LAPACK 3.3.1 Linear Algebra PACKage

# slatrs.f

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```00001       SUBROUTINE SLATRS( UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE,
00002      \$                   CNORM, INFO )
00003 *
00004 *  -- LAPACK auxiliary routine (version 3.2) --
00005 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00006 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00007 *     November 2006
00008 *
00009 *     .. Scalar Arguments ..
00010       CHARACTER          DIAG, NORMIN, TRANS, UPLO
00011       INTEGER            INFO, LDA, N
00012       REAL               SCALE
00013 *     ..
00014 *     .. Array Arguments ..
00015       REAL               A( LDA, * ), CNORM( * ), X( * )
00016 *     ..
00017 *
00018 *  Purpose
00019 *  =======
00020 *
00021 *  SLATRS solves one of the triangular systems
00022 *
00023 *     A *x = s*b  or  A**T*x = s*b
00024 *
00025 *  with scaling to prevent overflow.  Here A is an upper or lower
00026 *  triangular matrix, A**T denotes the transpose of A, x and b are
00027 *  n-element vectors, and s is a scaling factor, usually less than
00028 *  or equal to 1, chosen so that the components of x will be less than
00029 *  the overflow threshold.  If the unscaled problem will not cause
00030 *  overflow, the Level 2 BLAS routine STRSV is called.  If the matrix A
00031 *  is singular (A(j,j) = 0 for some j), then s is set to 0 and a
00032 *  non-trivial solution to A*x = 0 is returned.
00033 *
00034 *  Arguments
00035 *  =========
00036 *
00037 *  UPLO    (input) CHARACTER*1
00038 *          Specifies whether the matrix A is upper or lower triangular.
00039 *          = 'U':  Upper triangular
00040 *          = 'L':  Lower triangular
00041 *
00042 *  TRANS   (input) CHARACTER*1
00043 *          Specifies the operation applied to A.
00044 *          = 'N':  Solve A * x = s*b  (No transpose)
00045 *          = 'T':  Solve A**T* x = s*b  (Transpose)
00046 *          = 'C':  Solve A**T* x = s*b  (Conjugate transpose = Transpose)
00047 *
00048 *  DIAG    (input) CHARACTER*1
00049 *          Specifies whether or not the matrix A is unit triangular.
00050 *          = 'N':  Non-unit triangular
00051 *          = 'U':  Unit triangular
00052 *
00053 *  NORMIN  (input) CHARACTER*1
00054 *          Specifies whether CNORM has been set or not.
00055 *          = 'Y':  CNORM contains the column norms on entry
00056 *          = 'N':  CNORM is not set on entry.  On exit, the norms will
00057 *                  be computed and stored in CNORM.
00058 *
00059 *  N       (input) INTEGER
00060 *          The order of the matrix A.  N >= 0.
00061 *
00062 *  A       (input) REAL array, dimension (LDA,N)
00063 *          The triangular matrix A.  If UPLO = 'U', the leading n by n
00064 *          upper triangular part of the array A contains the upper
00065 *          triangular matrix, and the strictly lower triangular part of
00066 *          A is not referenced.  If UPLO = 'L', the leading n by n lower
00067 *          triangular part of the array A contains the lower triangular
00068 *          matrix, and the strictly upper triangular part of A is not
00069 *          referenced.  If DIAG = 'U', the diagonal elements of A are
00070 *          also not referenced and are assumed to be 1.
00071 *
00072 *  LDA     (input) INTEGER
00073 *          The leading dimension of the array A.  LDA >= max (1,N).
00074 *
00075 *  X       (input/output) REAL array, dimension (N)
00076 *          On entry, the right hand side b of the triangular system.
00077 *          On exit, X is overwritten by the solution vector x.
00078 *
00079 *  SCALE   (output) REAL
00080 *          The scaling factor s for the triangular system
00081 *             A * x = s*b  or  A**T* x = s*b.
00082 *          If SCALE = 0, the matrix A is singular or badly scaled, and
00083 *          the vector x is an exact or approximate solution to A*x = 0.
00084 *
00085 *  CNORM   (input or output) REAL array, dimension (N)
00086 *
00087 *          If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
00088 *          contains the norm of the off-diagonal part of the j-th column
00089 *          of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
00090 *          to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
00091 *          must be greater than or equal to the 1-norm.
00092 *
00093 *          If NORMIN = 'N', CNORM is an output argument and CNORM(j)
00094 *          returns the 1-norm of the offdiagonal part of the j-th column
00095 *          of A.
00096 *
00097 *  INFO    (output) INTEGER
00098 *          = 0:  successful exit
00099 *          < 0:  if INFO = -k, the k-th argument had an illegal value
00100 *
00101 *  Further Details
00102 *  ======= =======
00103 *
00104 *  A rough bound on x is computed; if that is less than overflow, STRSV
00105 *  is called, otherwise, specific code is used which checks for possible
00106 *  overflow or divide-by-zero at every operation.
00107 *
00108 *  A columnwise scheme is used for solving A*x = b.  The basic algorithm
00109 *  if A is lower triangular is
00110 *
00111 *       x[1:n] := b[1:n]
00112 *       for j = 1, ..., n
00113 *            x(j) := x(j) / A(j,j)
00114 *            x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
00115 *       end
00116 *
00117 *  Define bounds on the components of x after j iterations of the loop:
00118 *     M(j) = bound on x[1:j]
00119 *     G(j) = bound on x[j+1:n]
00120 *  Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
00121 *
00122 *  Then for iteration j+1 we have
00123 *     M(j+1) <= G(j) / | A(j+1,j+1) |
00124 *     G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
00125 *            <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
00126 *
00127 *  where CNORM(j+1) is greater than or equal to the infinity-norm of
00128 *  column j+1 of A, not counting the diagonal.  Hence
00129 *
00130 *     G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
00131 *                  1<=i<=j
00132 *  and
00133 *
00134 *     |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
00135 *                                   1<=i< j
00136 *
00137 *  Since |x(j)| <= M(j), we use the Level 2 BLAS routine STRSV if the
00138 *  reciprocal of the largest M(j), j=1,..,n, is larger than
00139 *  max(underflow, 1/overflow).
00140 *
00141 *  The bound on x(j) is also used to determine when a step in the
00142 *  columnwise method can be performed without fear of overflow.  If
00143 *  the computed bound is greater than a large constant, x is scaled to
00144 *  prevent overflow, but if the bound overflows, x is set to 0, x(j) to
00145 *  1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
00146 *
00147 *  Similarly, a row-wise scheme is used to solve A**T*x = b.  The basic
00148 *  algorithm for A upper triangular is
00149 *
00150 *       for j = 1, ..., n
00151 *            x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j)
00152 *       end
00153 *
00154 *  We simultaneously compute two bounds
00155 *       G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j
00156 *       M(j) = bound on x(i), 1<=i<=j
00157 *
00158 *  The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
00159 *  add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
00160 *  Then the bound on x(j) is
00161 *
00162 *       M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
00163 *
00164 *            <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
00165 *                      1<=i<=j
00166 *
00167 *  and we can safely call STRSV if 1/M(n) and 1/G(n) are both greater
00168 *  than max(underflow, 1/overflow).
00169 *
00170 *  =====================================================================
00171 *
00172 *     .. Parameters ..
00173       REAL               ZERO, HALF, ONE
00174       PARAMETER          ( ZERO = 0.0E+0, HALF = 0.5E+0, ONE = 1.0E+0 )
00175 *     ..
00176 *     .. Local Scalars ..
00177       LOGICAL            NOTRAN, NOUNIT, UPPER
00178       INTEGER            I, IMAX, J, JFIRST, JINC, JLAST
00179       REAL               BIGNUM, GROW, REC, SMLNUM, SUMJ, TJJ, TJJS,
00180      \$                   TMAX, TSCAL, USCAL, XBND, XJ, XMAX
00181 *     ..
00182 *     .. External Functions ..
00183       LOGICAL            LSAME
00184       INTEGER            ISAMAX
00185       REAL               SASUM, SDOT, SLAMCH
00186       EXTERNAL           LSAME, ISAMAX, SASUM, SDOT, SLAMCH
00187 *     ..
00188 *     .. External Subroutines ..
00189       EXTERNAL           SAXPY, SSCAL, STRSV, XERBLA
00190 *     ..
00191 *     .. Intrinsic Functions ..
00192       INTRINSIC          ABS, MAX, MIN
00193 *     ..
00194 *     .. Executable Statements ..
00195 *
00196       INFO = 0
00197       UPPER = LSAME( UPLO, 'U' )
00198       NOTRAN = LSAME( TRANS, 'N' )
00199       NOUNIT = LSAME( DIAG, 'N' )
00200 *
00201 *     Test the input parameters.
00202 *
00203       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
00204          INFO = -1
00205       ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
00206      \$         LSAME( TRANS, 'C' ) ) THEN
00207          INFO = -2
00208       ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
00209          INFO = -3
00210       ELSE IF( .NOT.LSAME( NORMIN, 'Y' ) .AND. .NOT.
00211      \$         LSAME( NORMIN, 'N' ) ) THEN
00212          INFO = -4
00213       ELSE IF( N.LT.0 ) THEN
00214          INFO = -5
00215       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
00216          INFO = -7
00217       END IF
00218       IF( INFO.NE.0 ) THEN
00219          CALL XERBLA( 'SLATRS', -INFO )
00220          RETURN
00221       END IF
00222 *
00223 *     Quick return if possible
00224 *
00225       IF( N.EQ.0 )
00226      \$   RETURN
00227 *
00228 *     Determine machine dependent parameters to control overflow.
00229 *
00230       SMLNUM = SLAMCH( 'Safe minimum' ) / SLAMCH( 'Precision' )
00231       BIGNUM = ONE / SMLNUM
00232       SCALE = ONE
00233 *
00234       IF( LSAME( NORMIN, 'N' ) ) THEN
00235 *
00236 *        Compute the 1-norm of each column, not including the diagonal.
00237 *
00238          IF( UPPER ) THEN
00239 *
00240 *           A is upper triangular.
00241 *
00242             DO 10 J = 1, N
00243                CNORM( J ) = SASUM( J-1, A( 1, J ), 1 )
00244    10       CONTINUE
00245          ELSE
00246 *
00247 *           A is lower triangular.
00248 *
00249             DO 20 J = 1, N - 1
00250                CNORM( J ) = SASUM( N-J, A( J+1, J ), 1 )
00251    20       CONTINUE
00252             CNORM( N ) = ZERO
00253          END IF
00254       END IF
00255 *
00256 *     Scale the column norms by TSCAL if the maximum element in CNORM is
00257 *     greater than BIGNUM.
00258 *
00259       IMAX = ISAMAX( N, CNORM, 1 )
00260       TMAX = CNORM( IMAX )
00261       IF( TMAX.LE.BIGNUM ) THEN
00262          TSCAL = ONE
00263       ELSE
00264          TSCAL = ONE / ( SMLNUM*TMAX )
00265          CALL SSCAL( N, TSCAL, CNORM, 1 )
00266       END IF
00267 *
00268 *     Compute a bound on the computed solution vector to see if the
00269 *     Level 2 BLAS routine STRSV can be used.
00270 *
00271       J = ISAMAX( N, X, 1 )
00272       XMAX = ABS( X( J ) )
00273       XBND = XMAX
00274       IF( NOTRAN ) THEN
00275 *
00276 *        Compute the growth in A * x = b.
00277 *
00278          IF( UPPER ) THEN
00279             JFIRST = N
00280             JLAST = 1
00281             JINC = -1
00282          ELSE
00283             JFIRST = 1
00284             JLAST = N
00285             JINC = 1
00286          END IF
00287 *
00288          IF( TSCAL.NE.ONE ) THEN
00289             GROW = ZERO
00290             GO TO 50
00291          END IF
00292 *
00293          IF( NOUNIT ) THEN
00294 *
00295 *           A is non-unit triangular.
00296 *
00297 *           Compute GROW = 1/G(j) and XBND = 1/M(j).
00298 *           Initially, G(0) = max{x(i), i=1,...,n}.
00299 *
00300             GROW = ONE / MAX( XBND, SMLNUM )
00301             XBND = GROW
00302             DO 30 J = JFIRST, JLAST, JINC
00303 *
00304 *              Exit the loop if the growth factor is too small.
00305 *
00306                IF( GROW.LE.SMLNUM )
00307      \$            GO TO 50
00308 *
00309 *              M(j) = G(j-1) / abs(A(j,j))
00310 *
00311                TJJ = ABS( A( J, J ) )
00312                XBND = MIN( XBND, MIN( ONE, TJJ )*GROW )
00313                IF( TJJ+CNORM( J ).GE.SMLNUM ) THEN
00314 *
00315 *                 G(j) = G(j-1)*( 1 + CNORM(j) / abs(A(j,j)) )
00316 *
00317                   GROW = GROW*( TJJ / ( TJJ+CNORM( J ) ) )
00318                ELSE
00319 *
00320 *                 G(j) could overflow, set GROW to 0.
00321 *
00322                   GROW = ZERO
00323                END IF
00324    30       CONTINUE
00325             GROW = XBND
00326          ELSE
00327 *
00328 *           A is unit triangular.
00329 *
00330 *           Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
00331 *
00332             GROW = MIN( ONE, ONE / MAX( XBND, SMLNUM ) )
00333             DO 40 J = JFIRST, JLAST, JINC
00334 *
00335 *              Exit the loop if the growth factor is too small.
00336 *
00337                IF( GROW.LE.SMLNUM )
00338      \$            GO TO 50
00339 *
00340 *              G(j) = G(j-1)*( 1 + CNORM(j) )
00341 *
00342                GROW = GROW*( ONE / ( ONE+CNORM( J ) ) )
00343    40       CONTINUE
00344          END IF
00345    50    CONTINUE
00346 *
00347       ELSE
00348 *
00349 *        Compute the growth in A**T * x = b.
00350 *
00351          IF( UPPER ) THEN
00352             JFIRST = 1
00353             JLAST = N
00354             JINC = 1
00355          ELSE
00356             JFIRST = N
00357             JLAST = 1
00358             JINC = -1
00359          END IF
00360 *
00361          IF( TSCAL.NE.ONE ) THEN
00362             GROW = ZERO
00363             GO TO 80
00364          END IF
00365 *
00366          IF( NOUNIT ) THEN
00367 *
00368 *           A is non-unit triangular.
00369 *
00370 *           Compute GROW = 1/G(j) and XBND = 1/M(j).
00371 *           Initially, M(0) = max{x(i), i=1,...,n}.
00372 *
00373             GROW = ONE / MAX( XBND, SMLNUM )
00374             XBND = GROW
00375             DO 60 J = JFIRST, JLAST, JINC
00376 *
00377 *              Exit the loop if the growth factor is too small.
00378 *
00379                IF( GROW.LE.SMLNUM )
00380      \$            GO TO 80
00381 *
00382 *              G(j) = max( G(j-1), M(j-1)*( 1 + CNORM(j) ) )
00383 *
00384                XJ = ONE + CNORM( J )
00385                GROW = MIN( GROW, XBND / XJ )
00386 *
00387 *              M(j) = M(j-1)*( 1 + CNORM(j) ) / abs(A(j,j))
00388 *
00389                TJJ = ABS( A( J, J ) )
00390                IF( XJ.GT.TJJ )
00391      \$            XBND = XBND*( TJJ / XJ )
00392    60       CONTINUE
00393             GROW = MIN( GROW, XBND )
00394          ELSE
00395 *
00396 *           A is unit triangular.
00397 *
00398 *           Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
00399 *
00400             GROW = MIN( ONE, ONE / MAX( XBND, SMLNUM ) )
00401             DO 70 J = JFIRST, JLAST, JINC
00402 *
00403 *              Exit the loop if the growth factor is too small.
00404 *
00405                IF( GROW.LE.SMLNUM )
00406      \$            GO TO 80
00407 *
00408 *              G(j) = ( 1 + CNORM(j) )*G(j-1)
00409 *
00410                XJ = ONE + CNORM( J )
00411                GROW = GROW / XJ
00412    70       CONTINUE
00413          END IF
00414    80    CONTINUE
00415       END IF
00416 *
00417       IF( ( GROW*TSCAL ).GT.SMLNUM ) THEN
00418 *
00419 *        Use the Level 2 BLAS solve if the reciprocal of the bound on
00420 *        elements of X is not too small.
00421 *
00422          CALL STRSV( UPLO, TRANS, DIAG, N, A, LDA, X, 1 )
00423       ELSE
00424 *
00425 *        Use a Level 1 BLAS solve, scaling intermediate results.
00426 *
00427          IF( XMAX.GT.BIGNUM ) THEN
00428 *
00429 *           Scale X so that its components are less than or equal to
00430 *           BIGNUM in absolute value.
00431 *
00432             SCALE = BIGNUM / XMAX
00433             CALL SSCAL( N, SCALE, X, 1 )
00434             XMAX = BIGNUM
00435          END IF
00436 *
00437          IF( NOTRAN ) THEN
00438 *
00439 *           Solve A * x = b
00440 *
00441             DO 100 J = JFIRST, JLAST, JINC
00442 *
00443 *              Compute x(j) = b(j) / A(j,j), scaling x if necessary.
00444 *
00445                XJ = ABS( X( J ) )
00446                IF( NOUNIT ) THEN
00447                   TJJS = A( J, J )*TSCAL
00448                ELSE
00449                   TJJS = TSCAL
00450                   IF( TSCAL.EQ.ONE )
00451      \$               GO TO 95
00452                END IF
00453                   TJJ = ABS( TJJS )
00454                   IF( TJJ.GT.SMLNUM ) THEN
00455 *
00456 *                    abs(A(j,j)) > SMLNUM:
00457 *
00458                      IF( TJJ.LT.ONE ) THEN
00459                         IF( XJ.GT.TJJ*BIGNUM ) THEN
00460 *
00461 *                          Scale x by 1/b(j).
00462 *
00463                            REC = ONE / XJ
00464                            CALL SSCAL( N, REC, X, 1 )
00465                            SCALE = SCALE*REC
00466                            XMAX = XMAX*REC
00467                         END IF
00468                      END IF
00469                      X( J ) = X( J ) / TJJS
00470                      XJ = ABS( X( J ) )
00471                   ELSE IF( TJJ.GT.ZERO ) THEN
00472 *
00473 *                    0 < abs(A(j,j)) <= SMLNUM:
00474 *
00475                      IF( XJ.GT.TJJ*BIGNUM ) THEN
00476 *
00477 *                       Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM
00478 *                       to avoid overflow when dividing by A(j,j).
00479 *
00480                         REC = ( TJJ*BIGNUM ) / XJ
00481                         IF( CNORM( J ).GT.ONE ) THEN
00482 *
00483 *                          Scale by 1/CNORM(j) to avoid overflow when
00484 *                          multiplying x(j) times column j.
00485 *
00486                            REC = REC / CNORM( J )
00487                         END IF
00488                         CALL SSCAL( N, REC, X, 1 )
00489                         SCALE = SCALE*REC
00490                         XMAX = XMAX*REC
00491                      END IF
00492                      X( J ) = X( J ) / TJJS
00493                      XJ = ABS( X( J ) )
00494                   ELSE
00495 *
00496 *                    A(j,j) = 0:  Set x(1:n) = 0, x(j) = 1, and
00497 *                    scale = 0, and compute a solution to A*x = 0.
00498 *
00499                      DO 90 I = 1, N
00500                         X( I ) = ZERO
00501    90                CONTINUE
00502                      X( J ) = ONE
00503                      XJ = ONE
00504                      SCALE = ZERO
00505                      XMAX = ZERO
00506                   END IF
00507    95          CONTINUE
00508 *
00509 *              Scale x if necessary to avoid overflow when adding a
00510 *              multiple of column j of A.
00511 *
00512                IF( XJ.GT.ONE ) THEN
00513                   REC = ONE / XJ
00514                   IF( CNORM( J ).GT.( BIGNUM-XMAX )*REC ) THEN
00515 *
00516 *                    Scale x by 1/(2*abs(x(j))).
00517 *
00518                      REC = REC*HALF
00519                      CALL SSCAL( N, REC, X, 1 )
00520                      SCALE = SCALE*REC
00521                   END IF
00522                ELSE IF( XJ*CNORM( J ).GT.( BIGNUM-XMAX ) ) THEN
00523 *
00524 *                 Scale x by 1/2.
00525 *
00526                   CALL SSCAL( N, HALF, X, 1 )
00527                   SCALE = SCALE*HALF
00528                END IF
00529 *
00530                IF( UPPER ) THEN
00531                   IF( J.GT.1 ) THEN
00532 *
00533 *                    Compute the update
00534 *                       x(1:j-1) := x(1:j-1) - x(j) * A(1:j-1,j)
00535 *
00536                      CALL SAXPY( J-1, -X( J )*TSCAL, A( 1, J ), 1, X,
00537      \$                           1 )
00538                      I = ISAMAX( J-1, X, 1 )
00539                      XMAX = ABS( X( I ) )
00540                   END IF
00541                ELSE
00542                   IF( J.LT.N ) THEN
00543 *
00544 *                    Compute the update
00545 *                       x(j+1:n) := x(j+1:n) - x(j) * A(j+1:n,j)
00546 *
00547                      CALL SAXPY( N-J, -X( J )*TSCAL, A( J+1, J ), 1,
00548      \$                           X( J+1 ), 1 )
00549                      I = J + ISAMAX( N-J, X( J+1 ), 1 )
00550                      XMAX = ABS( X( I ) )
00551                   END IF
00552                END IF
00553   100       CONTINUE
00554 *
00555          ELSE
00556 *
00557 *           Solve A**T * x = b
00558 *
00559             DO 140 J = JFIRST, JLAST, JINC
00560 *
00561 *              Compute x(j) = b(j) - sum A(k,j)*x(k).
00562 *                                    k<>j
00563 *
00564                XJ = ABS( X( J ) )
00565                USCAL = TSCAL
00566                REC = ONE / MAX( XMAX, ONE )
00567                IF( CNORM( J ).GT.( BIGNUM-XJ )*REC ) THEN
00568 *
00569 *                 If x(j) could overflow, scale x by 1/(2*XMAX).
00570 *
00571                   REC = REC*HALF
00572                   IF( NOUNIT ) THEN
00573                      TJJS = A( J, J )*TSCAL
00574                   ELSE
00575                      TJJS = TSCAL
00576                   END IF
00577                      TJJ = ABS( TJJS )
00578                      IF( TJJ.GT.ONE ) THEN
00579 *
00580 *                       Divide by A(j,j) when scaling x if A(j,j) > 1.
00581 *
00582                         REC = MIN( ONE, REC*TJJ )
00583                         USCAL = USCAL / TJJS
00584                      END IF
00585                   IF( REC.LT.ONE ) THEN
00586                      CALL SSCAL( N, REC, X, 1 )
00587                      SCALE = SCALE*REC
00588                      XMAX = XMAX*REC
00589                   END IF
00590                END IF
00591 *
00592                SUMJ = ZERO
00593                IF( USCAL.EQ.ONE ) THEN
00594 *
00595 *                 If the scaling needed for A in the dot product is 1,
00596 *                 call SDOT to perform the dot product.
00597 *
00598                   IF( UPPER ) THEN
00599                      SUMJ = SDOT( J-1, A( 1, J ), 1, X, 1 )
00600                   ELSE IF( J.LT.N ) THEN
00601                      SUMJ = SDOT( N-J, A( J+1, J ), 1, X( J+1 ), 1 )
00602                   END IF
00603                ELSE
00604 *
00605 *                 Otherwise, use in-line code for the dot product.
00606 *
00607                   IF( UPPER ) THEN
00608                      DO 110 I = 1, J - 1
00609                         SUMJ = SUMJ + ( A( I, J )*USCAL )*X( I )
00610   110                CONTINUE
00611                   ELSE IF( J.LT.N ) THEN
00612                      DO 120 I = J + 1, N
00613                         SUMJ = SUMJ + ( A( I, J )*USCAL )*X( I )
00614   120                CONTINUE
00615                   END IF
00616                END IF
00617 *
00618                IF( USCAL.EQ.TSCAL ) THEN
00619 *
00620 *                 Compute x(j) := ( x(j) - sumj ) / A(j,j) if 1/A(j,j)
00621 *                 was not used to scale the dotproduct.
00622 *
00623                   X( J ) = X( J ) - SUMJ
00624                   XJ = ABS( X( J ) )
00625                   IF( NOUNIT ) THEN
00626                      TJJS = A( J, J )*TSCAL
00627                   ELSE
00628                      TJJS = TSCAL
00629                      IF( TSCAL.EQ.ONE )
00630      \$                  GO TO 135
00631                   END IF
00632 *
00633 *                    Compute x(j) = x(j) / A(j,j), scaling if necessary.
00634 *
00635                      TJJ = ABS( TJJS )
00636                      IF( TJJ.GT.SMLNUM ) THEN
00637 *
00638 *                       abs(A(j,j)) > SMLNUM:
00639 *
00640                         IF( TJJ.LT.ONE ) THEN
00641                            IF( XJ.GT.TJJ*BIGNUM ) THEN
00642 *
00643 *                             Scale X by 1/abs(x(j)).
00644 *
00645                               REC = ONE / XJ
00646                               CALL SSCAL( N, REC, X, 1 )
00647                               SCALE = SCALE*REC
00648                               XMAX = XMAX*REC
00649                            END IF
00650                         END IF
00651                         X( J ) = X( J ) / TJJS
00652                      ELSE IF( TJJ.GT.ZERO ) THEN
00653 *
00654 *                       0 < abs(A(j,j)) <= SMLNUM:
00655 *
00656                         IF( XJ.GT.TJJ*BIGNUM ) THEN
00657 *
00658 *                          Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM.
00659 *
00660                            REC = ( TJJ*BIGNUM ) / XJ
00661                            CALL SSCAL( N, REC, X, 1 )
00662                            SCALE = SCALE*REC
00663                            XMAX = XMAX*REC
00664                         END IF
00665                         X( J ) = X( J ) / TJJS
00666                      ELSE
00667 *
00668 *                       A(j,j) = 0:  Set x(1:n) = 0, x(j) = 1, and
00669 *                       scale = 0, and compute a solution to A**T*x = 0.
00670 *
00671                         DO 130 I = 1, N
00672                            X( I ) = ZERO
00673   130                   CONTINUE
00674                         X( J ) = ONE
00675                         SCALE = ZERO
00676                         XMAX = ZERO
00677                      END IF
00678   135             CONTINUE
00679                ELSE
00680 *
00681 *                 Compute x(j) := x(j) / A(j,j)  - sumj if the dot
00682 *                 product has already been divided by 1/A(j,j).
00683 *
00684                   X( J ) = X( J ) / TJJS - SUMJ
00685                END IF
00686                XMAX = MAX( XMAX, ABS( X( J ) ) )
00687   140       CONTINUE
00688          END IF
00689          SCALE = SCALE / TSCAL
00690       END IF
00691 *
00692 *     Scale the column norms by 1/TSCAL for return.
00693 *
00694       IF( TSCAL.NE.ONE ) THEN
00695          CALL SSCAL( N, ONE / TSCAL, CNORM, 1 )
00696       END IF
00697 *
00698       RETURN
00699 *
00700 *     End of SLATRS
00701 *
00702       END
```