Annotation of rpl/lapack/lapack/dsytf2_rk.f, revision 1.1
1.1 ! bertrand 1: *> \brief \b DSYTF2_RK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 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_RK + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsytf2_rk.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsytf2_rk.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsytf2_rk.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE DSYTF2_RK( UPLO, N, A, LDA, E, 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, * ), E ( * )
! 30: * ..
! 31: *
! 32: *
! 33: *> \par Purpose:
! 34: * =============
! 35: *>
! 36: *> \verbatim
! 37: *> DSYTF2_RK computes the factorization of a real symmetric matrix A
! 38: *> using the bounded Bunch-Kaufman (rook) diagonal pivoting method:
! 39: *>
! 40: *> A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
! 41: *>
! 42: *> where U (or L) is unit upper (or lower) triangular matrix,
! 43: *> U**T (or L**T) is the transpose of U (or L), P is a permutation
! 44: *> matrix, P**T is the transpose of P, and D is symmetric and block
! 45: *> 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: *> For more information see Further Details section.
! 49: *> \endverbatim
! 50: *
! 51: * Arguments:
! 52: * ==========
! 53: *
! 54: *> \param[in] UPLO
! 55: *> \verbatim
! 56: *> UPLO is CHARACTER*1
! 57: *> Specifies whether the upper or lower triangular part of the
! 58: *> symmetric matrix A is stored:
! 59: *> = 'U': Upper triangular
! 60: *> = 'L': Lower triangular
! 61: *> \endverbatim
! 62: *>
! 63: *> \param[in] N
! 64: *> \verbatim
! 65: *> N is INTEGER
! 66: *> The order of the matrix A. N >= 0.
! 67: *> \endverbatim
! 68: *>
! 69: *> \param[in,out] A
! 70: *> \verbatim
! 71: *> A is DOUBLE PRECISION array, dimension (LDA,N)
! 72: *> On entry, the symmetric matrix A.
! 73: *> If UPLO = 'U': the leading N-by-N upper triangular part
! 74: *> of A contains the upper triangular part of the matrix A,
! 75: *> and the strictly lower triangular part of A is not
! 76: *> referenced.
! 77: *>
! 78: *> If UPLO = 'L': the leading N-by-N lower triangular part
! 79: *> of A contains the lower triangular part of the matrix A,
! 80: *> and the strictly upper triangular part of A is not
! 81: *> referenced.
! 82: *>
! 83: *> On exit, contains:
! 84: *> a) ONLY diagonal elements of the symmetric block diagonal
! 85: *> matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
! 86: *> (superdiagonal (or subdiagonal) elements of D
! 87: *> are stored on exit in array E), and
! 88: *> b) If UPLO = 'U': factor U in the superdiagonal part of A.
! 89: *> If UPLO = 'L': factor L in the subdiagonal part of A.
! 90: *> \endverbatim
! 91: *>
! 92: *> \param[in] LDA
! 93: *> \verbatim
! 94: *> LDA is INTEGER
! 95: *> The leading dimension of the array A. LDA >= max(1,N).
! 96: *> \endverbatim
! 97: *>
! 98: *> \param[out] E
! 99: *> \verbatim
! 100: *> E is DOUBLE PRECISION array, dimension (N)
! 101: *> On exit, contains the superdiagonal (or subdiagonal)
! 102: *> elements of the symmetric block diagonal matrix D
! 103: *> with 1-by-1 or 2-by-2 diagonal blocks, where
! 104: *> If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
! 105: *> If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
! 106: *>
! 107: *> NOTE: For 1-by-1 diagonal block D(k), where
! 108: *> 1 <= k <= N, the element E(k) is set to 0 in both
! 109: *> UPLO = 'U' or UPLO = 'L' cases.
! 110: *> \endverbatim
! 111: *>
! 112: *> \param[out] IPIV
! 113: *> \verbatim
! 114: *> IPIV is INTEGER array, dimension (N)
! 115: *> IPIV describes the permutation matrix P in the factorization
! 116: *> of matrix A as follows. The absolute value of IPIV(k)
! 117: *> represents the index of row and column that were
! 118: *> interchanged with the k-th row and column. The value of UPLO
! 119: *> describes the order in which the interchanges were applied.
! 120: *> Also, the sign of IPIV represents the block structure of
! 121: *> the symmetric block diagonal matrix D with 1-by-1 or 2-by-2
! 122: *> diagonal blocks which correspond to 1 or 2 interchanges
! 123: *> at each factorization step. For more info see Further
! 124: *> Details section.
! 125: *>
! 126: *> If UPLO = 'U',
! 127: *> ( in factorization order, k decreases from N to 1 ):
! 128: *> a) A single positive entry IPIV(k) > 0 means:
! 129: *> D(k,k) is a 1-by-1 diagonal block.
! 130: *> If IPIV(k) != k, rows and columns k and IPIV(k) were
! 131: *> interchanged in the matrix A(1:N,1:N);
! 132: *> If IPIV(k) = k, no interchange occurred.
! 133: *>
! 134: *> b) A pair of consecutive negative entries
! 135: *> IPIV(k) < 0 and IPIV(k-1) < 0 means:
! 136: *> D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
! 137: *> (NOTE: negative entries in IPIV appear ONLY in pairs).
! 138: *> 1) If -IPIV(k) != k, rows and columns
! 139: *> k and -IPIV(k) were interchanged
! 140: *> in the matrix A(1:N,1:N).
! 141: *> If -IPIV(k) = k, no interchange occurred.
! 142: *> 2) If -IPIV(k-1) != k-1, rows and columns
! 143: *> k-1 and -IPIV(k-1) were interchanged
! 144: *> in the matrix A(1:N,1:N).
! 145: *> If -IPIV(k-1) = k-1, no interchange occurred.
! 146: *>
! 147: *> c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
! 148: *>
! 149: *> d) NOTE: Any entry IPIV(k) is always NONZERO on output.
! 150: *>
! 151: *> If UPLO = 'L',
! 152: *> ( in factorization order, k increases from 1 to N ):
! 153: *> a) A single positive entry IPIV(k) > 0 means:
! 154: *> D(k,k) is a 1-by-1 diagonal block.
! 155: *> If IPIV(k) != k, rows and columns k and IPIV(k) were
! 156: *> interchanged in the matrix A(1:N,1:N).
! 157: *> If IPIV(k) = k, no interchange occurred.
! 158: *>
! 159: *> b) A pair of consecutive negative entries
! 160: *> IPIV(k) < 0 and IPIV(k+1) < 0 means:
! 161: *> D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
! 162: *> (NOTE: negative entries in IPIV appear ONLY in pairs).
! 163: *> 1) If -IPIV(k) != k, rows and columns
! 164: *> k and -IPIV(k) were interchanged
! 165: *> in the matrix A(1:N,1:N).
! 166: *> If -IPIV(k) = k, no interchange occurred.
! 167: *> 2) If -IPIV(k+1) != k+1, rows and columns
! 168: *> k-1 and -IPIV(k-1) were interchanged
! 169: *> in the matrix A(1:N,1:N).
! 170: *> If -IPIV(k+1) = k+1, no interchange occurred.
! 171: *>
! 172: *> c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
! 173: *>
! 174: *> d) NOTE: Any entry IPIV(k) is always NONZERO on output.
! 175: *> \endverbatim
! 176: *>
! 177: *> \param[out] INFO
! 178: *> \verbatim
! 179: *> INFO is INTEGER
! 180: *> = 0: successful exit
! 181: *>
! 182: *> < 0: If INFO = -k, the k-th argument had an illegal value
! 183: *>
! 184: *> > 0: If INFO = k, the matrix A is singular, because:
! 185: *> If UPLO = 'U': column k in the upper
! 186: *> triangular part of A contains all zeros.
! 187: *> If UPLO = 'L': column k in the lower
! 188: *> triangular part of A contains all zeros.
! 189: *>
! 190: *> Therefore D(k,k) is exactly zero, and superdiagonal
! 191: *> elements of column k of U (or subdiagonal elements of
! 192: *> column k of L ) are all zeros. The factorization has
! 193: *> been completed, but the block diagonal matrix D is
! 194: *> exactly singular, and division by zero will occur if
! 195: *> it is used to solve a system of equations.
! 196: *>
! 197: *> NOTE: INFO only stores the first occurrence of
! 198: *> a singularity, any subsequent occurrence of singularity
! 199: *> is not stored in INFO even though the factorization
! 200: *> always completes.
! 201: *> \endverbatim
! 202: *
! 203: * Authors:
! 204: * ========
! 205: *
! 206: *> \author Univ. of Tennessee
! 207: *> \author Univ. of California Berkeley
! 208: *> \author Univ. of Colorado Denver
! 209: *> \author NAG Ltd.
! 210: *
! 211: *> \date December 2016
! 212: *
! 213: *> \ingroup doubleSYcomputational
! 214: *
! 215: *> \par Further Details:
! 216: * =====================
! 217: *>
! 218: *> \verbatim
! 219: *> TODO: put further details
! 220: *> \endverbatim
! 221: *
! 222: *> \par Contributors:
! 223: * ==================
! 224: *>
! 225: *> \verbatim
! 226: *>
! 227: *> December 2016, Igor Kozachenko,
! 228: *> Computer Science Division,
! 229: *> University of California, Berkeley
! 230: *>
! 231: *> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
! 232: *> School of Mathematics,
! 233: *> University of Manchester
! 234: *>
! 235: *> 01-01-96 - Based on modifications by
! 236: *> J. Lewis, Boeing Computer Services Company
! 237: *> A. Petitet, Computer Science Dept.,
! 238: *> Univ. of Tenn., Knoxville abd , USA
! 239: *> \endverbatim
! 240: *
! 241: * =====================================================================
! 242: SUBROUTINE DSYTF2_RK( UPLO, N, A, LDA, E, IPIV, INFO )
! 243: *
! 244: * -- LAPACK computational routine (version 3.7.0) --
! 245: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 246: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 247: * December 2016
! 248: *
! 249: * .. Scalar Arguments ..
! 250: CHARACTER UPLO
! 251: INTEGER INFO, LDA, N
! 252: * ..
! 253: * .. Array Arguments ..
! 254: INTEGER IPIV( * )
! 255: DOUBLE PRECISION A( LDA, * ), E( * )
! 256: * ..
! 257: *
! 258: * =====================================================================
! 259: *
! 260: * .. Parameters ..
! 261: DOUBLE PRECISION ZERO, ONE
! 262: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
! 263: DOUBLE PRECISION EIGHT, SEVTEN
! 264: PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
! 265: * ..
! 266: * .. Local Scalars ..
! 267: LOGICAL UPPER, DONE
! 268: INTEGER I, IMAX, J, JMAX, ITEMP, K, KK, KP, KSTEP,
! 269: $ P, II
! 270: DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22,
! 271: $ ROWMAX, DTEMP, T, WK, WKM1, WKP1, SFMIN
! 272: * ..
! 273: * .. External Functions ..
! 274: LOGICAL LSAME
! 275: INTEGER IDAMAX
! 276: DOUBLE PRECISION DLAMCH
! 277: EXTERNAL LSAME, IDAMAX, DLAMCH
! 278: * ..
! 279: * .. External Subroutines ..
! 280: EXTERNAL DSCAL, DSWAP, DSYR, XERBLA
! 281: * ..
! 282: * .. Intrinsic Functions ..
! 283: INTRINSIC ABS, MAX, SQRT
! 284: * ..
! 285: * .. Executable Statements ..
! 286: *
! 287: * Test the input parameters.
! 288: *
! 289: INFO = 0
! 290: UPPER = LSAME( UPLO, 'U' )
! 291: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
! 292: INFO = -1
! 293: ELSE IF( N.LT.0 ) THEN
! 294: INFO = -2
! 295: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
! 296: INFO = -4
! 297: END IF
! 298: IF( INFO.NE.0 ) THEN
! 299: CALL XERBLA( 'DSYTF2_RK', -INFO )
! 300: RETURN
! 301: END IF
! 302: *
! 303: * Initialize ALPHA for use in choosing pivot block size.
! 304: *
! 305: ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
! 306: *
! 307: * Compute machine safe minimum
! 308: *
! 309: SFMIN = DLAMCH( 'S' )
! 310: *
! 311: IF( UPPER ) THEN
! 312: *
! 313: * Factorize A as U*D*U**T using the upper triangle of A
! 314: *
! 315: * Initilize the first entry of array E, where superdiagonal
! 316: * elements of D are stored
! 317: *
! 318: E( 1 ) = ZERO
! 319: *
! 320: * K is the main loop index, decreasing from N to 1 in steps of
! 321: * 1 or 2
! 322: *
! 323: K = N
! 324: 10 CONTINUE
! 325: *
! 326: * If K < 1, exit from loop
! 327: *
! 328: IF( K.LT.1 )
! 329: $ GO TO 34
! 330: KSTEP = 1
! 331: P = K
! 332: *
! 333: * Determine rows and columns to be interchanged and whether
! 334: * a 1-by-1 or 2-by-2 pivot block will be used
! 335: *
! 336: ABSAKK = ABS( A( K, K ) )
! 337: *
! 338: * IMAX is the row-index of the largest off-diagonal element in
! 339: * column K, and COLMAX is its absolute value.
! 340: * Determine both COLMAX and IMAX.
! 341: *
! 342: IF( K.GT.1 ) THEN
! 343: IMAX = IDAMAX( K-1, A( 1, K ), 1 )
! 344: COLMAX = ABS( A( IMAX, K ) )
! 345: ELSE
! 346: COLMAX = ZERO
! 347: END IF
! 348: *
! 349: IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) ) THEN
! 350: *
! 351: * Column K is zero or underflow: set INFO and continue
! 352: *
! 353: IF( INFO.EQ.0 )
! 354: $ INFO = K
! 355: KP = K
! 356: *
! 357: * Set E( K ) to zero
! 358: *
! 359: IF( K.GT.1 )
! 360: $ E( K ) = ZERO
! 361: *
! 362: ELSE
! 363: *
! 364: * Test for interchange
! 365: *
! 366: * Equivalent to testing for (used to handle NaN and Inf)
! 367: * ABSAKK.GE.ALPHA*COLMAX
! 368: *
! 369: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
! 370: *
! 371: * no interchange,
! 372: * use 1-by-1 pivot block
! 373: *
! 374: KP = K
! 375: ELSE
! 376: *
! 377: DONE = .FALSE.
! 378: *
! 379: * Loop until pivot found
! 380: *
! 381: 12 CONTINUE
! 382: *
! 383: * Begin pivot search loop body
! 384: *
! 385: * JMAX is the column-index of the largest off-diagonal
! 386: * element in row IMAX, and ROWMAX is its absolute value.
! 387: * Determine both ROWMAX and JMAX.
! 388: *
! 389: IF( IMAX.NE.K ) THEN
! 390: JMAX = IMAX + IDAMAX( K-IMAX, A( IMAX, IMAX+1 ),
! 391: $ LDA )
! 392: ROWMAX = ABS( A( IMAX, JMAX ) )
! 393: ELSE
! 394: ROWMAX = ZERO
! 395: END IF
! 396: *
! 397: IF( IMAX.GT.1 ) THEN
! 398: ITEMP = IDAMAX( IMAX-1, A( 1, IMAX ), 1 )
! 399: DTEMP = ABS( A( ITEMP, IMAX ) )
! 400: IF( DTEMP.GT.ROWMAX ) THEN
! 401: ROWMAX = DTEMP
! 402: JMAX = ITEMP
! 403: END IF
! 404: END IF
! 405: *
! 406: * Equivalent to testing for (used to handle NaN and Inf)
! 407: * ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX
! 408: *
! 409: IF( .NOT.( ABS( A( IMAX, IMAX ) ).LT.ALPHA*ROWMAX ) )
! 410: $ THEN
! 411: *
! 412: * interchange rows and columns K and IMAX,
! 413: * use 1-by-1 pivot block
! 414: *
! 415: KP = IMAX
! 416: DONE = .TRUE.
! 417: *
! 418: * Equivalent to testing for ROWMAX .EQ. COLMAX,
! 419: * used to handle NaN and Inf
! 420: *
! 421: ELSE IF( ( P.EQ.JMAX ).OR.( ROWMAX.LE.COLMAX ) ) THEN
! 422: *
! 423: * interchange rows and columns K+1 and IMAX,
! 424: * use 2-by-2 pivot block
! 425: *
! 426: KP = IMAX
! 427: KSTEP = 2
! 428: DONE = .TRUE.
! 429: ELSE
! 430: *
! 431: * Pivot NOT found, set variables and repeat
! 432: *
! 433: P = IMAX
! 434: COLMAX = ROWMAX
! 435: IMAX = JMAX
! 436: END IF
! 437: *
! 438: * End pivot search loop body
! 439: *
! 440: IF( .NOT. DONE ) GOTO 12
! 441: *
! 442: END IF
! 443: *
! 444: * Swap TWO rows and TWO columns
! 445: *
! 446: * First swap
! 447: *
! 448: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
! 449: *
! 450: * Interchange rows and column K and P in the leading
! 451: * submatrix A(1:k,1:k) if we have a 2-by-2 pivot
! 452: *
! 453: IF( P.GT.1 )
! 454: $ CALL DSWAP( P-1, A( 1, K ), 1, A( 1, P ), 1 )
! 455: IF( P.LT.(K-1) )
! 456: $ CALL DSWAP( K-P-1, A( P+1, K ), 1, A( P, P+1 ),
! 457: $ LDA )
! 458: T = A( K, K )
! 459: A( K, K ) = A( P, P )
! 460: A( P, P ) = T
! 461: *
! 462: * Convert upper triangle of A into U form by applying
! 463: * the interchanges in columns k+1:N.
! 464: *
! 465: IF( K.LT.N )
! 466: $ CALL DSWAP( N-K, A( K, K+1 ), LDA, A( P, K+1 ), LDA )
! 467: *
! 468: END IF
! 469: *
! 470: * Second swap
! 471: *
! 472: KK = K - KSTEP + 1
! 473: IF( KP.NE.KK ) THEN
! 474: *
! 475: * Interchange rows and columns KK and KP in the leading
! 476: * submatrix A(1:k,1:k)
! 477: *
! 478: IF( KP.GT.1 )
! 479: $ CALL DSWAP( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
! 480: IF( ( KK.GT.1 ) .AND. ( KP.LT.(KK-1) ) )
! 481: $ CALL DSWAP( KK-KP-1, A( KP+1, KK ), 1, A( KP, KP+1 ),
! 482: $ LDA )
! 483: T = A( KK, KK )
! 484: A( KK, KK ) = A( KP, KP )
! 485: A( KP, KP ) = T
! 486: IF( KSTEP.EQ.2 ) THEN
! 487: T = A( K-1, K )
! 488: A( K-1, K ) = A( KP, K )
! 489: A( KP, K ) = T
! 490: END IF
! 491: *
! 492: * Convert upper triangle of A into U form by applying
! 493: * the interchanges in columns k+1:N.
! 494: *
! 495: IF( K.LT.N )
! 496: $ CALL DSWAP( N-K, A( KK, K+1 ), LDA, A( KP, K+1 ),
! 497: $ LDA )
! 498: *
! 499: END IF
! 500: *
! 501: * Update the leading submatrix
! 502: *
! 503: IF( KSTEP.EQ.1 ) THEN
! 504: *
! 505: * 1-by-1 pivot block D(k): column k now holds
! 506: *
! 507: * W(k) = U(k)*D(k)
! 508: *
! 509: * where U(k) is the k-th column of U
! 510: *
! 511: IF( K.GT.1 ) THEN
! 512: *
! 513: * Perform a rank-1 update of A(1:k-1,1:k-1) and
! 514: * store U(k) in column k
! 515: *
! 516: IF( ABS( A( K, K ) ).GE.SFMIN ) THEN
! 517: *
! 518: * Perform a rank-1 update of A(1:k-1,1:k-1) as
! 519: * A := A - U(k)*D(k)*U(k)**T
! 520: * = A - W(k)*1/D(k)*W(k)**T
! 521: *
! 522: D11 = ONE / A( K, K )
! 523: CALL DSYR( UPLO, K-1, -D11, A( 1, K ), 1, A, LDA )
! 524: *
! 525: * Store U(k) in column k
! 526: *
! 527: CALL DSCAL( K-1, D11, A( 1, K ), 1 )
! 528: ELSE
! 529: *
! 530: * Store L(k) in column K
! 531: *
! 532: D11 = A( K, K )
! 533: DO 16 II = 1, K - 1
! 534: A( II, K ) = A( II, K ) / D11
! 535: 16 CONTINUE
! 536: *
! 537: * Perform a rank-1 update of A(k+1:n,k+1:n) as
! 538: * A := A - U(k)*D(k)*U(k)**T
! 539: * = A - W(k)*(1/D(k))*W(k)**T
! 540: * = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T
! 541: *
! 542: CALL DSYR( UPLO, K-1, -D11, A( 1, K ), 1, A, LDA )
! 543: END IF
! 544: *
! 545: * Store the superdiagonal element of D in array E
! 546: *
! 547: E( K ) = ZERO
! 548: *
! 549: END IF
! 550: *
! 551: ELSE
! 552: *
! 553: * 2-by-2 pivot block D(k): columns k and k-1 now hold
! 554: *
! 555: * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
! 556: *
! 557: * where U(k) and U(k-1) are the k-th and (k-1)-th columns
! 558: * of U
! 559: *
! 560: * Perform a rank-2 update of A(1:k-2,1:k-2) as
! 561: *
! 562: * A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T
! 563: * = A - ( ( A(k-1)A(k) )*inv(D(k)) ) * ( A(k-1)A(k) )**T
! 564: *
! 565: * and store L(k) and L(k+1) in columns k and k+1
! 566: *
! 567: IF( K.GT.2 ) THEN
! 568: *
! 569: D12 = A( K-1, K )
! 570: D22 = A( K-1, K-1 ) / D12
! 571: D11 = A( K, K ) / D12
! 572: T = ONE / ( D11*D22-ONE )
! 573: *
! 574: DO 30 J = K - 2, 1, -1
! 575: *
! 576: WKM1 = T*( D11*A( J, K-1 )-A( J, K ) )
! 577: WK = T*( D22*A( J, K )-A( J, K-1 ) )
! 578: *
! 579: DO 20 I = J, 1, -1
! 580: A( I, J ) = A( I, J ) - (A( I, K ) / D12 )*WK -
! 581: $ ( A( I, K-1 ) / D12 )*WKM1
! 582: 20 CONTINUE
! 583: *
! 584: * Store U(k) and U(k-1) in cols k and k-1 for row J
! 585: *
! 586: A( J, K ) = WK / D12
! 587: A( J, K-1 ) = WKM1 / D12
! 588: *
! 589: 30 CONTINUE
! 590: *
! 591: END IF
! 592: *
! 593: * Copy superdiagonal elements of D(K) to E(K) and
! 594: * ZERO out superdiagonal entry of A
! 595: *
! 596: E( K ) = A( K-1, K )
! 597: E( K-1 ) = ZERO
! 598: A( K-1, K ) = ZERO
! 599: *
! 600: END IF
! 601: *
! 602: * End column K is nonsingular
! 603: *
! 604: END IF
! 605: *
! 606: * Store details of the interchanges in IPIV
! 607: *
! 608: IF( KSTEP.EQ.1 ) THEN
! 609: IPIV( K ) = KP
! 610: ELSE
! 611: IPIV( K ) = -P
! 612: IPIV( K-1 ) = -KP
! 613: END IF
! 614: *
! 615: * Decrease K and return to the start of the main loop
! 616: *
! 617: K = K - KSTEP
! 618: GO TO 10
! 619: *
! 620: 34 CONTINUE
! 621: *
! 622: ELSE
! 623: *
! 624: * Factorize A as L*D*L**T using the lower triangle of A
! 625: *
! 626: * Initilize the unused last entry of the subdiagonal array E.
! 627: *
! 628: E( N ) = ZERO
! 629: *
! 630: * K is the main loop index, increasing from 1 to N in steps of
! 631: * 1 or 2
! 632: *
! 633: K = 1
! 634: 40 CONTINUE
! 635: *
! 636: * If K > N, exit from loop
! 637: *
! 638: IF( K.GT.N )
! 639: $ GO TO 64
! 640: KSTEP = 1
! 641: P = K
! 642: *
! 643: * Determine rows and columns to be interchanged and whether
! 644: * a 1-by-1 or 2-by-2 pivot block will be used
! 645: *
! 646: ABSAKK = ABS( A( K, K ) )
! 647: *
! 648: * IMAX is the row-index of the largest off-diagonal element in
! 649: * column K, and COLMAX is its absolute value.
! 650: * Determine both COLMAX and IMAX.
! 651: *
! 652: IF( K.LT.N ) THEN
! 653: IMAX = K + IDAMAX( N-K, A( K+1, K ), 1 )
! 654: COLMAX = ABS( A( IMAX, K ) )
! 655: ELSE
! 656: COLMAX = ZERO
! 657: END IF
! 658: *
! 659: IF( ( MAX( ABSAKK, COLMAX ).EQ.ZERO ) ) THEN
! 660: *
! 661: * Column K is zero or underflow: set INFO and continue
! 662: *
! 663: IF( INFO.EQ.0 )
! 664: $ INFO = K
! 665: KP = K
! 666: *
! 667: * Set E( K ) to zero
! 668: *
! 669: IF( K.LT.N )
! 670: $ E( K ) = ZERO
! 671: *
! 672: ELSE
! 673: *
! 674: * Test for interchange
! 675: *
! 676: * Equivalent to testing for (used to handle NaN and Inf)
! 677: * ABSAKK.GE.ALPHA*COLMAX
! 678: *
! 679: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
! 680: *
! 681: * no interchange, use 1-by-1 pivot block
! 682: *
! 683: KP = K
! 684: *
! 685: ELSE
! 686: *
! 687: DONE = .FALSE.
! 688: *
! 689: * Loop until pivot found
! 690: *
! 691: 42 CONTINUE
! 692: *
! 693: * Begin pivot search loop body
! 694: *
! 695: * JMAX is the column-index of the largest off-diagonal
! 696: * element in row IMAX, and ROWMAX is its absolute value.
! 697: * Determine both ROWMAX and JMAX.
! 698: *
! 699: IF( IMAX.NE.K ) THEN
! 700: JMAX = K - 1 + IDAMAX( IMAX-K, A( IMAX, K ), LDA )
! 701: ROWMAX = ABS( A( IMAX, JMAX ) )
! 702: ELSE
! 703: ROWMAX = ZERO
! 704: END IF
! 705: *
! 706: IF( IMAX.LT.N ) THEN
! 707: ITEMP = IMAX + IDAMAX( N-IMAX, A( IMAX+1, IMAX ),
! 708: $ 1 )
! 709: DTEMP = ABS( A( ITEMP, IMAX ) )
! 710: IF( DTEMP.GT.ROWMAX ) THEN
! 711: ROWMAX = DTEMP
! 712: JMAX = ITEMP
! 713: END IF
! 714: END IF
! 715: *
! 716: * Equivalent to testing for (used to handle NaN and Inf)
! 717: * ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX
! 718: *
! 719: IF( .NOT.( ABS( A( IMAX, IMAX ) ).LT.ALPHA*ROWMAX ) )
! 720: $ THEN
! 721: *
! 722: * interchange rows and columns K and IMAX,
! 723: * use 1-by-1 pivot block
! 724: *
! 725: KP = IMAX
! 726: DONE = .TRUE.
! 727: *
! 728: * Equivalent to testing for ROWMAX .EQ. COLMAX,
! 729: * used to handle NaN and Inf
! 730: *
! 731: ELSE IF( ( P.EQ.JMAX ).OR.( ROWMAX.LE.COLMAX ) ) THEN
! 732: *
! 733: * interchange rows and columns K+1 and IMAX,
! 734: * use 2-by-2 pivot block
! 735: *
! 736: KP = IMAX
! 737: KSTEP = 2
! 738: DONE = .TRUE.
! 739: ELSE
! 740: *
! 741: * Pivot NOT found, set variables and repeat
! 742: *
! 743: P = IMAX
! 744: COLMAX = ROWMAX
! 745: IMAX = JMAX
! 746: END IF
! 747: *
! 748: * End pivot search loop body
! 749: *
! 750: IF( .NOT. DONE ) GOTO 42
! 751: *
! 752: END IF
! 753: *
! 754: * Swap TWO rows and TWO columns
! 755: *
! 756: * First swap
! 757: *
! 758: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
! 759: *
! 760: * Interchange rows and column K and P in the trailing
! 761: * submatrix A(k:n,k:n) if we have a 2-by-2 pivot
! 762: *
! 763: IF( P.LT.N )
! 764: $ CALL DSWAP( N-P, A( P+1, K ), 1, A( P+1, P ), 1 )
! 765: IF( P.GT.(K+1) )
! 766: $ CALL DSWAP( P-K-1, A( K+1, K ), 1, A( P, K+1 ), LDA )
! 767: T = A( K, K )
! 768: A( K, K ) = A( P, P )
! 769: A( P, P ) = T
! 770: *
! 771: * Convert lower triangle of A into L form by applying
! 772: * the interchanges in columns 1:k-1.
! 773: *
! 774: IF ( K.GT.1 )
! 775: $ CALL DSWAP( K-1, A( K, 1 ), LDA, A( P, 1 ), LDA )
! 776: *
! 777: END IF
! 778: *
! 779: * Second swap
! 780: *
! 781: KK = K + KSTEP - 1
! 782: IF( KP.NE.KK ) THEN
! 783: *
! 784: * Interchange rows and columns KK and KP in the trailing
! 785: * submatrix A(k:n,k:n)
! 786: *
! 787: IF( KP.LT.N )
! 788: $ CALL DSWAP( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 )
! 789: IF( ( KK.LT.N ) .AND. ( KP.GT.(KK+1) ) )
! 790: $ CALL DSWAP( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ),
! 791: $ LDA )
! 792: T = A( KK, KK )
! 793: A( KK, KK ) = A( KP, KP )
! 794: A( KP, KP ) = T
! 795: IF( KSTEP.EQ.2 ) THEN
! 796: T = A( K+1, K )
! 797: A( K+1, K ) = A( KP, K )
! 798: A( KP, K ) = T
! 799: END IF
! 800: *
! 801: * Convert lower triangle of A into L form by applying
! 802: * the interchanges in columns 1:k-1.
! 803: *
! 804: IF ( K.GT.1 )
! 805: $ CALL DSWAP( K-1, A( KK, 1 ), LDA, A( KP, 1 ), LDA )
! 806: *
! 807: END IF
! 808: *
! 809: * Update the trailing submatrix
! 810: *
! 811: IF( KSTEP.EQ.1 ) THEN
! 812: *
! 813: * 1-by-1 pivot block D(k): column k now holds
! 814: *
! 815: * W(k) = L(k)*D(k)
! 816: *
! 817: * where L(k) is the k-th column of L
! 818: *
! 819: IF( K.LT.N ) THEN
! 820: *
! 821: * Perform a rank-1 update of A(k+1:n,k+1:n) and
! 822: * store L(k) in column k
! 823: *
! 824: IF( ABS( A( K, K ) ).GE.SFMIN ) THEN
! 825: *
! 826: * Perform a rank-1 update of A(k+1:n,k+1:n) as
! 827: * A := A - L(k)*D(k)*L(k)**T
! 828: * = A - W(k)*(1/D(k))*W(k)**T
! 829: *
! 830: D11 = ONE / A( K, K )
! 831: CALL DSYR( UPLO, N-K, -D11, A( K+1, K ), 1,
! 832: $ A( K+1, K+1 ), LDA )
! 833: *
! 834: * Store L(k) in column k
! 835: *
! 836: CALL DSCAL( N-K, D11, A( K+1, K ), 1 )
! 837: ELSE
! 838: *
! 839: * Store L(k) in column k
! 840: *
! 841: D11 = A( K, K )
! 842: DO 46 II = K + 1, N
! 843: A( II, K ) = A( II, K ) / D11
! 844: 46 CONTINUE
! 845: *
! 846: * Perform a rank-1 update of A(k+1:n,k+1:n) as
! 847: * A := A - L(k)*D(k)*L(k)**T
! 848: * = A - W(k)*(1/D(k))*W(k)**T
! 849: * = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T
! 850: *
! 851: CALL DSYR( UPLO, N-K, -D11, A( K+1, K ), 1,
! 852: $ A( K+1, K+1 ), LDA )
! 853: END IF
! 854: *
! 855: * Store the subdiagonal element of D in array E
! 856: *
! 857: E( K ) = ZERO
! 858: *
! 859: END IF
! 860: *
! 861: ELSE
! 862: *
! 863: * 2-by-2 pivot block D(k): columns k and k+1 now hold
! 864: *
! 865: * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
! 866: *
! 867: * where L(k) and L(k+1) are the k-th and (k+1)-th columns
! 868: * of L
! 869: *
! 870: *
! 871: * Perform a rank-2 update of A(k+2:n,k+2:n) as
! 872: *
! 873: * A := A - ( L(k) L(k+1) ) * D(k) * ( L(k) L(k+1) )**T
! 874: * = A - ( ( A(k)A(k+1) )*inv(D(k) ) * ( A(k)A(k+1) )**T
! 875: *
! 876: * and store L(k) and L(k+1) in columns k and k+1
! 877: *
! 878: IF( K.LT.N-1 ) THEN
! 879: *
! 880: D21 = A( K+1, K )
! 881: D11 = A( K+1, K+1 ) / D21
! 882: D22 = A( K, K ) / D21
! 883: T = ONE / ( D11*D22-ONE )
! 884: *
! 885: DO 60 J = K + 2, N
! 886: *
! 887: * Compute D21 * ( W(k)W(k+1) ) * inv(D(k)) for row J
! 888: *
! 889: WK = T*( D11*A( J, K )-A( J, K+1 ) )
! 890: WKP1 = T*( D22*A( J, K+1 )-A( J, K ) )
! 891: *
! 892: * Perform a rank-2 update of A(k+2:n,k+2:n)
! 893: *
! 894: DO 50 I = J, N
! 895: A( I, J ) = A( I, J ) - ( A( I, K ) / D21 )*WK -
! 896: $ ( A( I, K+1 ) / D21 )*WKP1
! 897: 50 CONTINUE
! 898: *
! 899: * Store L(k) and L(k+1) in cols k and k+1 for row J
! 900: *
! 901: A( J, K ) = WK / D21
! 902: A( J, K+1 ) = WKP1 / D21
! 903: *
! 904: 60 CONTINUE
! 905: *
! 906: END IF
! 907: *
! 908: * Copy subdiagonal elements of D(K) to E(K) and
! 909: * ZERO out subdiagonal entry of A
! 910: *
! 911: E( K ) = A( K+1, K )
! 912: E( K+1 ) = ZERO
! 913: A( K+1, K ) = ZERO
! 914: *
! 915: END IF
! 916: *
! 917: * End column K is nonsingular
! 918: *
! 919: END IF
! 920: *
! 921: * Store details of the interchanges in IPIV
! 922: *
! 923: IF( KSTEP.EQ.1 ) THEN
! 924: IPIV( K ) = KP
! 925: ELSE
! 926: IPIV( K ) = -P
! 927: IPIV( K+1 ) = -KP
! 928: END IF
! 929: *
! 930: * Increase K and return to the start of the main loop
! 931: *
! 932: K = K + KSTEP
! 933: GO TO 40
! 934: *
! 935: 64 CONTINUE
! 936: *
! 937: END IF
! 938: *
! 939: RETURN
! 940: *
! 941: * End of DSYTF2_RK
! 942: *
! 943: END
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