Annotation of rpl/lapack/lapack/dsytf2_rook.f, revision 1.1
1.1 ! bertrand 1: *> \brief \b DSYTF2_ROOK computes the factorization of a real symmetric 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 DSYTF2_ROOK + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsytf2_rook.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsytf2_rook.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsytf2_rook.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE DSYTF2_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: * DOUBLE PRECISION A( LDA, * )
! 30: * ..
! 31: *
! 32: *
! 33: *> \par Purpose:
! 34: * =============
! 35: *>
! 36: *> \verbatim
! 37: *>
! 38: *> DSYTF2_ROOK computes the factorization of a real symmetric matrix A
! 39: *> using the bounded Bunch-Kaufman ("rook") diagonal pivoting method:
! 40: *>
! 41: *> A = U*D*U**T or A = L*D*L**T
! 42: *>
! 43: *> where U (or L) is a product of permutation and unit upper (lower)
! 44: *> triangular matrices, U**T is the transpose of U, and D is symmetric and
! 45: *> 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: *> symmetric 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 DOUBLE PRECISION array, dimension (LDA,N)
! 71: *> On entry, the symmetric 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)
! 96: *> were 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 doubleSYcomputational
! 135: *
! 136: *> \par Further Details:
! 137: * =====================
! 138: *>
! 139: *> \verbatim
! 140: *>
! 141: *> If UPLO = 'U', then A = U*D*U**T, 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**T, 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 abd , USA
! 192: *> \endverbatim
! 193: *
! 194: * =====================================================================
! 195: SUBROUTINE DSYTF2_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: DOUBLE PRECISION 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 UPPER, DONE
! 221: INTEGER I, IMAX, J, JMAX, ITEMP, K, KK, KP, KSTEP,
! 222: $ P, II
! 223: DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22,
! 224: $ ROWMAX, DTEMP, T, WK, WKM1, WKP1, SFMIN
! 225: * ..
! 226: * .. External Functions ..
! 227: LOGICAL LSAME
! 228: INTEGER IDAMAX
! 229: DOUBLE PRECISION DLAMCH
! 230: EXTERNAL LSAME, IDAMAX, DLAMCH
! 231: * ..
! 232: * .. External Subroutines ..
! 233: EXTERNAL DSCAL, DSWAP, DSYR, XERBLA
! 234: * ..
! 235: * .. Intrinsic Functions ..
! 236: INTRINSIC ABS, MAX, SQRT
! 237: * ..
! 238: * .. Executable Statements ..
! 239: *
! 240: * Test the input parameters.
! 241: *
! 242: INFO = 0
! 243: UPPER = LSAME( UPLO, 'U' )
! 244: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
! 245: INFO = -1
! 246: ELSE IF( N.LT.0 ) THEN
! 247: INFO = -2
! 248: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
! 249: INFO = -4
! 250: END IF
! 251: IF( INFO.NE.0 ) THEN
! 252: CALL XERBLA( 'DSYTF2_ROOK', -INFO )
! 253: RETURN
! 254: END IF
! 255: *
! 256: * Initialize ALPHA for use in choosing pivot block size.
! 257: *
! 258: ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
! 259: *
! 260: * Compute machine safe minimum
! 261: *
! 262: SFMIN = DLAMCH( 'S' )
! 263: *
! 264: IF( UPPER ) THEN
! 265: *
! 266: * Factorize A as U*D*U**T using the upper triangle of A
! 267: *
! 268: * K is the main loop index, decreasing from N to 1 in steps of
! 269: * 1 or 2
! 270: *
! 271: K = N
! 272: 10 CONTINUE
! 273: *
! 274: * If K < 1, exit from loop
! 275: *
! 276: IF( K.LT.1 )
! 277: $ GO TO 70
! 278: KSTEP = 1
! 279: P = K
! 280: *
! 281: * Determine rows and columns to be interchanged and whether
! 282: * a 1-by-1 or 2-by-2 pivot block will be used
! 283: *
! 284: ABSAKK = ABS( A( K, K ) )
! 285: *
! 286: * IMAX is the row-index of the largest off-diagonal element in
! 287: * column K, and COLMAX is its absolute value.
! 288: * Determine both COLMAX and IMAX.
! 289: *
! 290: IF( K.GT.1 ) THEN
! 291: IMAX = IDAMAX( K-1, A( 1, K ), 1 )
! 292: COLMAX = ABS( A( IMAX, K ) )
! 293: ELSE
! 294: COLMAX = ZERO
! 295: END IF
! 296: *
! 297: IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) ) THEN
! 298: *
! 299: * Column K is zero or underflow: set INFO and continue
! 300: *
! 301: IF( INFO.EQ.0 )
! 302: $ INFO = K
! 303: KP = K
! 304: ELSE
! 305: *
! 306: * Test for interchange
! 307: *
! 308: * Equivalent to testing for (used to handle NaN and Inf)
! 309: * ABSAKK.GE.ALPHA*COLMAX
! 310: *
! 311: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
! 312: *
! 313: * no interchange,
! 314: * use 1-by-1 pivot block
! 315: *
! 316: KP = K
! 317: ELSE
! 318: *
! 319: DONE = .FALSE.
! 320: *
! 321: * Loop until pivot found
! 322: *
! 323: 12 CONTINUE
! 324: *
! 325: * Begin pivot search loop body
! 326: *
! 327: * JMAX is the column-index of the largest off-diagonal
! 328: * element in row IMAX, and ROWMAX is its absolute value.
! 329: * Determine both ROWMAX and JMAX.
! 330: *
! 331: IF( IMAX.NE.K ) THEN
! 332: JMAX = IMAX + IDAMAX( K-IMAX, A( IMAX, IMAX+1 ),
! 333: $ LDA )
! 334: ROWMAX = ABS( A( IMAX, JMAX ) )
! 335: ELSE
! 336: ROWMAX = ZERO
! 337: END IF
! 338: *
! 339: IF( IMAX.GT.1 ) THEN
! 340: ITEMP = IDAMAX( IMAX-1, A( 1, IMAX ), 1 )
! 341: DTEMP = ABS( A( ITEMP, IMAX ) )
! 342: IF( DTEMP.GT.ROWMAX ) THEN
! 343: ROWMAX = DTEMP
! 344: JMAX = ITEMP
! 345: END IF
! 346: END IF
! 347: *
! 348: * Equivalent to testing for (used to handle NaN and Inf)
! 349: * ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX
! 350: *
! 351: IF( .NOT.( ABS( A( IMAX, IMAX ) ).LT.ALPHA*ROWMAX ) )
! 352: $ THEN
! 353: *
! 354: * interchange rows and columns K and IMAX,
! 355: * use 1-by-1 pivot block
! 356: *
! 357: KP = IMAX
! 358: DONE = .TRUE.
! 359: *
! 360: * Equivalent to testing for ROWMAX .EQ. COLMAX,
! 361: * used to handle NaN and Inf
! 362: *
! 363: ELSE IF( ( P.EQ.JMAX ).OR.( ROWMAX.LE.COLMAX ) ) THEN
! 364: *
! 365: * interchange rows and columns K+1 and IMAX,
! 366: * use 2-by-2 pivot block
! 367: *
! 368: KP = IMAX
! 369: KSTEP = 2
! 370: DONE = .TRUE.
! 371: ELSE
! 372: *
! 373: * Pivot NOT found, set variables and repeat
! 374: *
! 375: P = IMAX
! 376: COLMAX = ROWMAX
! 377: IMAX = JMAX
! 378: END IF
! 379: *
! 380: * End pivot search loop body
! 381: *
! 382: IF( .NOT. DONE ) GOTO 12
! 383: *
! 384: END IF
! 385: *
! 386: * Swap TWO rows and TWO columns
! 387: *
! 388: * First swap
! 389: *
! 390: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
! 391: *
! 392: * Interchange rows and column K and P in the leading
! 393: * submatrix A(1:k,1:k) if we have a 2-by-2 pivot
! 394: *
! 395: IF( P.GT.1 )
! 396: $ CALL DSWAP( P-1, A( 1, K ), 1, A( 1, P ), 1 )
! 397: IF( P.LT.(K-1) )
! 398: $ CALL DSWAP( K-P-1, A( P+1, K ), 1, A( P, P+1 ),
! 399: $ LDA )
! 400: T = A( K, K )
! 401: A( K, K ) = A( P, P )
! 402: A( P, P ) = T
! 403: END IF
! 404: *
! 405: * Second swap
! 406: *
! 407: KK = K - KSTEP + 1
! 408: IF( KP.NE.KK ) THEN
! 409: *
! 410: * Interchange rows and columns KK and KP in the leading
! 411: * submatrix A(1:k,1:k)
! 412: *
! 413: IF( KP.GT.1 )
! 414: $ CALL DSWAP( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
! 415: IF( ( KK.GT.1 ) .AND. ( KP.LT.(KK-1) ) )
! 416: $ CALL DSWAP( KK-KP-1, A( KP+1, KK ), 1, A( KP, KP+1 ),
! 417: $ LDA )
! 418: T = A( KK, KK )
! 419: A( KK, KK ) = A( KP, KP )
! 420: A( KP, KP ) = T
! 421: IF( KSTEP.EQ.2 ) THEN
! 422: T = A( K-1, K )
! 423: A( K-1, K ) = A( KP, K )
! 424: A( KP, K ) = T
! 425: END IF
! 426: END IF
! 427: *
! 428: * Update the leading submatrix
! 429: *
! 430: IF( KSTEP.EQ.1 ) THEN
! 431: *
! 432: * 1-by-1 pivot block D(k): column k now holds
! 433: *
! 434: * W(k) = U(k)*D(k)
! 435: *
! 436: * where U(k) is the k-th column of U
! 437: *
! 438: IF( K.GT.1 ) THEN
! 439: *
! 440: * Perform a rank-1 update of A(1:k-1,1:k-1) and
! 441: * store U(k) in column k
! 442: *
! 443: IF( ABS( A( K, K ) ).GE.SFMIN ) THEN
! 444: *
! 445: * Perform a rank-1 update of A(1:k-1,1:k-1) as
! 446: * A := A - U(k)*D(k)*U(k)**T
! 447: * = A - W(k)*1/D(k)*W(k)**T
! 448: *
! 449: D11 = ONE / A( K, K )
! 450: CALL DSYR( UPLO, K-1, -D11, A( 1, K ), 1, A, LDA )
! 451: *
! 452: * Store U(k) in column k
! 453: *
! 454: CALL DSCAL( K-1, D11, A( 1, K ), 1 )
! 455: ELSE
! 456: *
! 457: * Store L(k) in column K
! 458: *
! 459: D11 = A( K, K )
! 460: DO 16 II = 1, K - 1
! 461: A( II, K ) = A( II, K ) / D11
! 462: 16 CONTINUE
! 463: *
! 464: * Perform a rank-1 update of A(k+1:n,k+1:n) as
! 465: * A := A - U(k)*D(k)*U(k)**T
! 466: * = A - W(k)*(1/D(k))*W(k)**T
! 467: * = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T
! 468: *
! 469: CALL DSYR( UPLO, K-1, -D11, A( 1, K ), 1, A, LDA )
! 470: END IF
! 471: END IF
! 472: *
! 473: ELSE
! 474: *
! 475: * 2-by-2 pivot block D(k): columns k and k-1 now hold
! 476: *
! 477: * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
! 478: *
! 479: * where U(k) and U(k-1) are the k-th and (k-1)-th columns
! 480: * of U
! 481: *
! 482: * Perform a rank-2 update of A(1:k-2,1:k-2) as
! 483: *
! 484: * A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T
! 485: * = A - ( ( A(k-1)A(k) )*inv(D(k)) ) * ( A(k-1)A(k) )**T
! 486: *
! 487: * and store L(k) and L(k+1) in columns k and k+1
! 488: *
! 489: IF( K.GT.2 ) THEN
! 490: *
! 491: D12 = A( K-1, K )
! 492: D22 = A( K-1, K-1 ) / D12
! 493: D11 = A( K, K ) / D12
! 494: T = ONE / ( D11*D22-ONE )
! 495: *
! 496: DO 30 J = K - 2, 1, -1
! 497: *
! 498: WKM1 = T*( D11*A( J, K-1 )-A( J, K ) )
! 499: WK = T*( D22*A( J, K )-A( J, K-1 ) )
! 500: *
! 501: DO 20 I = J, 1, -1
! 502: A( I, J ) = A( I, J ) - (A( I, K ) / D12 )*WK -
! 503: $ ( A( I, K-1 ) / D12 )*WKM1
! 504: 20 CONTINUE
! 505: *
! 506: * Store U(k) and U(k-1) in cols k and k-1 for row J
! 507: *
! 508: A( J, K ) = WK / D12
! 509: A( J, K-1 ) = WKM1 / D12
! 510: *
! 511: 30 CONTINUE
! 512: *
! 513: END IF
! 514: *
! 515: END IF
! 516: END IF
! 517: *
! 518: * Store details of the interchanges in IPIV
! 519: *
! 520: IF( KSTEP.EQ.1 ) THEN
! 521: IPIV( K ) = KP
! 522: ELSE
! 523: IPIV( K ) = -P
! 524: IPIV( K-1 ) = -KP
! 525: END IF
! 526: *
! 527: * Decrease K and return to the start of the main loop
! 528: *
! 529: K = K - KSTEP
! 530: GO TO 10
! 531: *
! 532: ELSE
! 533: *
! 534: * Factorize A as L*D*L**T using the lower triangle of A
! 535: *
! 536: * K is the main loop index, increasing from 1 to N in steps of
! 537: * 1 or 2
! 538: *
! 539: K = 1
! 540: 40 CONTINUE
! 541: *
! 542: * If K > N, exit from loop
! 543: *
! 544: IF( K.GT.N )
! 545: $ GO TO 70
! 546: KSTEP = 1
! 547: P = K
! 548: *
! 549: * Determine rows and columns to be interchanged and whether
! 550: * a 1-by-1 or 2-by-2 pivot block will be used
! 551: *
! 552: ABSAKK = ABS( A( K, K ) )
! 553: *
! 554: * IMAX is the row-index of the largest off-diagonal element in
! 555: * column K, and COLMAX is its absolute value.
! 556: * Determine both COLMAX and IMAX.
! 557: *
! 558: IF( K.LT.N ) THEN
! 559: IMAX = K + IDAMAX( N-K, A( K+1, K ), 1 )
! 560: COLMAX = ABS( A( IMAX, K ) )
! 561: ELSE
! 562: COLMAX = ZERO
! 563: END IF
! 564: *
! 565: IF( ( MAX( ABSAKK, COLMAX ).EQ.ZERO ) ) THEN
! 566: *
! 567: * Column K is zero or underflow: set INFO and continue
! 568: *
! 569: IF( INFO.EQ.0 )
! 570: $ INFO = K
! 571: KP = K
! 572: ELSE
! 573: *
! 574: * Test for interchange
! 575: *
! 576: * Equivalent to testing for (used to handle NaN and Inf)
! 577: * ABSAKK.GE.ALPHA*COLMAX
! 578: *
! 579: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
! 580: *
! 581: * no interchange, use 1-by-1 pivot block
! 582: *
! 583: KP = K
! 584: ELSE
! 585: *
! 586: DONE = .FALSE.
! 587: *
! 588: * Loop until pivot found
! 589: *
! 590: 42 CONTINUE
! 591: *
! 592: * Begin pivot search loop body
! 593: *
! 594: * JMAX is the column-index of the largest off-diagonal
! 595: * element in row IMAX, and ROWMAX is its absolute value.
! 596: * Determine both ROWMAX and JMAX.
! 597: *
! 598: IF( IMAX.NE.K ) THEN
! 599: JMAX = K - 1 + IDAMAX( IMAX-K, A( IMAX, K ), LDA )
! 600: ROWMAX = ABS( A( IMAX, JMAX ) )
! 601: ELSE
! 602: ROWMAX = ZERO
! 603: END IF
! 604: *
! 605: IF( IMAX.LT.N ) THEN
! 606: ITEMP = IMAX + IDAMAX( N-IMAX, A( IMAX+1, IMAX ),
! 607: $ 1 )
! 608: DTEMP = ABS( A( ITEMP, IMAX ) )
! 609: IF( DTEMP.GT.ROWMAX ) THEN
! 610: ROWMAX = DTEMP
! 611: JMAX = ITEMP
! 612: END IF
! 613: END IF
! 614: *
! 615: * Equivalent to testing for (used to handle NaN and Inf)
! 616: * ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX
! 617: *
! 618: IF( .NOT.( ABS( A( IMAX, IMAX ) ).LT.ALPHA*ROWMAX ) )
! 619: $ THEN
! 620: *
! 621: * interchange rows and columns K and IMAX,
! 622: * use 1-by-1 pivot block
! 623: *
! 624: KP = IMAX
! 625: DONE = .TRUE.
! 626: *
! 627: * Equivalent to testing for ROWMAX .EQ. COLMAX,
! 628: * used to handle NaN and Inf
! 629: *
! 630: ELSE IF( ( P.EQ.JMAX ).OR.( ROWMAX.LE.COLMAX ) ) THEN
! 631: *
! 632: * interchange rows and columns K+1 and IMAX,
! 633: * use 2-by-2 pivot block
! 634: *
! 635: KP = IMAX
! 636: KSTEP = 2
! 637: DONE = .TRUE.
! 638: ELSE
! 639: *
! 640: * Pivot NOT found, set variables and repeat
! 641: *
! 642: P = IMAX
! 643: COLMAX = ROWMAX
! 644: IMAX = JMAX
! 645: END IF
! 646: *
! 647: * End pivot search loop body
! 648: *
! 649: IF( .NOT. DONE ) GOTO 42
! 650: *
! 651: END IF
! 652: *
! 653: * Swap TWO rows and TWO columns
! 654: *
! 655: * First swap
! 656: *
! 657: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
! 658: *
! 659: * Interchange rows and column K and P in the trailing
! 660: * submatrix A(k:n,k:n) if we have a 2-by-2 pivot
! 661: *
! 662: IF( P.LT.N )
! 663: $ CALL DSWAP( N-P, A( P+1, K ), 1, A( P+1, P ), 1 )
! 664: IF( P.GT.(K+1) )
! 665: $ CALL DSWAP( P-K-1, A( K+1, K ), 1, A( P, K+1 ), LDA )
! 666: T = A( K, K )
! 667: A( K, K ) = A( P, P )
! 668: A( P, P ) = T
! 669: END IF
! 670: *
! 671: * Second swap
! 672: *
! 673: KK = K + KSTEP - 1
! 674: IF( KP.NE.KK ) THEN
! 675: *
! 676: * Interchange rows and columns KK and KP in the trailing
! 677: * submatrix A(k:n,k:n)
! 678: *
! 679: IF( KP.LT.N )
! 680: $ CALL DSWAP( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 )
! 681: IF( ( KK.LT.N ) .AND. ( KP.GT.(KK+1) ) )
! 682: $ CALL DSWAP( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ),
! 683: $ LDA )
! 684: T = A( KK, KK )
! 685: A( KK, KK ) = A( KP, KP )
! 686: A( KP, KP ) = T
! 687: IF( KSTEP.EQ.2 ) THEN
! 688: T = A( K+1, K )
! 689: A( K+1, K ) = A( KP, K )
! 690: A( KP, K ) = T
! 691: END IF
! 692: END IF
! 693: *
! 694: * Update the trailing submatrix
! 695: *
! 696: IF( KSTEP.EQ.1 ) THEN
! 697: *
! 698: * 1-by-1 pivot block D(k): column k now holds
! 699: *
! 700: * W(k) = L(k)*D(k)
! 701: *
! 702: * where L(k) is the k-th column of L
! 703: *
! 704: IF( K.LT.N ) THEN
! 705: *
! 706: * Perform a rank-1 update of A(k+1:n,k+1:n) and
! 707: * store L(k) in column k
! 708: *
! 709: IF( ABS( A( K, K ) ).GE.SFMIN ) THEN
! 710: *
! 711: * Perform a rank-1 update of A(k+1:n,k+1:n) as
! 712: * A := A - L(k)*D(k)*L(k)**T
! 713: * = A - W(k)*(1/D(k))*W(k)**T
! 714: *
! 715: D11 = ONE / A( K, K )
! 716: CALL DSYR( UPLO, N-K, -D11, A( K+1, K ), 1,
! 717: $ A( K+1, K+1 ), LDA )
! 718: *
! 719: * Store L(k) in column k
! 720: *
! 721: CALL DSCAL( N-K, D11, A( K+1, K ), 1 )
! 722: ELSE
! 723: *
! 724: * Store L(k) in column k
! 725: *
! 726: D11 = A( K, K )
! 727: DO 46 II = K + 1, N
! 728: A( II, K ) = A( II, K ) / D11
! 729: 46 CONTINUE
! 730: *
! 731: * Perform a rank-1 update of A(k+1:n,k+1:n) as
! 732: * A := A - L(k)*D(k)*L(k)**T
! 733: * = A - W(k)*(1/D(k))*W(k)**T
! 734: * = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T
! 735: *
! 736: CALL DSYR( UPLO, N-K, -D11, A( K+1, K ), 1,
! 737: $ A( K+1, K+1 ), LDA )
! 738: END IF
! 739: END IF
! 740: *
! 741: ELSE
! 742: *
! 743: * 2-by-2 pivot block D(k): columns k and k+1 now hold
! 744: *
! 745: * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
! 746: *
! 747: * where L(k) and L(k+1) are the k-th and (k+1)-th columns
! 748: * of L
! 749: *
! 750: *
! 751: * Perform a rank-2 update of A(k+2:n,k+2:n) as
! 752: *
! 753: * A := A - ( L(k) L(k+1) ) * D(k) * ( L(k) L(k+1) )**T
! 754: * = A - ( ( A(k)A(k+1) )*inv(D(k) ) * ( A(k)A(k+1) )**T
! 755: *
! 756: * and store L(k) and L(k+1) in columns k and k+1
! 757: *
! 758: IF( K.LT.N-1 ) THEN
! 759: *
! 760: D21 = A( K+1, K )
! 761: D11 = A( K+1, K+1 ) / D21
! 762: D22 = A( K, K ) / D21
! 763: T = ONE / ( D11*D22-ONE )
! 764: *
! 765: DO 60 J = K + 2, N
! 766: *
! 767: * Compute D21 * ( W(k)W(k+1) ) * inv(D(k)) for row J
! 768: *
! 769: WK = T*( D11*A( J, K )-A( J, K+1 ) )
! 770: WKP1 = T*( D22*A( J, K+1 )-A( J, K ) )
! 771: *
! 772: * Perform a rank-2 update of A(k+2:n,k+2:n)
! 773: *
! 774: DO 50 I = J, N
! 775: A( I, J ) = A( I, J ) - ( A( I, K ) / D21 )*WK -
! 776: $ ( A( I, K+1 ) / D21 )*WKP1
! 777: 50 CONTINUE
! 778: *
! 779: * Store L(k) and L(k+1) in cols k and k+1 for row J
! 780: *
! 781: A( J, K ) = WK / D21
! 782: A( J, K+1 ) = WKP1 / D21
! 783: *
! 784: 60 CONTINUE
! 785: *
! 786: END IF
! 787: *
! 788: END IF
! 789: END IF
! 790: *
! 791: * Store details of the interchanges in IPIV
! 792: *
! 793: IF( KSTEP.EQ.1 ) THEN
! 794: IPIV( K ) = KP
! 795: ELSE
! 796: IPIV( K ) = -P
! 797: IPIV( K+1 ) = -KP
! 798: END IF
! 799: *
! 800: * Increase K and return to the start of the main loop
! 801: *
! 802: K = K + KSTEP
! 803: GO TO 40
! 804: *
! 805: END IF
! 806: *
! 807: 70 CONTINUE
! 808: *
! 809: RETURN
! 810: *
! 811: * End of DSYTF2_ROOK
! 812: *
! 813: END
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