Annotation of rpl/lapack/lapack/zlahef.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZLAHEF( 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: * ZLAHEF computes a partial factorization of a complex Hermitian
! 21: * matrix A using the Bunch-Kaufman diagonal pivoting method. The
! 22: * partial 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 conjugate transpose of U.
! 33: *
! 34: * ZLAHEF is an auxiliary routine called by ZHETRF. 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: * Hermitian 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 Hermitian 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: COMPLEX*16 CONE
! 102: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
! 103: DOUBLE PRECISION EIGHT, SEVTEN
! 104: PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+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, R1, ROWMAX, T
! 110: COMPLEX*16 D11, D21, D22, Z
! 111: * ..
! 112: * .. External Functions ..
! 113: LOGICAL LSAME
! 114: INTEGER IZAMAX
! 115: EXTERNAL LSAME, IZAMAX
! 116: * ..
! 117: * .. External Subroutines ..
! 118: EXTERNAL ZCOPY, ZDSCAL, ZGEMM, ZGEMV, ZLACGV, ZSWAP
! 119: * ..
! 120: * .. Intrinsic Functions ..
! 121: INTRINSIC ABS, DBLE, DCONJG, 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 (note that conjg(W) is actually stored)
! 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-1, A( 1, K ), 1, W( 1, KW ), 1 )
! 159: W( K, KW ) = DBLE( A( K, K ) )
! 160: IF( K.LT.N ) THEN
! 161: CALL ZGEMV( 'No transpose', K, N-K, -CONE, A( 1, K+1 ), LDA,
! 162: $ W( K, KW+1 ), LDW, CONE, W( 1, KW ), 1 )
! 163: W( K, KW ) = DBLE( W( K, KW ) )
! 164: END IF
! 165: *
! 166: KSTEP = 1
! 167: *
! 168: * Determine rows and columns to be interchanged and whether
! 169: * a 1-by-1 or 2-by-2 pivot block will be used
! 170: *
! 171: ABSAKK = ABS( DBLE( W( K, KW ) ) )
! 172: *
! 173: * IMAX is the row-index of the largest off-diagonal element in
! 174: * column K, and COLMAX is its absolute value
! 175: *
! 176: IF( K.GT.1 ) THEN
! 177: IMAX = IZAMAX( K-1, W( 1, KW ), 1 )
! 178: COLMAX = CABS1( W( IMAX, KW ) )
! 179: ELSE
! 180: COLMAX = ZERO
! 181: END IF
! 182: *
! 183: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
! 184: *
! 185: * Column K is zero: set INFO and continue
! 186: *
! 187: IF( INFO.EQ.0 )
! 188: $ INFO = K
! 189: KP = K
! 190: A( K, K ) = DBLE( A( K, K ) )
! 191: ELSE
! 192: IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
! 193: *
! 194: * no interchange, use 1-by-1 pivot block
! 195: *
! 196: KP = K
! 197: ELSE
! 198: *
! 199: * Copy column IMAX to column KW-1 of W and update it
! 200: *
! 201: CALL ZCOPY( IMAX-1, A( 1, IMAX ), 1, W( 1, KW-1 ), 1 )
! 202: W( IMAX, KW-1 ) = DBLE( A( IMAX, IMAX ) )
! 203: CALL ZCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA,
! 204: $ W( IMAX+1, KW-1 ), 1 )
! 205: CALL ZLACGV( K-IMAX, W( IMAX+1, KW-1 ), 1 )
! 206: IF( K.LT.N ) THEN
! 207: CALL ZGEMV( 'No transpose', K, N-K, -CONE,
! 208: $ A( 1, K+1 ), LDA, W( IMAX, KW+1 ), LDW,
! 209: $ CONE, W( 1, KW-1 ), 1 )
! 210: W( IMAX, KW-1 ) = DBLE( W( IMAX, KW-1 ) )
! 211: END IF
! 212: *
! 213: * JMAX is the column-index of the largest off-diagonal
! 214: * element in row IMAX, and ROWMAX is its absolute value
! 215: *
! 216: JMAX = IMAX + IZAMAX( K-IMAX, W( IMAX+1, KW-1 ), 1 )
! 217: ROWMAX = CABS1( W( JMAX, KW-1 ) )
! 218: IF( IMAX.GT.1 ) THEN
! 219: JMAX = IZAMAX( IMAX-1, W( 1, KW-1 ), 1 )
! 220: ROWMAX = MAX( ROWMAX, CABS1( W( JMAX, KW-1 ) ) )
! 221: END IF
! 222: *
! 223: IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
! 224: *
! 225: * no interchange, use 1-by-1 pivot block
! 226: *
! 227: KP = K
! 228: ELSE IF( ABS( DBLE( W( IMAX, KW-1 ) ) ).GE.ALPHA*ROWMAX )
! 229: $ THEN
! 230: *
! 231: * interchange rows and columns K and IMAX, use 1-by-1
! 232: * pivot block
! 233: *
! 234: KP = IMAX
! 235: *
! 236: * copy column KW-1 of W to column KW
! 237: *
! 238: CALL ZCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
! 239: ELSE
! 240: *
! 241: * interchange rows and columns K-1 and IMAX, use 2-by-2
! 242: * pivot block
! 243: *
! 244: KP = IMAX
! 245: KSTEP = 2
! 246: END IF
! 247: END IF
! 248: *
! 249: KK = K - KSTEP + 1
! 250: KKW = NB + KK - N
! 251: *
! 252: * Updated column KP is already stored in column KKW of W
! 253: *
! 254: IF( KP.NE.KK ) THEN
! 255: *
! 256: * Copy non-updated column KK to column KP
! 257: *
! 258: A( KP, KP ) = DBLE( A( KK, KK ) )
! 259: CALL ZCOPY( KK-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ),
! 260: $ LDA )
! 261: CALL ZLACGV( KK-1-KP, A( KP, KP+1 ), LDA )
! 262: CALL ZCOPY( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
! 263: *
! 264: * Interchange rows KK and KP in last KK columns of A and W
! 265: *
! 266: IF( KK.LT.N )
! 267: $ CALL ZSWAP( N-KK, A( KK, KK+1 ), LDA, A( KP, KK+1 ),
! 268: $ LDA )
! 269: CALL ZSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ),
! 270: $ LDW )
! 271: END IF
! 272: *
! 273: IF( KSTEP.EQ.1 ) THEN
! 274: *
! 275: * 1-by-1 pivot block D(k): column KW of W now holds
! 276: *
! 277: * W(k) = U(k)*D(k)
! 278: *
! 279: * where U(k) is the k-th column of U
! 280: *
! 281: * Store U(k) in column k of A
! 282: *
! 283: CALL ZCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
! 284: R1 = ONE / DBLE( A( K, K ) )
! 285: CALL ZDSCAL( K-1, R1, A( 1, K ), 1 )
! 286: *
! 287: * Conjugate W(k)
! 288: *
! 289: CALL ZLACGV( K-1, W( 1, KW ), 1 )
! 290: ELSE
! 291: *
! 292: * 2-by-2 pivot block D(k): columns KW and KW-1 of W now
! 293: * hold
! 294: *
! 295: * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
! 296: *
! 297: * where U(k) and U(k-1) are the k-th and (k-1)-th columns
! 298: * of U
! 299: *
! 300: IF( K.GT.2 ) THEN
! 301: *
! 302: * Store U(k) and U(k-1) in columns k and k-1 of A
! 303: *
! 304: D21 = W( K-1, KW )
! 305: D11 = W( K, KW ) / DCONJG( D21 )
! 306: D22 = W( K-1, KW-1 ) / D21
! 307: T = ONE / ( DBLE( D11*D22 )-ONE )
! 308: D21 = T / D21
! 309: DO 20 J = 1, K - 2
! 310: A( J, K-1 ) = D21*( D11*W( J, KW-1 )-W( J, KW ) )
! 311: A( J, K ) = DCONJG( D21 )*
! 312: $ ( D22*W( J, KW )-W( J, KW-1 ) )
! 313: 20 CONTINUE
! 314: END IF
! 315: *
! 316: * Copy D(k) to A
! 317: *
! 318: A( K-1, K-1 ) = W( K-1, KW-1 )
! 319: A( K-1, K ) = W( K-1, KW )
! 320: A( K, K ) = W( K, KW )
! 321: *
! 322: * Conjugate W(k) and W(k-1)
! 323: *
! 324: CALL ZLACGV( K-1, W( 1, KW ), 1 )
! 325: CALL ZLACGV( K-2, W( 1, KW-1 ), 1 )
! 326: END IF
! 327: END IF
! 328: *
! 329: * Store details of the interchanges in IPIV
! 330: *
! 331: IF( KSTEP.EQ.1 ) THEN
! 332: IPIV( K ) = KP
! 333: ELSE
! 334: IPIV( K ) = -KP
! 335: IPIV( K-1 ) = -KP
! 336: END IF
! 337: *
! 338: * Decrease K and return to the start of the main loop
! 339: *
! 340: K = K - KSTEP
! 341: GO TO 10
! 342: *
! 343: 30 CONTINUE
! 344: *
! 345: * Update the upper triangle of A11 (= A(1:k,1:k)) as
! 346: *
! 347: * A11 := A11 - U12*D*U12' = A11 - U12*W'
! 348: *
! 349: * computing blocks of NB columns at a time (note that conjg(W) is
! 350: * actually stored)
! 351: *
! 352: DO 50 J = ( ( K-1 ) / NB )*NB + 1, 1, -NB
! 353: JB = MIN( NB, K-J+1 )
! 354: *
! 355: * Update the upper triangle of the diagonal block
! 356: *
! 357: DO 40 JJ = J, J + JB - 1
! 358: A( JJ, JJ ) = DBLE( A( JJ, JJ ) )
! 359: CALL ZGEMV( 'No transpose', JJ-J+1, N-K, -CONE,
! 360: $ A( J, K+1 ), LDA, W( JJ, KW+1 ), LDW, CONE,
! 361: $ A( J, JJ ), 1 )
! 362: A( JJ, JJ ) = DBLE( A( JJ, JJ ) )
! 363: 40 CONTINUE
! 364: *
! 365: * Update the rectangular superdiagonal block
! 366: *
! 367: CALL ZGEMM( 'No transpose', 'Transpose', J-1, JB, N-K,
! 368: $ -CONE, A( 1, K+1 ), LDA, W( J, KW+1 ), LDW,
! 369: $ CONE, A( 1, J ), LDA )
! 370: 50 CONTINUE
! 371: *
! 372: * Put U12 in standard form by partially undoing the interchanges
! 373: * in columns k+1:n
! 374: *
! 375: J = K + 1
! 376: 60 CONTINUE
! 377: JJ = J
! 378: JP = IPIV( J )
! 379: IF( JP.LT.0 ) THEN
! 380: JP = -JP
! 381: J = J + 1
! 382: END IF
! 383: J = J + 1
! 384: IF( JP.NE.JJ .AND. J.LE.N )
! 385: $ CALL ZSWAP( N-J+1, A( JP, J ), LDA, A( JJ, J ), LDA )
! 386: IF( J.LE.N )
! 387: $ GO TO 60
! 388: *
! 389: * Set KB to the number of columns factorized
! 390: *
! 391: KB = N - K
! 392: *
! 393: ELSE
! 394: *
! 395: * Factorize the leading columns of A using the lower triangle
! 396: * of A and working forwards, and compute the matrix W = L21*D
! 397: * for use in updating A22 (note that conjg(W) is actually stored)
! 398: *
! 399: * K is the main loop index, increasing from 1 in steps of 1 or 2
! 400: *
! 401: K = 1
! 402: 70 CONTINUE
! 403: *
! 404: * Exit from loop
! 405: *
! 406: IF( ( K.GE.NB .AND. NB.LT.N ) .OR. K.GT.N )
! 407: $ GO TO 90
! 408: *
! 409: * Copy column K of A to column K of W and update it
! 410: *
! 411: W( K, K ) = DBLE( A( K, K ) )
! 412: IF( K.LT.N )
! 413: $ CALL ZCOPY( N-K, A( K+1, K ), 1, W( K+1, K ), 1 )
! 414: CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE, A( K, 1 ), LDA,
! 415: $ W( K, 1 ), LDW, CONE, W( K, K ), 1 )
! 416: W( K, K ) = DBLE( W( K, K ) )
! 417: *
! 418: KSTEP = 1
! 419: *
! 420: * Determine rows and columns to be interchanged and whether
! 421: * a 1-by-1 or 2-by-2 pivot block will be used
! 422: *
! 423: ABSAKK = ABS( DBLE( W( K, K ) ) )
! 424: *
! 425: * IMAX is the row-index of the largest off-diagonal element in
! 426: * column K, and COLMAX is its absolute value
! 427: *
! 428: IF( K.LT.N ) THEN
! 429: IMAX = K + IZAMAX( N-K, W( K+1, K ), 1 )
! 430: COLMAX = CABS1( W( IMAX, K ) )
! 431: ELSE
! 432: COLMAX = ZERO
! 433: END IF
! 434: *
! 435: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
! 436: *
! 437: * Column K is zero: set INFO and continue
! 438: *
! 439: IF( INFO.EQ.0 )
! 440: $ INFO = K
! 441: KP = K
! 442: A( K, K ) = DBLE( A( K, K ) )
! 443: ELSE
! 444: IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
! 445: *
! 446: * no interchange, use 1-by-1 pivot block
! 447: *
! 448: KP = K
! 449: ELSE
! 450: *
! 451: * Copy column IMAX to column K+1 of W and update it
! 452: *
! 453: CALL ZCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1 )
! 454: CALL ZLACGV( IMAX-K, W( K, K+1 ), 1 )
! 455: W( IMAX, K+1 ) = DBLE( A( IMAX, IMAX ) )
! 456: IF( IMAX.LT.N )
! 457: $ CALL ZCOPY( N-IMAX, A( IMAX+1, IMAX ), 1,
! 458: $ W( IMAX+1, K+1 ), 1 )
! 459: CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE, A( K, 1 ),
! 460: $ LDA, W( IMAX, 1 ), LDW, CONE, W( K, K+1 ),
! 461: $ 1 )
! 462: W( IMAX, K+1 ) = DBLE( W( IMAX, K+1 ) )
! 463: *
! 464: * JMAX is the column-index of the largest off-diagonal
! 465: * element in row IMAX, and ROWMAX is its absolute value
! 466: *
! 467: JMAX = K - 1 + IZAMAX( IMAX-K, W( K, K+1 ), 1 )
! 468: ROWMAX = CABS1( W( JMAX, K+1 ) )
! 469: IF( IMAX.LT.N ) THEN
! 470: JMAX = IMAX + IZAMAX( N-IMAX, W( IMAX+1, K+1 ), 1 )
! 471: ROWMAX = MAX( ROWMAX, CABS1( W( JMAX, K+1 ) ) )
! 472: END IF
! 473: *
! 474: IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
! 475: *
! 476: * no interchange, use 1-by-1 pivot block
! 477: *
! 478: KP = K
! 479: ELSE IF( ABS( DBLE( W( IMAX, K+1 ) ) ).GE.ALPHA*ROWMAX )
! 480: $ THEN
! 481: *
! 482: * interchange rows and columns K and IMAX, use 1-by-1
! 483: * pivot block
! 484: *
! 485: KP = IMAX
! 486: *
! 487: * copy column K+1 of W to column K
! 488: *
! 489: CALL ZCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
! 490: ELSE
! 491: *
! 492: * interchange rows and columns K+1 and IMAX, use 2-by-2
! 493: * pivot block
! 494: *
! 495: KP = IMAX
! 496: KSTEP = 2
! 497: END IF
! 498: END IF
! 499: *
! 500: KK = K + KSTEP - 1
! 501: *
! 502: * Updated column KP is already stored in column KK of W
! 503: *
! 504: IF( KP.NE.KK ) THEN
! 505: *
! 506: * Copy non-updated column KK to column KP
! 507: *
! 508: A( KP, KP ) = DBLE( A( KK, KK ) )
! 509: CALL ZCOPY( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ),
! 510: $ LDA )
! 511: CALL ZLACGV( KP-KK-1, A( KP, KK+1 ), LDA )
! 512: IF( KP.LT.N )
! 513: $ CALL ZCOPY( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 )
! 514: *
! 515: * Interchange rows KK and KP in first KK columns of A and W
! 516: *
! 517: CALL ZSWAP( KK-1, A( KK, 1 ), LDA, A( KP, 1 ), LDA )
! 518: CALL ZSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW )
! 519: END IF
! 520: *
! 521: IF( KSTEP.EQ.1 ) THEN
! 522: *
! 523: * 1-by-1 pivot block D(k): column k of W now holds
! 524: *
! 525: * W(k) = L(k)*D(k)
! 526: *
! 527: * where L(k) is the k-th column of L
! 528: *
! 529: * Store L(k) in column k of A
! 530: *
! 531: CALL ZCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
! 532: IF( K.LT.N ) THEN
! 533: R1 = ONE / DBLE( A( K, K ) )
! 534: CALL ZDSCAL( N-K, R1, A( K+1, K ), 1 )
! 535: *
! 536: * Conjugate W(k)
! 537: *
! 538: CALL ZLACGV( N-K, W( K+1, K ), 1 )
! 539: END IF
! 540: ELSE
! 541: *
! 542: * 2-by-2 pivot block D(k): columns k and k+1 of W now hold
! 543: *
! 544: * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
! 545: *
! 546: * where L(k) and L(k+1) are the k-th and (k+1)-th columns
! 547: * of L
! 548: *
! 549: IF( K.LT.N-1 ) THEN
! 550: *
! 551: * Store L(k) and L(k+1) in columns k and k+1 of A
! 552: *
! 553: D21 = W( K+1, K )
! 554: D11 = W( K+1, K+1 ) / D21
! 555: D22 = W( K, K ) / DCONJG( D21 )
! 556: T = ONE / ( DBLE( D11*D22 )-ONE )
! 557: D21 = T / D21
! 558: DO 80 J = K + 2, N
! 559: A( J, K ) = DCONJG( D21 )*
! 560: $ ( D11*W( J, K )-W( J, K+1 ) )
! 561: A( J, K+1 ) = D21*( D22*W( J, K+1 )-W( J, K ) )
! 562: 80 CONTINUE
! 563: END IF
! 564: *
! 565: * Copy D(k) to A
! 566: *
! 567: A( K, K ) = W( K, K )
! 568: A( K+1, K ) = W( K+1, K )
! 569: A( K+1, K+1 ) = W( K+1, K+1 )
! 570: *
! 571: * Conjugate W(k) and W(k+1)
! 572: *
! 573: CALL ZLACGV( N-K, W( K+1, K ), 1 )
! 574: CALL ZLACGV( N-K-1, W( K+2, K+1 ), 1 )
! 575: END IF
! 576: END IF
! 577: *
! 578: * Store details of the interchanges in IPIV
! 579: *
! 580: IF( KSTEP.EQ.1 ) THEN
! 581: IPIV( K ) = KP
! 582: ELSE
! 583: IPIV( K ) = -KP
! 584: IPIV( K+1 ) = -KP
! 585: END IF
! 586: *
! 587: * Increase K and return to the start of the main loop
! 588: *
! 589: K = K + KSTEP
! 590: GO TO 70
! 591: *
! 592: 90 CONTINUE
! 593: *
! 594: * Update the lower triangle of A22 (= A(k:n,k:n)) as
! 595: *
! 596: * A22 := A22 - L21*D*L21' = A22 - L21*W'
! 597: *
! 598: * computing blocks of NB columns at a time (note that conjg(W) is
! 599: * actually stored)
! 600: *
! 601: DO 110 J = K, N, NB
! 602: JB = MIN( NB, N-J+1 )
! 603: *
! 604: * Update the lower triangle of the diagonal block
! 605: *
! 606: DO 100 JJ = J, J + JB - 1
! 607: A( JJ, JJ ) = DBLE( A( JJ, JJ ) )
! 608: CALL ZGEMV( 'No transpose', J+JB-JJ, K-1, -CONE,
! 609: $ A( JJ, 1 ), LDA, W( JJ, 1 ), LDW, CONE,
! 610: $ A( JJ, JJ ), 1 )
! 611: A( JJ, JJ ) = DBLE( A( JJ, JJ ) )
! 612: 100 CONTINUE
! 613: *
! 614: * Update the rectangular subdiagonal block
! 615: *
! 616: IF( J+JB.LE.N )
! 617: $ CALL ZGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB,
! 618: $ K-1, -CONE, A( J+JB, 1 ), LDA, W( J, 1 ),
! 619: $ LDW, CONE, A( J+JB, J ), LDA )
! 620: 110 CONTINUE
! 621: *
! 622: * Put L21 in standard form by partially undoing the interchanges
! 623: * in columns 1:k-1
! 624: *
! 625: J = K - 1
! 626: 120 CONTINUE
! 627: JJ = J
! 628: JP = IPIV( J )
! 629: IF( JP.LT.0 ) THEN
! 630: JP = -JP
! 631: J = J - 1
! 632: END IF
! 633: J = J - 1
! 634: IF( JP.NE.JJ .AND. J.GE.1 )
! 635: $ CALL ZSWAP( J, A( JP, 1 ), LDA, A( JJ, 1 ), LDA )
! 636: IF( J.GE.1 )
! 637: $ GO TO 120
! 638: *
! 639: * Set KB to the number of columns factorized
! 640: *
! 641: KB = K - 1
! 642: *
! 643: END IF
! 644: RETURN
! 645: *
! 646: * End of ZLAHEF
! 647: *
! 648: END
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