTINUE
! 881: *
! 882: END IF
! 883: *
! 884: END IF
! 885: *
! 886: END IF
! 887: *
! 888: * Store details of the interchanges in IPIV
! 889: *
! 890: IF( KSTEP.EQ.1 ) THEN
! 891: IPIV( K ) = KP
! 892: ELSE
! 893: IPIV( K ) = -P
! 894: IPIV( K+1 ) = -KP
! 895: END IF
! 896: *
! 897: * Increase K and return to the start of the main loop
! 898: *
! 899: K = K + KSTEP
! 900: GO TO 40
! 901: *
! 902: END IF
! 903: *
! 904: 70 CONTINUE
! 905: *
! 906: RETURN
! 907: *
! 908: * End of ZHETF2_ROOK
! 909: *
! 910: END
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