Annotation of rpl/lapack/lapack/dlatps.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DLATPS( UPLO, TRANS, DIAG, NORMIN, N, AP, 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, N
! 12: DOUBLE PRECISION SCALE
! 13: * ..
! 14: * .. Array Arguments ..
! 15: DOUBLE PRECISION AP( * ), CNORM( * ), X( * )
! 16: * ..
! 17: *
! 18: * Purpose
! 19: * =======
! 20: *
! 21: * DLATPS solves one of the triangular systems
! 22: *
! 23: * A *x = s*b or A'*x = s*b
! 24: *
! 25: * with scaling to prevent overflow, where A is an upper or lower
! 26: * triangular matrix stored in packed form. Here A' denotes the
! 27: * transpose of A, x and b are n-element vectors, and s is a scaling
! 28: * factor, usually less than or equal to 1, chosen so that the
! 29: * components of x will be less than the overflow threshold. If the
! 30: * unscaled problem will not cause overflow, the Level 2 BLAS routine
! 31: * DTPSV is called. If the matrix A is singular (A(j,j) = 0 for some j),
! 32: * then s is set to 0 and a 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: * AP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
! 63: * The upper or lower triangular matrix A, packed columnwise in
! 64: * a linear array. The j-th column of A is stored in the array
! 65: * AP as follows:
! 66: * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
! 67: * if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
! 68: *
! 69: * X (input/output) DOUBLE PRECISION array, dimension (N)
! 70: * On entry, the right hand side b of the triangular system.
! 71: * On exit, X is overwritten by the solution vector x.
! 72: *
! 73: * SCALE (output) DOUBLE PRECISION
! 74: * The scaling factor s for the triangular system
! 75: * A * x = s*b or A'* x = s*b.
! 76: * If SCALE = 0, the matrix A is singular or badly scaled, and
! 77: * the vector x is an exact or approximate solution to A*x = 0.
! 78: *
! 79: * CNORM (input or output) DOUBLE PRECISION array, dimension (N)
! 80: *
! 81: * If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
! 82: * contains the norm of the off-diagonal part of the j-th column
! 83: * of A. If TRANS = 'N', CNORM(j) must be greater than or equal
! 84: * to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
! 85: * must be greater than or equal to the 1-norm.
! 86: *
! 87: * If NORMIN = 'N', CNORM is an output argument and CNORM(j)
! 88: * returns the 1-norm of the offdiagonal part of the j-th column
! 89: * of A.
! 90: *
! 91: * INFO (output) INTEGER
! 92: * = 0: successful exit
! 93: * < 0: if INFO = -k, the k-th argument had an illegal value
! 94: *
! 95: * Further Details
! 96: * ======= =======
! 97: *
! 98: * A rough bound on x is computed; if that is less than overflow, DTPSV
! 99: * is called, otherwise, specific code is used which checks for possible
! 100: * overflow or divide-by-zero at every operation.
! 101: *
! 102: * A columnwise scheme is used for solving A*x = b. The basic algorithm
! 103: * if A is lower triangular is
! 104: *
! 105: * x[1:n] := b[1:n]
! 106: * for j = 1, ..., n
! 107: * x(j) := x(j) / A(j,j)
! 108: * x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
! 109: * end
! 110: *
! 111: * Define bounds on the components of x after j iterations of the loop:
! 112: * M(j) = bound on x[1:j]
! 113: * G(j) = bound on x[j+1:n]
! 114: * Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
! 115: *
! 116: * Then for iteration j+1 we have
! 117: * M(j+1) <= G(j) / | A(j+1,j+1) |
! 118: * G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
! 119: * <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
! 120: *
! 121: * where CNORM(j+1) is greater than or equal to the infinity-norm of
! 122: * column j+1 of A, not counting the diagonal. Hence
! 123: *
! 124: * G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
! 125: * 1<=i<=j
! 126: * and
! 127: *
! 128: * |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
! 129: * 1<=i< j
! 130: *
! 131: * Since |x(j)| <= M(j), we use the Level 2 BLAS routine DTPSV if the
! 132: * reciprocal of the largest M(j), j=1,..,n, is larger than
! 133: * max(underflow, 1/overflow).
! 134: *
! 135: * The bound on x(j) is also used to determine when a step in the
! 136: * columnwise method can be performed without fear of overflow. If
! 137: * the computed bound is greater than a large constant, x is scaled to
! 138: * prevent overflow, but if the bound overflows, x is set to 0, x(j) to
! 139: * 1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
! 140: *
! 141: * Similarly, a row-wise scheme is used to solve A'*x = b. The basic
! 142: * algorithm for A upper triangular is
! 143: *
! 144: * for j = 1, ..., n
! 145: * x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
! 146: * end
! 147: *
! 148: * We simultaneously compute two bounds
! 149: * G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
! 150: * M(j) = bound on x(i), 1<=i<=j
! 151: *
! 152: * The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
! 153: * add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
! 154: * Then the bound on x(j) is
! 155: *
! 156: * M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
! 157: *
! 158: * <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
! 159: * 1<=i<=j
! 160: *
! 161: * and we can safely call DTPSV if 1/M(n) and 1/G(n) are both greater
! 162: * than max(underflow, 1/overflow).
! 163: *
! 164: * =====================================================================
! 165: *
! 166: * .. Parameters ..
! 167: DOUBLE PRECISION ZERO, HALF, ONE
! 168: PARAMETER ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0 )
! 169: * ..
! 170: * .. Local Scalars ..
! 171: LOGICAL NOTRAN, NOUNIT, UPPER
! 172: INTEGER I, IMAX, IP, J, JFIRST, JINC, JLAST, JLEN
! 173: DOUBLE PRECISION BIGNUM, GROW, REC, SMLNUM, SUMJ, TJJ, TJJS,
! 174: $ TMAX, TSCAL, USCAL, XBND, XJ, XMAX
! 175: * ..
! 176: * .. External Functions ..
! 177: LOGICAL LSAME
! 178: INTEGER IDAMAX
! 179: DOUBLE PRECISION DASUM, DDOT, DLAMCH
! 180: EXTERNAL LSAME, IDAMAX, DASUM, DDOT, DLAMCH
! 181: * ..
! 182: * .. External Subroutines ..
! 183: EXTERNAL DAXPY, DSCAL, DTPSV, XERBLA
! 184: * ..
! 185: * .. Intrinsic Functions ..
! 186: INTRINSIC ABS, MAX, MIN
! 187: * ..
! 188: * .. Executable Statements ..
! 189: *
! 190: INFO = 0
! 191: UPPER = LSAME( UPLO, 'U' )
! 192: NOTRAN = LSAME( TRANS, 'N' )
! 193: NOUNIT = LSAME( DIAG, 'N' )
! 194: *
! 195: * Test the input parameters.
! 196: *
! 197: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
! 198: INFO = -1
! 199: ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
! 200: $ LSAME( TRANS, 'C' ) ) THEN
! 201: INFO = -2
! 202: ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
! 203: INFO = -3
! 204: ELSE IF( .NOT.LSAME( NORMIN, 'Y' ) .AND. .NOT.
! 205: $ LSAME( NORMIN, 'N' ) ) THEN
! 206: INFO = -4
! 207: ELSE IF( N.LT.0 ) THEN
! 208: INFO = -5
! 209: END IF
! 210: IF( INFO.NE.0 ) THEN
! 211: CALL XERBLA( 'DLATPS', -INFO )
! 212: RETURN
! 213: END IF
! 214: *
! 215: * Quick return if possible
! 216: *
! 217: IF( N.EQ.0 )
! 218: $ RETURN
! 219: *
! 220: * Determine machine dependent parameters to control overflow.
! 221: *
! 222: SMLNUM = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' )
! 223: BIGNUM = ONE / SMLNUM
! 224: SCALE = ONE
! 225: *
! 226: IF( LSAME( NORMIN, 'N' ) ) THEN
! 227: *
! 228: * Compute the 1-norm of each column, not including the diagonal.
! 229: *
! 230: IF( UPPER ) THEN
! 231: *
! 232: * A is upper triangular.
! 233: *
! 234: IP = 1
! 235: DO 10 J = 1, N
! 236: CNORM( J ) = DASUM( J-1, AP( IP ), 1 )
! 237: IP = IP + J
! 238: 10 CONTINUE
! 239: ELSE
! 240: *
! 241: * A is lower triangular.
! 242: *
! 243: IP = 1
! 244: DO 20 J = 1, N - 1
! 245: CNORM( J ) = DASUM( N-J, AP( IP+1 ), 1 )
! 246: IP = IP + N - J + 1
! 247: 20 CONTINUE
! 248: CNORM( N ) = ZERO
! 249: END IF
! 250: END IF
! 251: *
! 252: * Scale the column norms by TSCAL if the maximum element in CNORM is
! 253: * greater than BIGNUM.
! 254: *
! 255: IMAX = IDAMAX( N, CNORM, 1 )
! 256: TMAX = CNORM( IMAX )
! 257: IF( TMAX.LE.BIGNUM ) THEN
! 258: TSCAL = ONE
! 259: ELSE
! 260: TSCAL = ONE / ( SMLNUM*TMAX )
! 261: CALL DSCAL( N, TSCAL, CNORM, 1 )
! 262: END IF
! 263: *
! 264: * Compute a bound on the computed solution vector to see if the
! 265: * Level 2 BLAS routine DTPSV can be used.
! 266: *
! 267: J = IDAMAX( N, X, 1 )
! 268: XMAX = ABS( X( J ) )
! 269: XBND = XMAX
! 270: IF( NOTRAN ) THEN
! 271: *
! 272: * Compute the growth in A * x = b.
! 273: *
! 274: IF( UPPER ) THEN
! 275: JFIRST = N
! 276: JLAST = 1
! 277: JINC = -1
! 278: ELSE
! 279: JFIRST = 1
! 280: JLAST = N
! 281: JINC = 1
! 282: END IF
! 283: *
! 284: IF( TSCAL.NE.ONE ) THEN
! 285: GROW = ZERO
! 286: GO TO 50
! 287: END IF
! 288: *
! 289: IF( NOUNIT ) THEN
! 290: *
! 291: * A is non-unit triangular.
! 292: *
! 293: * Compute GROW = 1/G(j) and XBND = 1/M(j).
! 294: * Initially, G(0) = max{x(i), i=1,...,n}.
! 295: *
! 296: GROW = ONE / MAX( XBND, SMLNUM )
! 297: XBND = GROW
! 298: IP = JFIRST*( JFIRST+1 ) / 2
! 299: JLEN = N
! 300: DO 30 J = JFIRST, JLAST, JINC
! 301: *
! 302: * Exit the loop if the growth factor is too small.
! 303: *
! 304: IF( GROW.LE.SMLNUM )
! 305: $ GO TO 50
! 306: *
! 307: * M(j) = G(j-1) / abs(A(j,j))
! 308: *
! 309: TJJ = ABS( AP( IP ) )
! 310: XBND = MIN( XBND, MIN( ONE, TJJ )*GROW )
! 311: IF( TJJ+CNORM( J ).GE.SMLNUM ) THEN
! 312: *
! 313: * G(j) = G(j-1)*( 1 + CNORM(j) / abs(A(j,j)) )
! 314: *
! 315: GROW = GROW*( TJJ / ( TJJ+CNORM( J ) ) )
! 316: ELSE
! 317: *
! 318: * G(j) could overflow, set GROW to 0.
! 319: *
! 320: GROW = ZERO
! 321: END IF
! 322: IP = IP + JINC*JLEN
! 323: JLEN = JLEN - 1
! 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: IP = JFIRST*( JFIRST+1 ) / 2
! 376: JLEN = 1
! 377: DO 60 J = JFIRST, JLAST, JINC
! 378: *
! 379: * Exit the loop if the growth factor is too small.
! 380: *
! 381: IF( GROW.LE.SMLNUM )
! 382: $ GO TO 80
! 383: *
! 384: * G(j) = max( G(j-1), M(j-1)*( 1 + CNORM(j) ) )
! 385: *
! 386: XJ = ONE + CNORM( J )
! 387: GROW = MIN( GROW, XBND / XJ )
! 388: *
! 389: * M(j) = M(j-1)*( 1 + CNORM(j) ) / abs(A(j,j))
! 390: *
! 391: TJJ = ABS( AP( IP ) )
! 392: IF( XJ.GT.TJJ )
! 393: $ XBND = XBND*( TJJ / XJ )
! 394: JLEN = JLEN + 1
! 395: IP = IP + JINC*JLEN
! 396: 60 CONTINUE
! 397: GROW = MIN( GROW, XBND )
! 398: ELSE
! 399: *
! 400: * A is unit triangular.
! 401: *
! 402: * Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
! 403: *
! 404: GROW = MIN( ONE, ONE / MAX( XBND, SMLNUM ) )
! 405: DO 70 J = JFIRST, JLAST, JINC
! 406: *
! 407: * Exit the loop if the growth factor is too small.
! 408: *
! 409: IF( GROW.LE.SMLNUM )
! 410: $ GO TO 80
! 411: *
! 412: * G(j) = ( 1 + CNORM(j) )*G(j-1)
! 413: *
! 414: XJ = ONE + CNORM( J )
! 415: GROW = GROW / XJ
! 416: 70 CONTINUE
! 417: END IF
! 418: 80 CONTINUE
! 419: END IF
! 420: *
! 421: IF( ( GROW*TSCAL ).GT.SMLNUM ) THEN
! 422: *
! 423: * Use the Level 2 BLAS solve if the reciprocal of the bound on
! 424: * elements of X is not too small.
! 425: *
! 426: CALL DTPSV( UPLO, TRANS, DIAG, N, AP, X, 1 )
! 427: ELSE
! 428: *
! 429: * Use a Level 1 BLAS solve, scaling intermediate results.
! 430: *
! 431: IF( XMAX.GT.BIGNUM ) THEN
! 432: *
! 433: * Scale X so that its components are less than or equal to
! 434: * BIGNUM in absolute value.
! 435: *
! 436: SCALE = BIGNUM / XMAX
! 437: CALL DSCAL( N, SCALE, X, 1 )
! 438: XMAX = BIGNUM
! 439: END IF
! 440: *
! 441: IF( NOTRAN ) THEN
! 442: *
! 443: * Solve A * x = b
! 444: *
! 445: IP = JFIRST*( JFIRST+1 ) / 2
! 446: DO 110 J = JFIRST, JLAST, JINC
! 447: *
! 448: * Compute x(j) = b(j) / A(j,j), scaling x if necessary.
! 449: *
! 450: XJ = ABS( X( J ) )
! 451: IF( NOUNIT ) THEN
! 452: TJJS = AP( IP )*TSCAL
! 453: ELSE
! 454: TJJS = TSCAL
! 455: IF( TSCAL.EQ.ONE )
! 456: $ GO TO 100
! 457: END IF
! 458: TJJ = ABS( TJJS )
! 459: IF( TJJ.GT.SMLNUM ) THEN
! 460: *
! 461: * abs(A(j,j)) > SMLNUM:
! 462: *
! 463: IF( TJJ.LT.ONE ) THEN
! 464: IF( XJ.GT.TJJ*BIGNUM ) THEN
! 465: *
! 466: * Scale x by 1/b(j).
! 467: *
! 468: REC = ONE / XJ
! 469: CALL DSCAL( N, REC, X, 1 )
! 470: SCALE = SCALE*REC
! 471: XMAX = XMAX*REC
! 472: END IF
! 473: END IF
! 474: X( J ) = X( J ) / TJJS
! 475: XJ = ABS( X( J ) )
! 476: ELSE IF( TJJ.GT.ZERO ) THEN
! 477: *
! 478: * 0 < abs(A(j,j)) <= SMLNUM:
! 479: *
! 480: IF( XJ.GT.TJJ*BIGNUM ) THEN
! 481: *
! 482: * Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM
! 483: * to avoid overflow when dividing by A(j,j).
! 484: *
! 485: REC = ( TJJ*BIGNUM ) / XJ
! 486: IF( CNORM( J ).GT.ONE ) THEN
! 487: *
! 488: * Scale by 1/CNORM(j) to avoid overflow when
! 489: * multiplying x(j) times column j.
! 490: *
! 491: REC = REC / CNORM( J )
! 492: END IF
! 493: CALL DSCAL( N, REC, X, 1 )
! 494: SCALE = SCALE*REC
! 495: XMAX = XMAX*REC
! 496: END IF
! 497: X( J ) = X( J ) / TJJS
! 498: XJ = ABS( X( J ) )
! 499: ELSE
! 500: *
! 501: * A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
! 502: * scale = 0, and compute a solution to A*x = 0.
! 503: *
! 504: DO 90 I = 1, N
! 505: X( I ) = ZERO
! 506: 90 CONTINUE
! 507: X( J ) = ONE
! 508: XJ = ONE
! 509: SCALE = ZERO
! 510: XMAX = ZERO
! 511: END IF
! 512: 100 CONTINUE
! 513: *
! 514: * Scale x if necessary to avoid overflow when adding a
! 515: * multiple of column j of A.
! 516: *
! 517: IF( XJ.GT.ONE ) THEN
! 518: REC = ONE / XJ
! 519: IF( CNORM( J ).GT.( BIGNUM-XMAX )*REC ) THEN
! 520: *
! 521: * Scale x by 1/(2*abs(x(j))).
! 522: *
! 523: REC = REC*HALF
! 524: CALL DSCAL( N, REC, X, 1 )
! 525: SCALE = SCALE*REC
! 526: END IF
! 527: ELSE IF( XJ*CNORM( J ).GT.( BIGNUM-XMAX ) ) THEN
! 528: *
! 529: * Scale x by 1/2.
! 530: *
! 531: CALL DSCAL( N, HALF, X, 1 )
! 532: SCALE = SCALE*HALF
! 533: END IF
! 534: *
! 535: IF( UPPER ) THEN
! 536: IF( J.GT.1 ) THEN
! 537: *
! 538: * Compute the update
! 539: * x(1:j-1) := x(1:j-1) - x(j) * A(1:j-1,j)
! 540: *
! 541: CALL DAXPY( J-1, -X( J )*TSCAL, AP( IP-J+1 ), 1, X,
! 542: $ 1 )
! 543: I = IDAMAX( J-1, X, 1 )
! 544: XMAX = ABS( X( I ) )
! 545: END IF
! 546: IP = IP - J
! 547: ELSE
! 548: IF( J.LT.N ) THEN
! 549: *
! 550: * Compute the update
! 551: * x(j+1:n) := x(j+1:n) - x(j) * A(j+1:n,j)
! 552: *
! 553: CALL DAXPY( N-J, -X( J )*TSCAL, AP( IP+1 ), 1,
! 554: $ X( J+1 ), 1 )
! 555: I = J + IDAMAX( N-J, X( J+1 ), 1 )
! 556: XMAX = ABS( X( I ) )
! 557: END IF
! 558: IP = IP + N - J + 1
! 559: END IF
! 560: 110 CONTINUE
! 561: *
! 562: ELSE
! 563: *
! 564: * Solve A' * x = b
! 565: *
! 566: IP = JFIRST*( JFIRST+1 ) / 2
! 567: JLEN = 1
! 568: DO 160 J = JFIRST, JLAST, JINC
! 569: *
! 570: * Compute x(j) = b(j) - sum A(k,j)*x(k).
! 571: * k<>j
! 572: *
! 573: XJ = ABS( X( J ) )
! 574: USCAL = TSCAL
! 575: REC = ONE / MAX( XMAX, ONE )
! 576: IF( CNORM( J ).GT.( BIGNUM-XJ )*REC ) THEN
! 577: *
! 578: * If x(j) could overflow, scale x by 1/(2*XMAX).
! 579: *
! 580: REC = REC*HALF
! 581: IF( NOUNIT ) THEN
! 582: TJJS = AP( IP )*TSCAL
! 583: ELSE
! 584: TJJS = TSCAL
! 585: END IF
! 586: TJJ = ABS( TJJS )
! 587: IF( TJJ.GT.ONE ) THEN
! 588: *
! 589: * Divide by A(j,j) when scaling x if A(j,j) > 1.
! 590: *
! 591: REC = MIN( ONE, REC*TJJ )
! 592: USCAL = USCAL / TJJS
! 593: END IF
! 594: IF( REC.LT.ONE ) THEN
! 595: CALL DSCAL( N, REC, X, 1 )
! 596: SCALE = SCALE*REC
! 597: XMAX = XMAX*REC
! 598: END IF
! 599: END IF
! 600: *
! 601: SUMJ = ZERO
! 602: IF( USCAL.EQ.ONE ) THEN
! 603: *
! 604: * If the scaling needed for A in the dot product is 1,
! 605: * call DDOT to perform the dot product.
! 606: *
! 607: IF( UPPER ) THEN
! 608: SUMJ = DDOT( J-1, AP( IP-J+1 ), 1, X, 1 )
! 609: ELSE IF( J.LT.N ) THEN
! 610: SUMJ = DDOT( N-J, AP( IP+1 ), 1, X( J+1 ), 1 )
! 611: END IF
! 612: ELSE
! 613: *
! 614: * Otherwise, use in-line code for the dot product.
! 615: *
! 616: IF( UPPER ) THEN
! 617: DO 120 I = 1, J - 1
! 618: SUMJ = SUMJ + ( AP( IP-J+I )*USCAL )*X( I )
! 619: 120 CONTINUE
! 620: ELSE IF( J.LT.N ) THEN
! 621: DO 130 I = 1, N - J
! 622: SUMJ = SUMJ + ( AP( IP+I )*USCAL )*X( J+I )
! 623: 130 CONTINUE
! 624: END IF
! 625: END IF
! 626: *
! 627: IF( USCAL.EQ.TSCAL ) THEN
! 628: *
! 629: * Compute x(j) := ( x(j) - sumj ) / A(j,j) if 1/A(j,j)
! 630: * was not used to scale the dotproduct.
! 631: *
! 632: X( J ) = X( J ) - SUMJ
! 633: XJ = ABS( X( J ) )
! 634: IF( NOUNIT ) THEN
! 635: *
! 636: * Compute x(j) = x(j) / A(j,j), scaling if necessary.
! 637: *
! 638: TJJS = AP( IP )*TSCAL
! 639: ELSE
! 640: TJJS = TSCAL
! 641: IF( TSCAL.EQ.ONE )
! 642: $ GO TO 150
! 643: END IF
! 644: TJJ = ABS( TJJS )
! 645: IF( TJJ.GT.SMLNUM ) THEN
! 646: *
! 647: * abs(A(j,j)) > SMLNUM:
! 648: *
! 649: IF( TJJ.LT.ONE ) THEN
! 650: IF( XJ.GT.TJJ*BIGNUM ) THEN
! 651: *
! 652: * Scale X by 1/abs(x(j)).
! 653: *
! 654: REC = ONE / XJ
! 655: CALL DSCAL( N, REC, X, 1 )
! 656: SCALE = SCALE*REC
! 657: XMAX = XMAX*REC
! 658: END IF
! 659: END IF
! 660: X( J ) = X( J ) / TJJS
! 661: ELSE IF( TJJ.GT.ZERO ) THEN
! 662: *
! 663: * 0 < abs(A(j,j)) <= SMLNUM:
! 664: *
! 665: IF( XJ.GT.TJJ*BIGNUM ) THEN
! 666: *
! 667: * Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM.
! 668: *
! 669: REC = ( TJJ*BIGNUM ) / XJ
! 670: CALL DSCAL( N, REC, X, 1 )
! 671: SCALE = SCALE*REC
! 672: XMAX = XMAX*REC
! 673: END IF
! 674: X( J ) = X( J ) / TJJS
! 675: ELSE
! 676: *
! 677: * A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
! 678: * scale = 0, and compute a solution to A'*x = 0.
! 679: *
! 680: DO 140 I = 1, N
! 681: X( I ) = ZERO
! 682: 140 CONTINUE
! 683: X( J ) = ONE
! 684: SCALE = ZERO
! 685: XMAX = ZERO
! 686: END IF
! 687: 150 CONTINUE
! 688: ELSE
! 689: *
! 690: * Compute x(j) := x(j) / A(j,j) - sumj if the dot
! 691: * product has already been divided by 1/A(j,j).
! 692: *
! 693: X( J ) = X( J ) / TJJS - SUMJ
! 694: END IF
! 695: XMAX = MAX( XMAX, ABS( X( J ) ) )
! 696: JLEN = JLEN + 1
! 697: IP = IP + JINC*JLEN
! 698: 160 CONTINUE
! 699: END IF
! 700: SCALE = SCALE / TSCAL
! 701: END IF
! 702: *
! 703: * Scale the column norms by 1/TSCAL for return.
! 704: *
! 705: IF( TSCAL.NE.ONE ) THEN
! 706: CALL DSCAL( N, ONE / TSCAL, CNORM, 1 )
! 707: END IF
! 708: *
! 709: RETURN
! 710: *
! 711: * End of DLATPS
! 712: *
! 713: END
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