Annotation of rpl/lapack/lapack/zsytf2.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZSYTF2( UPLO, N, A, LDA, IPIV, 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, LDA, N
! 11: * ..
! 12: * .. Array Arguments ..
! 13: INTEGER IPIV( * )
! 14: COMPLEX*16 A( LDA, * )
! 15: * ..
! 16: *
! 17: * Purpose
! 18: * =======
! 19: *
! 20: * ZSYTF2 computes the factorization of a complex symmetric matrix A
! 21: * using the Bunch-Kaufman diagonal pivoting method:
! 22: *
! 23: * A = U*D*U' or A = L*D*L'
! 24: *
! 25: * where U (or L) is a product of permutation and unit upper (lower)
! 26: * triangular matrices, U' is the transpose of U, and D is symmetric and
! 27: * block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
! 28: *
! 29: * This is the unblocked version of the algorithm, calling Level 2 BLAS.
! 30: *
! 31: * Arguments
! 32: * =========
! 33: *
! 34: * UPLO (input) CHARACTER*1
! 35: * Specifies whether the upper or lower triangular part of the
! 36: * symmetric matrix A is stored:
! 37: * = 'U': Upper triangular
! 38: * = 'L': Lower triangular
! 39: *
! 40: * N (input) INTEGER
! 41: * The order of the matrix A. N >= 0.
! 42: *
! 43: * A (input/output) COMPLEX*16 array, dimension (LDA,N)
! 44: * On entry, the symmetric matrix A. If UPLO = 'U', the leading
! 45: * n-by-n upper triangular part of A contains the upper
! 46: * triangular part of the matrix A, and the strictly lower
! 47: * triangular part of A is not referenced. If UPLO = 'L', the
! 48: * leading n-by-n lower triangular part of A contains the lower
! 49: * triangular part of the matrix A, and the strictly upper
! 50: * triangular part of A is not referenced.
! 51: *
! 52: * On exit, the block diagonal matrix D and the multipliers used
! 53: * to obtain the factor U or L (see below for further details).
! 54: *
! 55: * LDA (input) INTEGER
! 56: * The leading dimension of the array A. LDA >= max(1,N).
! 57: *
! 58: * IPIV (output) INTEGER array, dimension (N)
! 59: * Details of the interchanges and the block structure of D.
! 60: * If IPIV(k) > 0, then rows and columns k and IPIV(k) were
! 61: * interchanged and D(k,k) is a 1-by-1 diagonal block.
! 62: * If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
! 63: * columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
! 64: * is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
! 65: * IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
! 66: * interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
! 67: *
! 68: * INFO (output) INTEGER
! 69: * = 0: successful exit
! 70: * < 0: if INFO = -k, the k-th argument had an illegal value
! 71: * > 0: if INFO = k, D(k,k) is exactly zero. The factorization
! 72: * has been completed, but the block diagonal matrix D is
! 73: * exactly singular, and division by zero will occur if it
! 74: * is used to solve a system of equations.
! 75: *
! 76: * Further Details
! 77: * ===============
! 78: *
! 79: * 09-29-06 - patch from
! 80: * Bobby Cheng, MathWorks
! 81: *
! 82: * Replace l.209 and l.377
! 83: * IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
! 84: * by
! 85: * IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN
! 86: *
! 87: * 1-96 - Based on modifications by J. Lewis, Boeing Computer Services
! 88: * Company
! 89: *
! 90: * If UPLO = 'U', then A = U*D*U', where
! 91: * U = P(n)*U(n)* ... *P(k)U(k)* ...,
! 92: * i.e., U is a product of terms P(k)*U(k), where k decreases from n to
! 93: * 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
! 94: * and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
! 95: * defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
! 96: * that if the diagonal block D(k) is of order s (s = 1 or 2), then
! 97: *
! 98: * ( I v 0 ) k-s
! 99: * U(k) = ( 0 I 0 ) s
! 100: * ( 0 0 I ) n-k
! 101: * k-s s n-k
! 102: *
! 103: * If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
! 104: * If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
! 105: * and A(k,k), and v overwrites A(1:k-2,k-1:k).
! 106: *
! 107: * If UPLO = 'L', then A = L*D*L', where
! 108: * L = P(1)*L(1)* ... *P(k)*L(k)* ...,
! 109: * i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
! 110: * n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
! 111: * and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
! 112: * defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
! 113: * that if the diagonal block D(k) is of order s (s = 1 or 2), then
! 114: *
! 115: * ( I 0 0 ) k-1
! 116: * L(k) = ( 0 I 0 ) s
! 117: * ( 0 v I ) n-k-s+1
! 118: * k-1 s n-k-s+1
! 119: *
! 120: * If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
! 121: * If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
! 122: * and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
! 123: *
! 124: * =====================================================================
! 125: *
! 126: * .. Parameters ..
! 127: DOUBLE PRECISION ZERO, ONE
! 128: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
! 129: DOUBLE PRECISION EIGHT, SEVTEN
! 130: PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
! 131: COMPLEX*16 CONE
! 132: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
! 133: * ..
! 134: * .. Local Scalars ..
! 135: LOGICAL UPPER
! 136: INTEGER I, IMAX, J, JMAX, K, KK, KP, KSTEP
! 137: DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, ROWMAX
! 138: COMPLEX*16 D11, D12, D21, D22, R1, T, WK, WKM1, WKP1, Z
! 139: * ..
! 140: * .. External Functions ..
! 141: LOGICAL DISNAN, LSAME
! 142: INTEGER IZAMAX
! 143: EXTERNAL DISNAN, LSAME, IZAMAX
! 144: * ..
! 145: * .. External Subroutines ..
! 146: EXTERNAL XERBLA, ZSCAL, ZSWAP, ZSYR
! 147: * ..
! 148: * .. Intrinsic Functions ..
! 149: INTRINSIC ABS, DBLE, DIMAG, MAX, SQRT
! 150: * ..
! 151: * .. Statement Functions ..
! 152: DOUBLE PRECISION CABS1
! 153: * ..
! 154: * .. Statement Function definitions ..
! 155: CABS1( Z ) = ABS( DBLE( Z ) ) + ABS( DIMAG( Z ) )
! 156: * ..
! 157: * .. Executable Statements ..
! 158: *
! 159: * Test the input parameters.
! 160: *
! 161: INFO = 0
! 162: UPPER = LSAME( UPLO, 'U' )
! 163: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
! 164: INFO = -1
! 165: ELSE IF( N.LT.0 ) THEN
! 166: INFO = -2
! 167: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
! 168: INFO = -4
! 169: END IF
! 170: IF( INFO.NE.0 ) THEN
! 171: CALL XERBLA( 'ZSYTF2', -INFO )
! 172: RETURN
! 173: END IF
! 174: *
! 175: * Initialize ALPHA for use in choosing pivot block size.
! 176: *
! 177: ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
! 178: *
! 179: IF( UPPER ) THEN
! 180: *
! 181: * Factorize A as U*D*U' using the upper triangle of A
! 182: *
! 183: * K is the main loop index, decreasing from N to 1 in steps of
! 184: * 1 or 2
! 185: *
! 186: K = N
! 187: 10 CONTINUE
! 188: *
! 189: * If K < 1, exit from loop
! 190: *
! 191: IF( K.LT.1 )
! 192: $ GO TO 70
! 193: KSTEP = 1
! 194: *
! 195: * Determine rows and columns to be interchanged and whether
! 196: * a 1-by-1 or 2-by-2 pivot block will be used
! 197: *
! 198: ABSAKK = CABS1( A( K, K ) )
! 199: *
! 200: * IMAX is the row-index of the largest off-diagonal element in
! 201: * column K, and COLMAX is its absolute value
! 202: *
! 203: IF( K.GT.1 ) THEN
! 204: IMAX = IZAMAX( K-1, A( 1, K ), 1 )
! 205: COLMAX = CABS1( A( IMAX, K ) )
! 206: ELSE
! 207: COLMAX = ZERO
! 208: END IF
! 209: *
! 210: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO .OR. DISNAN(ABSAKK) ) THEN
! 211: *
! 212: * Column K is zero or contains a NaN: set INFO and continue
! 213: *
! 214: IF( INFO.EQ.0 )
! 215: $ INFO = K
! 216: KP = K
! 217: ELSE
! 218: IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
! 219: *
! 220: * no interchange, use 1-by-1 pivot block
! 221: *
! 222: KP = K
! 223: ELSE
! 224: *
! 225: * JMAX is the column-index of the largest off-diagonal
! 226: * element in row IMAX, and ROWMAX is its absolute value
! 227: *
! 228: JMAX = IMAX + IZAMAX( K-IMAX, A( IMAX, IMAX+1 ), LDA )
! 229: ROWMAX = CABS1( A( IMAX, JMAX ) )
! 230: IF( IMAX.GT.1 ) THEN
! 231: JMAX = IZAMAX( IMAX-1, A( 1, IMAX ), 1 )
! 232: ROWMAX = MAX( ROWMAX, CABS1( A( JMAX, IMAX ) ) )
! 233: END IF
! 234: *
! 235: IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
! 236: *
! 237: * no interchange, use 1-by-1 pivot block
! 238: *
! 239: KP = K
! 240: ELSE IF( CABS1( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX ) THEN
! 241: *
! 242: * interchange rows and columns K and IMAX, use 1-by-1
! 243: * pivot block
! 244: *
! 245: KP = IMAX
! 246: ELSE
! 247: *
! 248: * interchange rows and columns K-1 and IMAX, use 2-by-2
! 249: * pivot block
! 250: *
! 251: KP = IMAX
! 252: KSTEP = 2
! 253: END IF
! 254: END IF
! 255: *
! 256: KK = K - KSTEP + 1
! 257: IF( KP.NE.KK ) THEN
! 258: *
! 259: * Interchange rows and columns KK and KP in the leading
! 260: * submatrix A(1:k,1:k)
! 261: *
! 262: CALL ZSWAP( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
! 263: CALL ZSWAP( KK-KP-1, A( KP+1, KK ), 1, A( KP, KP+1 ),
! 264: $ LDA )
! 265: T = A( KK, KK )
! 266: A( KK, KK ) = A( KP, KP )
! 267: A( KP, KP ) = T
! 268: IF( KSTEP.EQ.2 ) THEN
! 269: T = A( K-1, K )
! 270: A( K-1, K ) = A( KP, K )
! 271: A( KP, K ) = T
! 272: END IF
! 273: END IF
! 274: *
! 275: * Update the leading submatrix
! 276: *
! 277: IF( KSTEP.EQ.1 ) THEN
! 278: *
! 279: * 1-by-1 pivot block D(k): column k now holds
! 280: *
! 281: * W(k) = U(k)*D(k)
! 282: *
! 283: * where U(k) is the k-th column of U
! 284: *
! 285: * Perform a rank-1 update of A(1:k-1,1:k-1) as
! 286: *
! 287: * A := A - U(k)*D(k)*U(k)' = A - W(k)*1/D(k)*W(k)'
! 288: *
! 289: R1 = CONE / A( K, K )
! 290: CALL ZSYR( UPLO, K-1, -R1, A( 1, K ), 1, A, LDA )
! 291: *
! 292: * Store U(k) in column k
! 293: *
! 294: CALL ZSCAL( K-1, R1, A( 1, K ), 1 )
! 295: ELSE
! 296: *
! 297: * 2-by-2 pivot block D(k): columns k and k-1 now hold
! 298: *
! 299: * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
! 300: *
! 301: * where U(k) and U(k-1) are the k-th and (k-1)-th columns
! 302: * of U
! 303: *
! 304: * Perform a rank-2 update of A(1:k-2,1:k-2) as
! 305: *
! 306: * A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )'
! 307: * = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )'
! 308: *
! 309: IF( K.GT.2 ) THEN
! 310: *
! 311: D12 = A( K-1, K )
! 312: D22 = A( K-1, K-1 ) / D12
! 313: D11 = A( K, K ) / D12
! 314: T = CONE / ( D11*D22-CONE )
! 315: D12 = T / D12
! 316: *
! 317: DO 30 J = K - 2, 1, -1
! 318: WKM1 = D12*( D11*A( J, K-1 )-A( J, K ) )
! 319: WK = D12*( D22*A( J, K )-A( J, K-1 ) )
! 320: DO 20 I = J, 1, -1
! 321: A( I, J ) = A( I, J ) - A( I, K )*WK -
! 322: $ A( I, K-1 )*WKM1
! 323: 20 CONTINUE
! 324: A( J, K ) = WK
! 325: A( J, K-1 ) = WKM1
! 326: 30 CONTINUE
! 327: *
! 328: END IF
! 329: *
! 330: END IF
! 331: END IF
! 332: *
! 333: * Store details of the interchanges in IPIV
! 334: *
! 335: IF( KSTEP.EQ.1 ) THEN
! 336: IPIV( K ) = KP
! 337: ELSE
! 338: IPIV( K ) = -KP
! 339: IPIV( K-1 ) = -KP
! 340: END IF
! 341: *
! 342: * Decrease K and return to the start of the main loop
! 343: *
! 344: K = K - KSTEP
! 345: GO TO 10
! 346: *
! 347: ELSE
! 348: *
! 349: * Factorize A as L*D*L' using the lower triangle of A
! 350: *
! 351: * K is the main loop index, increasing from 1 to N in steps of
! 352: * 1 or 2
! 353: *
! 354: K = 1
! 355: 40 CONTINUE
! 356: *
! 357: * If K > N, exit from loop
! 358: *
! 359: IF( K.GT.N )
! 360: $ GO TO 70
! 361: KSTEP = 1
! 362: *
! 363: * Determine rows and columns to be interchanged and whether
! 364: * a 1-by-1 or 2-by-2 pivot block will be used
! 365: *
! 366: ABSAKK = CABS1( A( K, K ) )
! 367: *
! 368: * IMAX is the row-index of the largest off-diagonal element in
! 369: * column K, and COLMAX is its absolute value
! 370: *
! 371: IF( K.LT.N ) THEN
! 372: IMAX = K + IZAMAX( N-K, A( K+1, K ), 1 )
! 373: COLMAX = CABS1( A( IMAX, K ) )
! 374: ELSE
! 375: COLMAX = ZERO
! 376: END IF
! 377: *
! 378: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO .OR. DISNAN(ABSAKK) ) THEN
! 379: *
! 380: * Column K is zero or contains a NaN: set INFO and continue
! 381: *
! 382: IF( INFO.EQ.0 )
! 383: $ INFO = K
! 384: KP = K
! 385: ELSE
! 386: IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
! 387: *
! 388: * no interchange, use 1-by-1 pivot block
! 389: *
! 390: KP = K
! 391: ELSE
! 392: *
! 393: * JMAX is the column-index of the largest off-diagonal
! 394: * element in row IMAX, and ROWMAX is its absolute value
! 395: *
! 396: JMAX = K - 1 + IZAMAX( IMAX-K, A( IMAX, K ), LDA )
! 397: ROWMAX = CABS1( A( IMAX, JMAX ) )
! 398: IF( IMAX.LT.N ) THEN
! 399: JMAX = IMAX + IZAMAX( N-IMAX, A( IMAX+1, IMAX ), 1 )
! 400: ROWMAX = MAX( ROWMAX, CABS1( A( JMAX, IMAX ) ) )
! 401: END IF
! 402: *
! 403: IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
! 404: *
! 405: * no interchange, use 1-by-1 pivot block
! 406: *
! 407: KP = K
! 408: ELSE IF( CABS1( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX ) THEN
! 409: *
! 410: * interchange rows and columns K and IMAX, use 1-by-1
! 411: * pivot block
! 412: *
! 413: KP = IMAX
! 414: ELSE
! 415: *
! 416: * interchange rows and columns K+1 and IMAX, use 2-by-2
! 417: * pivot block
! 418: *
! 419: KP = IMAX
! 420: KSTEP = 2
! 421: END IF
! 422: END IF
! 423: *
! 424: KK = K + KSTEP - 1
! 425: IF( KP.NE.KK ) THEN
! 426: *
! 427: * Interchange rows and columns KK and KP in the trailing
! 428: * submatrix A(k:n,k:n)
! 429: *
! 430: IF( KP.LT.N )
! 431: $ CALL ZSWAP( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 )
! 432: CALL ZSWAP( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ),
! 433: $ LDA )
! 434: T = A( KK, KK )
! 435: A( KK, KK ) = A( KP, KP )
! 436: A( KP, KP ) = T
! 437: IF( KSTEP.EQ.2 ) THEN
! 438: T = A( K+1, K )
! 439: A( K+1, K ) = A( KP, K )
! 440: A( KP, K ) = T
! 441: END IF
! 442: END IF
! 443: *
! 444: * Update the trailing submatrix
! 445: *
! 446: IF( KSTEP.EQ.1 ) THEN
! 447: *
! 448: * 1-by-1 pivot block D(k): column k now holds
! 449: *
! 450: * W(k) = L(k)*D(k)
! 451: *
! 452: * where L(k) is the k-th column of L
! 453: *
! 454: IF( K.LT.N ) THEN
! 455: *
! 456: * Perform a rank-1 update of A(k+1:n,k+1:n) as
! 457: *
! 458: * A := A - L(k)*D(k)*L(k)' = A - W(k)*(1/D(k))*W(k)'
! 459: *
! 460: R1 = CONE / A( K, K )
! 461: CALL ZSYR( UPLO, N-K, -R1, A( K+1, K ), 1,
! 462: $ A( K+1, K+1 ), LDA )
! 463: *
! 464: * Store L(k) in column K
! 465: *
! 466: CALL ZSCAL( N-K, R1, A( K+1, K ), 1 )
! 467: END IF
! 468: ELSE
! 469: *
! 470: * 2-by-2 pivot block D(k)
! 471: *
! 472: IF( K.LT.N-1 ) THEN
! 473: *
! 474: * Perform a rank-2 update of A(k+2:n,k+2:n) as
! 475: *
! 476: * A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )'
! 477: * = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )'
! 478: *
! 479: * where L(k) and L(k+1) are the k-th and (k+1)-th
! 480: * columns of L
! 481: *
! 482: D21 = A( K+1, K )
! 483: D11 = A( K+1, K+1 ) / D21
! 484: D22 = A( K, K ) / D21
! 485: T = CONE / ( D11*D22-CONE )
! 486: D21 = T / D21
! 487: *
! 488: DO 60 J = K + 2, N
! 489: WK = D21*( D11*A( J, K )-A( J, K+1 ) )
! 490: WKP1 = D21*( D22*A( J, K+1 )-A( J, K ) )
! 491: DO 50 I = J, N
! 492: A( I, J ) = A( I, J ) - A( I, K )*WK -
! 493: $ A( I, K+1 )*WKP1
! 494: 50 CONTINUE
! 495: A( J, K ) = WK
! 496: A( J, K+1 ) = WKP1
! 497: 60 CONTINUE
! 498: END IF
! 499: END IF
! 500: END IF
! 501: *
! 502: * Store details of the interchanges in IPIV
! 503: *
! 504: IF( KSTEP.EQ.1 ) THEN
! 505: IPIV( K ) = KP
! 506: ELSE
! 507: IPIV( K ) = -KP
! 508: IPIV( K+1 ) = -KP
! 509: END IF
! 510: *
! 511: * Increase K and return to the start of the main loop
! 512: *
! 513: K = K + KSTEP
! 514: GO TO 40
! 515: *
! 516: END IF
! 517: *
! 518: 70 CONTINUE
! 519: RETURN
! 520: *
! 521: * End of ZSYTF2
! 522: *
! 523: END
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