Annotation of rpl/lapack/lapack/dlatrs.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DLATRS( UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE,
! 2: $ CNORM, INFO )
! 3: *
! 4: * -- LAPACK auxiliary routine (version 3.2) --
! 5: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 7: * November 2006
! 8: *
! 9: * .. Scalar Arguments ..
! 10: CHARACTER DIAG, NORMIN, TRANS, UPLO
! 11: INTEGER INFO, LDA, N
! 12: DOUBLE PRECISION SCALE
! 13: * ..
! 14: * .. Array Arguments ..
! 15: DOUBLE PRECISION A( LDA, * ), CNORM( * ), X( * )
! 16: * ..
! 17: *
! 18: * Purpose
! 19: * =======
! 20: *
! 21: * DLATRS solves one of the triangular systems
! 22: *
! 23: * A *x = s*b or A'*x = s*b
! 24: *
! 25: * with scaling to prevent overflow. Here A is an upper or lower
! 26: * triangular matrix, A' denotes the transpose of A, x and b are
! 27: * n-element vectors, and s is a scaling factor, usually less than
! 28: * or equal to 1, chosen so that the components of x will be less than
! 29: * the overflow threshold. If the unscaled problem will not cause
! 30: * overflow, the Level 2 BLAS routine DTRSV is called. If the matrix A
! 31: * is singular (A(j,j) = 0 for some j), then s is set to 0 and a
! 32: * non-trivial solution to A*x = 0 is returned.
! 33: *
! 34: * Arguments
! 35: * =========
! 36: *
! 37: * UPLO (input) CHARACTER*1
! 38: * Specifies whether the matrix A is upper or lower triangular.
! 39: * = 'U': Upper triangular
! 40: * = 'L': Lower triangular
! 41: *
! 42: * TRANS (input) CHARACTER*1
! 43: * Specifies the operation applied to A.
! 44: * = 'N': Solve A * x = s*b (No transpose)
! 45: * = 'T': Solve A'* x = s*b (Transpose)
! 46: * = 'C': Solve A'* x = s*b (Conjugate transpose = Transpose)
! 47: *
! 48: * DIAG (input) CHARACTER*1
! 49: * Specifies whether or not the matrix A is unit triangular.
! 50: * = 'N': Non-unit triangular
! 51: * = 'U': Unit triangular
! 52: *
! 53: * NORMIN (input) CHARACTER*1
! 54: * Specifies whether CNORM has been set or not.
! 55: * = 'Y': CNORM contains the column norms on entry
! 56: * = 'N': CNORM is not set on entry. On exit, the norms will
! 57: * be computed and stored in CNORM.
! 58: *
! 59: * N (input) INTEGER
! 60: * The order of the matrix A. N >= 0.
! 61: *
! 62: * A (input) DOUBLE PRECISION array, dimension (LDA,N)
! 63: * The triangular matrix A. If UPLO = 'U', the leading n by n
! 64: * upper triangular part of the array A contains the upper
! 65: * triangular matrix, and the strictly lower triangular part of
! 66: * A is not referenced. If UPLO = 'L', the leading n by n lower
! 67: * triangular part of the array A contains the lower triangular
! 68: * matrix, and the strictly upper triangular part of A is not
! 69: * referenced. If DIAG = 'U', the diagonal elements of A are
! 70: * also not referenced and are assumed to be 1.
! 71: *
! 72: * LDA (input) INTEGER
! 73: * The leading dimension of the array A. LDA >= max (1,N).
! 74: *
! 75: * X (input/output) DOUBLE PRECISION array, dimension (N)
! 76: * On entry, the right hand side b of the triangular system.
! 77: * On exit, X is overwritten by the solution vector x.
! 78: *
! 79: * SCALE (output) DOUBLE PRECISION
! 80: * The scaling factor s for the triangular system
! 81: * A * x = s*b or A'* x = s*b.
! 82: * If SCALE = 0, the matrix A is singular or badly scaled, and
! 83: * the vector x is an exact or approximate solution to A*x = 0.
! 84: *
! 85: * CNORM (input or output) DOUBLE PRECISION array, dimension (N)
! 86: *
! 87: * If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
! 88: * contains the norm of the off-diagonal part of the j-th column
! 89: * of A. If TRANS = 'N', CNORM(j) must be greater than or equal
! 90: * to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
! 91: * must be greater than or equal to the 1-norm.
! 92: *
! 93: * If NORMIN = 'N', CNORM is an output argument and CNORM(j)
! 94: * returns the 1-norm of the offdiagonal part of the j-th column
! 95: * of A.
! 96: *
! 97: * INFO (output) INTEGER
! 98: * = 0: successful exit
! 99: * < 0: if INFO = -k, the k-th argument had an illegal value
! 100: *
! 101: * Further Details
! 102: * ======= =======
! 103: *
! 104: * A rough bound on x is computed; if that is less than overflow, DTRSV
! 105: * is called, otherwise, specific code is used which checks for possible
! 106: * overflow or divide-by-zero at every operation.
! 107: *
! 108: * A columnwise scheme is used for solving A*x = b. The basic algorithm
! 109: * if A is lower triangular is
! 110: *
! 111: * x[1:n] := b[1:n]
! 112: * for j = 1, ..., n
! 113: * x(j) := x(j) / A(j,j)
! 114: * x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
! 115: * end
! 116: *
! 117: * Define bounds on the components of x after j iterations of the loop:
! 118: * M(j) = bound on x[1:j]
! 119: * G(j) = bound on x[j+1:n]
! 120: * Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
! 121: *
! 122: * Then for iteration j+1 we have
! 123: * M(j+1) <= G(j) / | A(j+1,j+1) |
! 124: * G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
! 125: * <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
! 126: *
! 127: * where CNORM(j+1) is greater than or equal to the infinity-norm of
! 128: * column j+1 of A, not counting the diagonal. Hence
! 129: *
! 130: * G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
! 131: * 1<=i<=j
! 132: * and
! 133: *
! 134: * |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
! 135: * 1<=i< j
! 136: *
! 137: * Since |x(j)| <= M(j), we use the Level 2 BLAS routine DTRSV if the
! 138: * reciprocal of the largest M(j), j=1,..,n, is larger than
! 139: * max(underflow, 1/overflow).
! 140: *
! 141: * The bound on x(j) is also used to determine when a step in the
! 142: * columnwise method can be performed without fear of overflow. If
! 143: * the computed bound is greater than a large constant, x is scaled to
! 144: * prevent overflow, but if the bound overflows, x is set to 0, x(j) to
! 145: * 1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
! 146: *
! 147: * Similarly, a row-wise scheme is used to solve A'*x = b. The basic
! 148: * algorithm for A upper triangular is
! 149: *
! 150: * for j = 1, ..., n
! 151: * x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
! 152: * end
! 153: *
! 154: * We simultaneously compute two bounds
! 155: * G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
! 156: * M(j) = bound on x(i), 1<=i<=j
! 157: *
! 158: * The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
! 159: * add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
! 160: * Then the bound on x(j) is
! 161: *
! 162: * M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
! 163: *
! 164: * <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
! 165: * 1<=i<=j
! 166: *
! 167: * and we can safely call DTRSV if 1/M(n) and 1/G(n) are both greater
! 168: * than max(underflow, 1/overflow).
! 169: *
! 170: * =====================================================================
! 171: *
! 172: * .. Parameters ..
! 173: DOUBLE PRECISION ZERO, HALF, ONE
! 174: PARAMETER ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0 )
! 175: * ..
! 176: * .. Local Scalars ..
! 177: LOGICAL NOTRAN, NOUNIT, UPPER
! 178: INTEGER I, IMAX, J, JFIRST, JINC, JLAST
! 179: DOUBLE PRECISION BIGNUM, GROW, REC, SMLNUM, SUMJ, TJJ, TJJS,
! 180: $ TMAX, TSCAL, USCAL, XBND, XJ, XMAX
! 181: * ..
! 182: * .. External Functions ..
! 183: LOGICAL LSAME
! 184: INTEGER IDAMAX
! 185: DOUBLE PRECISION DASUM, DDOT, DLAMCH
! 186: EXTERNAL LSAME, IDAMAX, DASUM, DDOT, DLAMCH
! 187: * ..
! 188: * .. External Subroutines ..
! 189: EXTERNAL DAXPY, DSCAL, DTRSV, XERBLA
! 190: * ..
! 191: * .. Intrinsic Functions ..
! 192: INTRINSIC ABS, MAX, MIN
! 193: * ..
! 194: * .. Executable Statements ..
! 195: *
! 196: INFO = 0
! 197: UPPER = LSAME( UPLO, 'U' )
! 198: NOTRAN = LSAME( TRANS, 'N' )
! 199: NOUNIT = LSAME( DIAG, 'N' )
! 200: *
! 201: * Test the input parameters.
! 202: *
! 203: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
! 204: INFO = -1
! 205: ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
! 206: $ LSAME( TRANS, 'C' ) ) THEN
! 207: INFO = -2
! 208: ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
! 209: INFO = -3
! 210: ELSE IF( .NOT.LSAME( NORMIN, 'Y' ) .AND. .NOT.
! 211: $ LSAME( NORMIN, 'N' ) ) THEN
! 212: INFO = -4
! 213: ELSE IF( N.LT.0 ) THEN
! 214: INFO = -5
! 215: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
! 216: INFO = -7
! 217: END IF
! 218: IF( INFO.NE.0 ) THEN
! 219: CALL XERBLA( 'DLATRS', -INFO )
! 220: RETURN
! 221: END IF
! 222: *
! 223: * Quick return if possible
! 224: *
! 225: IF( N.EQ.0 )
! 226: $ RETURN
! 227: *
! 228: * Determine machine dependent parameters to control overflow.
! 229: *
! 230: SMLNUM = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' )
! 231: BIGNUM = ONE / SMLNUM
! 232: SCALE = ONE
! 233: *
! 234: IF( LSAME( NORMIN, 'N' ) ) THEN
! 235: *
! 236: * Compute the 1-norm of each column, not including the diagonal.
! 237: *
! 238: IF( UPPER ) THEN
! 239: *
! 240: * A is upper triangular.
! 241: *
! 242: DO 10 J = 1, N
! 243: CNORM( J ) = DASUM( J-1, A( 1, J ), 1 )
! 244: 10 CONTINUE
! 245: ELSE
! 246: *
! 247: * A is lower triangular.
! 248: *
! 249: DO 20 J = 1, N - 1
! 250: CNORM( J ) = DASUM( N-J, A( J+1, J ), 1 )
! 251: 20 CONTINUE
! 252: CNORM( N ) = ZERO
! 253: END IF
! 254: END IF
! 255: *
! 256: * Scale the column norms by TSCAL if the maximum element in CNORM is
! 257: * greater than BIGNUM.
! 258: *
! 259: IMAX = IDAMAX( N, CNORM, 1 )
! 260: TMAX = CNORM( IMAX )
! 261: IF( TMAX.LE.BIGNUM ) THEN
! 262: TSCAL = ONE
! 263: ELSE
! 264: TSCAL = ONE / ( SMLNUM*TMAX )
! 265: CALL DSCAL( N, TSCAL, CNORM, 1 )
! 266: END IF
! 267: *
! 268: * Compute a bound on the computed solution vector to see if the
! 269: * Level 2 BLAS routine DTRSV can be used.
! 270: *
! 271: J = IDAMAX( N, X, 1 )
! 272: XMAX = ABS( X( J ) )
! 273: XBND = XMAX
! 274: IF( NOTRAN ) THEN
! 275: *
! 276: * Compute the growth in A * x = b.
! 277: *
! 278: IF( UPPER ) THEN
! 279: JFIRST = N
! 280: JLAST = 1
! 281: JINC = -1
! 282: ELSE
! 283: JFIRST = 1
! 284: JLAST = N
! 285: JINC = 1
! 286: END IF
! 287: *
! 288: IF( TSCAL.NE.ONE ) THEN
! 289: GROW = ZERO
! 290: GO TO 50
! 291: END IF
! 292: *
! 293: IF( NOUNIT ) THEN
! 294: *
! 295: * A is non-unit triangular.
! 296: *
! 297: * Compute GROW = 1/G(j) and XBND = 1/M(j).
! 298: * Initially, G(0) = max{x(i), i=1,...,n}.
! 299: *
! 300: GROW = ONE / MAX( XBND, SMLNUM )
! 301: XBND = GROW
! 302: DO 30 J = JFIRST, JLAST, JINC
! 303: *
! 304: * Exit the loop if the growth factor is too small.
! 305: *
! 306: IF( GROW.LE.SMLNUM )
! 307: $ GO TO 50
! 308: *
! 309: * M(j) = G(j-1) / abs(A(j,j))
! 310: *
! 311: TJJ = ABS( A( J, J ) )
! 312: XBND = MIN( XBND, MIN( ONE, TJJ )*GROW )
! 313: IF( TJJ+CNORM( J ).GE.SMLNUM ) THEN
! 314: *
! 315: * G(j) = G(j-1)*( 1 + CNORM(j) / abs(A(j,j)) )
! 316: *
! 317: GROW = GROW*( TJJ / ( TJJ+CNORM( J ) ) )
! 318: ELSE
! 319: *
! 320: * G(j) could overflow, set GROW to 0.
! 321: *
! 322: GROW = ZERO
! 323: END IF
! 324: 30 CONTINUE
! 325: GROW = XBND
! 326: ELSE
! 327: *
! 328: * A is unit triangular.
! 329: *
! 330: * Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
! 331: *
! 332: GROW = MIN( ONE, ONE / MAX( XBND, SMLNUM ) )
! 333: DO 40 J = JFIRST, JLAST, JINC
! 334: *
! 335: * Exit the loop if the growth factor is too small.
! 336: *
! 337: IF( GROW.LE.SMLNUM )
! 338: $ GO TO 50
! 339: *
! 340: * G(j) = G(j-1)*( 1 + CNORM(j) )
! 341: *
! 342: GROW = GROW*( ONE / ( ONE+CNORM( J ) ) )
! 343: 40 CONTINUE
! 344: END IF
! 345: 50 CONTINUE
! 346: *
! 347: ELSE
! 348: *
! 349: * Compute the growth in A' * x = b.
! 350: *
! 351: IF( UPPER ) THEN
! 352: JFIRST = 1
! 353: JLAST = N
! 354: JINC = 1
! 355: ELSE
! 356: JFIRST = N
! 357: JLAST = 1
! 358: JINC = -1
! 359: END IF
! 360: *
! 361: IF( TSCAL.NE.ONE ) THEN
! 362: GROW = ZERO
! 363: GO TO 80
! 364: END IF
! 365: *
! 366: IF( NOUNIT ) THEN
! 367: *
! 368: * A is non-unit triangular.
! 369: *
! 370: * Compute GROW = 1/G(j) and XBND = 1/M(j).
! 371: * Initially, M(0) = max{x(i), i=1,...,n}.
! 372: *
! 373: GROW = ONE / MAX( XBND, SMLNUM )
! 374: XBND = GROW
! 375: DO 60 J = JFIRST, JLAST, JINC
! 376: *
! 377: * Exit the loop if the growth factor is too small.
! 378: *
! 379: IF( GROW.LE.SMLNUM )
! 380: $ GO TO 80
! 381: *
! 382: * G(j) = max( G(j-1), M(j-1)*( 1 + CNORM(j) ) )
! 383: *
! 384: XJ = ONE + CNORM( J )
! 385: GROW = MIN( GROW, XBND / XJ )
! 386: *
! 387: * M(j) = M(j-1)*( 1 + CNORM(j) ) / abs(A(j,j))
! 388: *
! 389: TJJ = ABS( A( J, J ) )
! 390: IF( XJ.GT.TJJ )
! 391: $ XBND = XBND*( TJJ / XJ )
! 392: 60 CONTINUE
! 393: GROW = MIN( GROW, XBND )
! 394: ELSE
! 395: *
! 396: * A is unit triangular.
! 397: *
! 398: * Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
! 399: *
! 400: GROW = MIN( ONE, ONE / MAX( XBND, SMLNUM ) )
! 401: DO 70 J = JFIRST, JLAST, JINC
! 402: *
! 403: * Exit the loop if the growth factor is too small.
! 404: *
! 405: IF( GROW.LE.SMLNUM )
! 406: $ GO TO 80
! 407: *
! 408: * G(j) = ( 1 + CNORM(j) )*G(j-1)
! 409: *
! 410: XJ = ONE + CNORM( J )
! 411: GROW = GROW / XJ
! 412: 70 CONTINUE
! 413: END IF
! 414: 80 CONTINUE
! 415: END IF
! 416: *
! 417: IF( ( GROW*TSCAL ).GT.SMLNUM ) THEN
! 418: *
! 419: * Use the Level 2 BLAS solve if the reciprocal of the bound on
! 420: * elements of X is not too small.
! 421: *
! 422: CALL DTRSV( UPLO, TRANS, DIAG, N, A, LDA, X, 1 )
! 423: ELSE
! 424: *
! 425: * Use a Level 1 BLAS solve, scaling intermediate results.
! 426: *
! 427: IF( XMAX.GT.BIGNUM ) THEN
! 428: *
! 429: * Scale X so that its components are less than or equal to
! 430: * BIGNUM in absolute value.
! 431: *
! 432: SCALE = BIGNUM / XMAX
! 433: CALL DSCAL( N, SCALE, X, 1 )
! 434: XMAX = BIGNUM
! 435: END IF
! 436: *
! 437: IF( NOTRAN ) THEN
! 438: *
! 439: * Solve A * x = b
! 440: *
! 441: DO 110 J = JFIRST, JLAST, JINC
! 442: *
! 443: * Compute x(j) = b(j) / A(j,j), scaling x if necessary.
! 444: *
! 445: XJ = ABS( X( J ) )
! 446: IF( NOUNIT ) THEN
! 447: TJJS = A( J, J )*TSCAL
! 448: ELSE
! 449: TJJS = TSCAL
! 450: IF( TSCAL.EQ.ONE )
! 451: $ GO TO 100
! 452: END IF
! 453: TJJ = ABS( TJJS )
! 454: IF( TJJ.GT.SMLNUM ) THEN
! 455: *
! 456: * abs(A(j,j)) > SMLNUM:
! 457: *
! 458: IF( TJJ.LT.ONE ) THEN
! 459: IF( XJ.GT.TJJ*BIGNUM ) THEN
! 460: *
! 461: * Scale x by 1/b(j).
! 462: *
! 463: REC = ONE / XJ
! 464: CALL DSCAL( N, REC, X, 1 )
! 465: SCALE = SCALE*REC
! 466: XMAX = XMAX*REC
! 467: END IF
! 468: END IF
! 469: X( J ) = X( J ) / TJJS
! 470: XJ = ABS( X( J ) )
! 471: ELSE IF( TJJ.GT.ZERO ) THEN
! 472: *
! 473: * 0 < abs(A(j,j)) <= SMLNUM:
! 474: *
! 475: IF( XJ.GT.TJJ*BIGNUM ) THEN
! 476: *
! 477: * Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM
! 478: * to avoid overflow when dividing by A(j,j).
! 479: *
! 480: REC = ( TJJ*BIGNUM ) / XJ
! 481: IF( CNORM( J ).GT.ONE ) THEN
! 482: *
! 483: * Scale by 1/CNORM(j) to avoid overflow when
! 484: * multiplying x(j) times column j.
! 485: *
! 486: REC = REC / CNORM( J )
! 487: END IF
! 488: CALL DSCAL( N, REC, X, 1 )
! 489: SCALE = SCALE*REC
! 490: XMAX = XMAX*REC
! 491: END IF
! 492: X( J ) = X( J ) / TJJS
! 493: XJ = ABS( X( J ) )
! 494: ELSE
! 495: *
! 496: * A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
! 497: * scale = 0, and compute a solution to A*x = 0.
! 498: *
! 499: DO 90 I = 1, N
! 500: X( I ) = ZERO
! 501: 90 CONTINUE
! 502: X( J ) = ONE
! 503: XJ = ONE
! 504: SCALE = ZERO
! 505: XMAX = ZERO
! 506: END IF
! 507: 100 CONTINUE
! 508: *
! 509: * Scale x if necessary to avoid overflow when adding a
! 510: * multiple of column j of A.
! 511: *
! 512: IF( XJ.GT.ONE ) THEN
! 513: REC = ONE / XJ
! 514: IF( CNORM( J ).GT.( BIGNUM-XMAX )*REC ) THEN
! 515: *
! 516: * Scale x by 1/(2*abs(x(j))).
! 517: *
! 518: REC = REC*HALF
! 519: CALL DSCAL( N, REC, X, 1 )
! 520: SCALE = SCALE*REC
! 521: END IF
! 522: ELSE IF( XJ*CNORM( J ).GT.( BIGNUM-XMAX ) ) THEN
! 523: *
! 524: * Scale x by 1/2.
! 525: *
! 526: CALL DSCAL( N, HALF, X, 1 )
! 527: SCALE = SCALE*HALF
! 528: END IF
! 529: *
! 530: IF( UPPER ) THEN
! 531: IF( J.GT.1 ) THEN
! 532: *
! 533: * Compute the update
! 534: * x(1:j-1) := x(1:j-1) - x(j) * A(1:j-1,j)
! 535: *
! 536: CALL DAXPY( J-1, -X( J )*TSCAL, A( 1, J ), 1, X,
! 537: $ 1 )
! 538: I = IDAMAX( J-1, X, 1 )
! 539: XMAX = ABS( X( I ) )
! 540: END IF
! 541: ELSE
! 542: IF( J.LT.N ) THEN
! 543: *
! 544: * Compute the update
! 545: * x(j+1:n) := x(j+1:n) - x(j) * A(j+1:n,j)
! 546: *
! 547: CALL DAXPY( N-J, -X( J )*TSCAL, A( J+1, J ), 1,
! 548: $ X( J+1 ), 1 )
! 549: I = J + IDAMAX( N-J, X( J+1 ), 1 )
! 550: XMAX = ABS( X( I ) )
! 551: END IF
! 552: END IF
! 553: 110 CONTINUE
! 554: *
! 555: ELSE
! 556: *
! 557: * Solve A' * x = b
! 558: *
! 559: DO 160 J = JFIRST, JLAST, JINC
! 560: *
! 561: * Compute x(j) = b(j) - sum A(k,j)*x(k).
! 562: * k<>j
! 563: *
! 564: XJ = ABS( X( J ) )
! 565: USCAL = TSCAL
! 566: REC = ONE / MAX( XMAX, ONE )
! 567: IF( CNORM( J ).GT.( BIGNUM-XJ )*REC ) THEN
! 568: *
! 569: * If x(j) could overflow, scale x by 1/(2*XMAX).
! 570: *
! 571: REC = REC*HALF
! 572: IF( NOUNIT ) THEN
! 573: TJJS = A( J, J )*TSCAL
! 574: ELSE
! 575: TJJS = TSCAL
! 576: END IF
! 577: TJJ = ABS( TJJS )
! 578: IF( TJJ.GT.ONE ) THEN
! 579: *
! 580: * Divide by A(j,j) when scaling x if A(j,j) > 1.
! 581: *
! 582: REC = MIN( ONE, REC*TJJ )
! 583: USCAL = USCAL / TJJS
! 584: END IF
! 585: IF( REC.LT.ONE ) THEN
! 586: CALL DSCAL( N, REC, X, 1 )
! 587: SCALE = SCALE*REC
! 588: XMAX = XMAX*REC
! 589: END IF
! 590: END IF
! 591: *
! 592: SUMJ = ZERO
! 593: IF( USCAL.EQ.ONE ) THEN
! 594: *
! 595: * If the scaling needed for A in the dot product is 1,
! 596: * call DDOT to perform the dot product.
! 597: *
! 598: IF( UPPER ) THEN
! 599: SUMJ = DDOT( J-1, A( 1, J ), 1, X, 1 )
! 600: ELSE IF( J.LT.N ) THEN
! 601: SUMJ = DDOT( N-J, A( J+1, J ), 1, X( J+1 ), 1 )
! 602: END IF
! 603: ELSE
! 604: *
! 605: * Otherwise, use in-line code for the dot product.
! 606: *
! 607: IF( UPPER ) THEN
! 608: DO 120 I = 1, J - 1
! 609: SUMJ = SUMJ + ( A( I, J )*USCAL )*X( I )
! 610: 120 CONTINUE
! 611: ELSE IF( J.LT.N ) THEN
! 612: DO 130 I = J + 1, N
! 613: SUMJ = SUMJ + ( A( I, J )*USCAL )*X( I )
! 614: 130 CONTINUE
! 615: END IF
! 616: END IF
! 617: *
! 618: IF( USCAL.EQ.TSCAL ) THEN
! 619: *
! 620: * Compute x(j) := ( x(j) - sumj ) / A(j,j) if 1/A(j,j)
! 621: * was not used to scale the dotproduct.
! 622: *
! 623: X( J ) = X( J ) - SUMJ
! 624: XJ = ABS( X( J ) )
! 625: IF( NOUNIT ) THEN
! 626: TJJS = A( J, J )*TSCAL
! 627: ELSE
! 628: TJJS = TSCAL
! 629: IF( TSCAL.EQ.ONE )
! 630: $ GO TO 150
! 631: END IF
! 632: *
! 633: * Compute x(j) = x(j) / A(j,j), scaling if necessary.
! 634: *
! 635: TJJ = ABS( TJJS )
! 636: IF( TJJ.GT.SMLNUM ) THEN
! 637: *
! 638: * abs(A(j,j)) > SMLNUM:
! 639: *
! 640: IF( TJJ.LT.ONE ) THEN
! 641: IF( XJ.GT.TJJ*BIGNUM ) THEN
! 642: *
! 643: * Scale X by 1/abs(x(j)).
! 644: *
! 645: REC = ONE / XJ
! 646: CALL DSCAL( N, REC, X, 1 )
! 647: SCALE = SCALE*REC
! 648: XMAX = XMAX*REC
! 649: END IF
! 650: END IF
! 651: X( J ) = X( J ) / TJJS
! 652: ELSE IF( TJJ.GT.ZERO ) THEN
! 653: *
! 654: * 0 < abs(A(j,j)) <= SMLNUM:
! 655: *
! 656: IF( XJ.GT.TJJ*BIGNUM ) THEN
! 657: *
! 658: * Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM.
! 659: *
! 660: REC = ( TJJ*BIGNUM ) / XJ
! 661: CALL DSCAL( N, REC, X, 1 )
! 662: SCALE = SCALE*REC
! 663: XMAX = XMAX*REC
! 664: END IF
! 665: X( J ) = X( J ) / TJJS
! 666: ELSE
! 667: *
! 668: * A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
! 669: * scale = 0, and compute a solution to A'*x = 0.
! 670: *
! 671: DO 140 I = 1, N
! 672: X( I ) = ZERO
! 673: 140 CONTINUE
! 674: X( J ) = ONE
! 675: SCALE = ZERO
! 676: XMAX = ZERO
! 677: END IF
! 678: 150 CONTINUE
! 679: ELSE
! 680: *
! 681: * Compute x(j) := x(j) / A(j,j) - sumj if the dot
! 682: * product has already been divided by 1/A(j,j).
! 683: *
! 684: X( J ) = X( J ) / TJJS - SUMJ
! 685: END IF
! 686: XMAX = MAX( XMAX, ABS( X( J ) ) )
! 687: 160 CONTINUE
! 688: END IF
! 689: SCALE = SCALE / TSCAL
! 690: END IF
! 691: *
! 692: * Scale the column norms by 1/TSCAL for return.
! 693: *
! 694: IF( TSCAL.NE.ONE ) THEN
! 695: CALL DSCAL( N, ONE / TSCAL, CNORM, 1 )
! 696: END IF
! 697: *
! 698: RETURN
! 699: *
! 700: * End of DLATRS
! 701: *
! 702: END
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