dlatps.f

          SUBROUTINE DLATPS( UPLO, TRANS, DIAG, NORMIN, N, AP, X, SCALE,
         $                   CNORM, INFO )
    *
    *  -- LAPACK auxiliary routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          CHARACTER          DIAG, NORMIN, TRANS, UPLO
          INTEGER            INFO, N
          DOUBLE PRECISION   SCALE
    *     ..
    *     .. Array Arguments ..
          DOUBLE PRECISION   AP( * ), CNORM( * ), X( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  DLATPS solves one of the triangular systems
    *
    *     A *x = s*b  or  A'*x = s*b
    *
    *  with scaling to prevent overflow, where A is an upper or lower
    *  triangular matrix stored in packed form.  Here A' denotes the
    *  transpose of A, x and b are n-element vectors, and s is a scaling
    *  factor, usually less than or equal to 1, chosen so that the
    *  components of x will be less than the overflow threshold.  If the
    *  unscaled problem will not cause overflow, the Level 2 BLAS routine
    *  DTPSV is called. If the matrix A is singular (A(j,j) = 0 for some j),
    *  then s is set to 0 and a non-trivial solution to A*x = 0 is returned.
    *
    *  Arguments
    *  =========
    *
    *  UPLO    (input) CHARACTER*1
    *          Specifies whether the matrix A is upper or lower triangular.
    *          = 'U':  Upper triangular
    *          = 'L':  Lower triangular
    *
    *  TRANS   (input) CHARACTER*1
    *          Specifies the operation applied to A.
    *          = 'N':  Solve A * x = s*b  (No transpose)
    *          = 'T':  Solve A'* x = s*b  (Transpose)
    *          = 'C':  Solve A'* x = s*b  (Conjugate transpose = Transpose)
    *
    *  DIAG    (input) CHARACTER*1
    *          Specifies whether or not the matrix A is unit triangular.
    *          = 'N':  Non-unit triangular
    *          = 'U':  Unit triangular
    *
    *  NORMIN  (input) CHARACTER*1
    *          Specifies whether CNORM has been set or not.
    *          = 'Y':  CNORM contains the column norms on entry
    *          = 'N':  CNORM is not set on entry.  On exit, the norms will
    *                  be computed and stored in CNORM.
    *
    *  N       (input) INTEGER
    *          The order of the matrix A.  N >= 0.
    *
    *  AP      (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
    *          The upper or lower triangular matrix A, packed columnwise in
    *          a linear array.  The j-th column of A is stored in the array
    *          AP as follows:
    *          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
    *          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
    *
    *  X       (input/output) DOUBLE PRECISION array, dimension (N)
    *          On entry, the right hand side b of the triangular system.
    *          On exit, X is overwritten by the solution vector x.
    *
    *  SCALE   (output) DOUBLE PRECISION
    *          The scaling factor s for the triangular system
    *             A * x = s*b  or  A'* x = s*b.
    *          If SCALE = 0, the matrix A is singular or badly scaled, and
    *          the vector x is an exact or approximate solution to A*x = 0.
    *
    *  CNORM   (input or output) DOUBLE PRECISION array, dimension (N)
    *
    *          If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
    *          contains the norm of the off-diagonal part of the j-th column
    *          of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
    *          to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
    *          must be greater than or equal to the 1-norm.
    *
    *          If NORMIN = 'N', CNORM is an output argument and CNORM(j)
    *          returns the 1-norm of the offdiagonal part of the j-th column
    *          of A.
    *
    *  INFO    (output) INTEGER
    *          = 0:  successful exit
    *          < 0:  if INFO = -k, the k-th argument had an illegal value
    *
    *  Further Details
    *  ======= =======
    *
    *  A rough bound on x is computed; if that is less than overflow, DTPSV
    *  is called, otherwise, specific code is used which checks for possible
    *  overflow or divide-by-zero at every operation.
    *
    *  A columnwise scheme is used for solving A*x = b.  The basic algorithm
    *  if A is lower triangular is
    *
    *       x[1:n] := b[1:n]
    *       for j = 1, ..., n
    *            x(j) := x(j) / A(j,j)
    *            x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
    *       end
    *
    *  Define bounds on the components of x after j iterations of the loop:
    *     M(j) = bound on x[1:j]
    *     G(j) = bound on x[j+1:n]
    *  Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
    *
    *  Then for iteration j+1 we have
    *     M(j+1) <= G(j) / | A(j+1,j+1) |
    *     G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
    *            <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
    *
    *  where CNORM(j+1) is greater than or equal to the infinity-norm of
    *  column j+1 of A, not counting the diagonal.  Hence
    *
    *     G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
    *                  1<=i<=j
    *  and
    *
    *     |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
    *                                   1<=i< j
    *
    *  Since |x(j)| <= M(j), we use the Level 2 BLAS routine DTPSV if the
    *  reciprocal of the largest M(j), j=1,..,n, is larger than
    *  max(underflow, 1/overflow).
    *
    *  The bound on x(j) is also used to determine when a step in the
    *  columnwise method can be performed without fear of overflow.  If
    *  the computed bound is greater than a large constant, x is scaled to
    *  prevent overflow, but if the bound overflows, x is set to 0, x(j) to
    *  1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
    *
    *  Similarly, a row-wise scheme is used to solve A'*x = b.  The basic
    *  algorithm for A upper triangular is
    *
    *       for j = 1, ..., n
    *            x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
    *       end
    *
    *  We simultaneously compute two bounds
    *       G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
    *       M(j) = bound on x(i), 1<=i<=j
    *
    *  The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
    *  add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
    *  Then the bound on x(j) is
    *
    *       M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
    *
    *            <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
    *                      1<=i<=j
    *
    *  and we can safely call DTPSV if 1/M(n) and 1/G(n) are both greater
    *  than max(underflow, 1/overflow).
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          DOUBLE PRECISION   ZERO, HALF, ONE
          PARAMETER          ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0 )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            NOTRAN, NOUNIT, UPPER
          INTEGER            I, IMAX, IP, J, JFIRST, JINC, JLAST, JLEN
          DOUBLE PRECISION   BIGNUM, GROW, REC, SMLNUM, SUMJ, TJJ, TJJS,
         $                   TMAX, TSCAL, USCAL, XBND, XJ, XMAX
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          INTEGER            IDAMAX
          DOUBLE PRECISION   DASUM, DDOT, DLAMCH
          EXTERNAL           LSAME, IDAMAX, DASUM, DDOT, DLAMCH
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           DAXPY, DSCAL, DTPSV, XERBLA
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          ABS, MAX, MIN
    *     ..
    *     .. Executable Statements ..
    *
          INFO = 0
          UPPER = LSAME( UPLO, 'U' )
          NOTRAN = LSAME( TRANS, 'N' )
          NOUNIT = LSAME( DIAG, 'N' )
    *
    *     Test the input parameters.
    *
          IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
             INFO = -1
          ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
         $         LSAME( TRANS, 'C' ) ) THEN
             INFO = -2
          ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
             INFO = -3
          ELSE IF( .NOT.LSAME( NORMIN, 'Y' ) .AND. .NOT.
         $         LSAME( NORMIN, 'N' ) ) THEN
             INFO = -4
          ELSE IF( N.LT.0 ) THEN
             INFO = -5
          END IF
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'DLATPS', -INFO )
             RETURN
          END IF
    *
    *     Quick return if possible
    *
          IF( N.EQ.0 )
         $   RETURN
    *
    *     Determine machine dependent parameters to control overflow.
    *
          SMLNUM = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' )
          BIGNUM = ONE / SMLNUM
          SCALE = ONE
    *
          IF( LSAME( NORMIN, 'N' ) ) THEN
    *
    *        Compute the 1-norm of each column, not including the diagonal.
    *
             IF( UPPER ) THEN
    *
    *           A is upper triangular.
    *
                IP = 1
                DO 10 J = 1, N
                   CNORM( J ) = DASUM( J-1, AP( IP ), 1 )
                   IP = IP + J
       10       CONTINUE
             ELSE
    *
    *           A is lower triangular.
    *
                IP = 1
                DO 20 J = 1, N - 1
                   CNORM( J ) = DASUM( N-J, AP( IP+1 ), 1 )
                   IP = IP + N - J + 1
       20       CONTINUE
                CNORM( N ) = ZERO
             END IF
          END IF
    *
    *     Scale the column norms by TSCAL if the maximum element in CNORM is
    *     greater than BIGNUM.
    *
          IMAX = IDAMAX( N, CNORM, 1 )
          TMAX = CNORM( IMAX )
          IF( TMAX.LE.BIGNUM ) THEN
             TSCAL = ONE
          ELSE
             TSCAL = ONE / ( SMLNUM*TMAX )
             CALL DSCAL( N, TSCAL, CNORM, 1 )
          END IF
    *
    *     Compute a bound on the computed solution vector to see if the
    *     Level 2 BLAS routine DTPSV can be used.
    *
          J = IDAMAX( N, X, 1 )
          XMAX = ABS( X( J ) )
          XBND = XMAX
          IF( NOTRAN ) THEN
    *
    *        Compute the growth in A * x = b.
    *
             IF( UPPER ) THEN
                JFIRST = N
                JLAST = 1
                JINC = -1
             ELSE
                JFIRST = 1
                JLAST = N
                JINC = 1
             END IF
    *
             IF( TSCAL.NE.ONE ) THEN
                GROW = ZERO
                GO TO 50
             END IF
    *
             IF( NOUNIT ) THEN
    *
    *           A is non-unit triangular.
    *
    *           Compute GROW = 1/G(j) and XBND = 1/M(j).
    *           Initially, G(0) = max{x(i), i=1,...,n}.
    *
                GROW = ONE / MAX( XBND, SMLNUM )
                XBND = GROW
                IP = JFIRST*( JFIRST+1 ) / 2
                JLEN = N
                DO 30 J = JFIRST, JLAST, JINC
    *
    *              Exit the loop if the growth factor is too small.
    *
                   IF( GROW.LE.SMLNUM )
         $            GO TO 50
    *
    *              M(j) = G(j-1) / abs(A(j,j))
    *
                   TJJ = ABS( AP( IP ) )
                   XBND = MIN( XBND, MIN( ONE, TJJ )*GROW )
                   IF( TJJ+CNORM( J ).GE.SMLNUM ) THEN
    *
    *                 G(j) = G(j-1)*( 1 + CNORM(j) / abs(A(j,j)) )
    *
                      GROW = GROW*( TJJ / ( TJJ+CNORM( J ) ) )
                   ELSE
    *
    *                 G(j) could overflow, set GROW to 0.
    *
                      GROW = ZERO
                   END IF
                   IP = IP + JINC*JLEN
                   JLEN = JLEN - 1
       30       CONTINUE
                GROW = XBND
             ELSE
    *
    *           A is unit triangular.
    *
    *           Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
    *
                GROW = MIN( ONE, ONE / MAX( XBND, SMLNUM ) )
                DO 40 J = JFIRST, JLAST, JINC
    *
    *              Exit the loop if the growth factor is too small.
    *
                   IF( GROW.LE.SMLNUM )
         $            GO TO 50
    *
    *              G(j) = G(j-1)*( 1 + CNORM(j) )
    *
                   GROW = GROW*( ONE / ( ONE+CNORM( J ) ) )
       40       CONTINUE
             END IF
       50    CONTINUE
    *
          ELSE
    *
    *        Compute the growth in A' * x = b.
    *
             IF( UPPER ) THEN
                JFIRST = 1
                JLAST = N
                JINC = 1
             ELSE
                JFIRST = N
                JLAST = 1
                JINC = -1
             END IF
    *
             IF( TSCAL.NE.ONE ) THEN
                GROW = ZERO
                GO TO 80
             END IF
    *
             IF( NOUNIT ) THEN
    *
    *           A is non-unit triangular.
    *
    *           Compute GROW = 1/G(j) and XBND = 1/M(j).
    *           Initially, M(0) = max{x(i), i=1,...,n}.
    *
                GROW = ONE / MAX( XBND, SMLNUM )
                XBND = GROW
                IP = JFIRST*( JFIRST+1 ) / 2
                JLEN = 1
                DO 60 J = JFIRST, JLAST, JINC
    *
    *              Exit the loop if the growth factor is too small.
    *
                   IF( GROW.LE.SMLNUM )
         $            GO TO 80
    *
    *              G(j) = max( G(j-1), M(j-1)*( 1 + CNORM(j) ) )
    *
                   XJ = ONE + CNORM( J )
                   GROW = MIN( GROW, XBND / XJ )
    *
    *              M(j) = M(j-1)*( 1 + CNORM(j) ) / abs(A(j,j))
    *
                   TJJ = ABS( AP( IP ) )
                   IF( XJ.GT.TJJ )
         $            XBND = XBND*( TJJ / XJ )
                   JLEN = JLEN + 1
                   IP = IP + JINC*JLEN
       60       CONTINUE
                GROW = MIN( GROW, XBND )
             ELSE
    *
    *           A is unit triangular.
    *
    *           Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
    *
                GROW = MIN( ONE, ONE / MAX( XBND, SMLNUM ) )
                DO 70 J = JFIRST, JLAST, JINC
    *
    *              Exit the loop if the growth factor is too small.
    *
                   IF( GROW.LE.SMLNUM )
         $            GO TO 80
    *
    *              G(j) = ( 1 + CNORM(j) )*G(j-1)
    *
                   XJ = ONE + CNORM( J )
                   GROW = GROW / XJ
       70       CONTINUE
             END IF
       80    CONTINUE
          END IF
    *
          IF( ( GROW*TSCAL ).GT.SMLNUM ) THEN
    *
    *        Use the Level 2 BLAS solve if the reciprocal of the bound on
    *        elements of X is not too small.
    *
             CALL DTPSV( UPLO, TRANS, DIAG, N, AP, X, 1 )
          ELSE
    *
    *        Use a Level 1 BLAS solve, scaling intermediate results.
    *
             IF( XMAX.GT.BIGNUM ) THEN
    *
    *           Scale X so that its components are less than or equal to
    *           BIGNUM in absolute value.
    *
                SCALE = BIGNUM / XMAX
                CALL DSCAL( N, SCALE, X, 1 )
                XMAX = BIGNUM
             END IF
    *
             IF( NOTRAN ) THEN
    *
    *           Solve A * x = b
    *
                IP = JFIRST*( JFIRST+1 ) / 2
                DO 110 J = JFIRST, JLAST, JINC
    *
    *              Compute x(j) = b(j) / A(j,j), scaling x if necessary.
    *
                   XJ = ABS( X( J ) )
                   IF( NOUNIT ) THEN
                      TJJS = AP( IP )*TSCAL
                   ELSE
                      TJJS = TSCAL
                      IF( TSCAL.EQ.ONE )
         $               GO TO 100
                   END IF
                   TJJ = ABS( TJJS )
                   IF( TJJ.GT.SMLNUM ) THEN
    *
    *                    abs(A(j,j)) > SMLNUM:
    *
                      IF( TJJ.LT.ONE ) THEN
                         IF( XJ.GT.TJJ*BIGNUM ) THEN
    *
    *                          Scale x by 1/b(j).
    *
                            REC = ONE / XJ
                            CALL DSCAL( N, REC, X, 1 )
                            SCALE = SCALE*REC
                            XMAX = XMAX*REC
                         END IF
                      END IF
                      X( J ) = X( J ) / TJJS
                      XJ = ABS( X( J ) )
                   ELSE IF( TJJ.GT.ZERO ) THEN
    *
    *                    0 < abs(A(j,j)) <= SMLNUM:
    *
                      IF( XJ.GT.TJJ*BIGNUM ) THEN
    *
    *                       Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM
    *                       to avoid overflow when dividing by A(j,j).
    *
                         REC = ( TJJ*BIGNUM ) / XJ
                         IF( CNORM( J ).GT.ONE ) THEN
    *
    *                          Scale by 1/CNORM(j) to avoid overflow when
    *                          multiplying x(j) times column j.
    *
                            REC = REC / CNORM( J )
                         END IF
                         CALL DSCAL( N, REC, X, 1 )
                         SCALE = SCALE*REC
                         XMAX = XMAX*REC
                      END IF
                      X( J ) = X( J ) / TJJS
                      XJ = ABS( X( J ) )
                   ELSE
    *
    *                    A(j,j) = 0:  Set x(1:n) = 0, x(j) = 1, and
    *                    scale = 0, and compute a solution to A*x = 0.
    *
                      DO 90 I = 1, N
                         X( I ) = ZERO
       90             CONTINUE
                      X( J ) = ONE
                      XJ = ONE
                      SCALE = ZERO
                      XMAX = ZERO
                   END IF
      100          CONTINUE
    *
    *              Scale x if necessary to avoid overflow when adding a
    *              multiple of column j of A.
    *
                   IF( XJ.GT.ONE ) THEN
                      REC = ONE / XJ
                      IF( CNORM( J ).GT.( BIGNUM-XMAX )*REC ) THEN
    *
    *                    Scale x by 1/(2*abs(x(j))).
    *
                         REC = REC*HALF
                         CALL DSCAL( N, REC, X, 1 )
                         SCALE = SCALE*REC
                      END IF
                   ELSE IF( XJ*CNORM( J ).GT.( BIGNUM-XMAX ) ) THEN
    *
    *                 Scale x by 1/2.
    *
                      CALL DSCAL( N, HALF, X, 1 )
                      SCALE = SCALE*HALF
                   END IF
    *
                   IF( UPPER ) THEN
                      IF( J.GT.1 ) THEN
    *
    *                    Compute the update
    *                       x(1:j-1) := x(1:j-1) - x(j) * A(1:j-1,j)
    *
                         CALL DAXPY( J-1, -X( J )*TSCAL, AP( IP-J+1 ), 1, X,
         $                           1 )
                         I = IDAMAX( J-1, X, 1 )
                         XMAX = ABS( X( I ) )
                      END IF
                      IP = IP - J
                   ELSE
                      IF( J.LT.N ) THEN
    *
    *                    Compute the update
    *                       x(j+1:n) := x(j+1:n) - x(j) * A(j+1:n,j)
    *
                         CALL DAXPY( N-J, -X( J )*TSCAL, AP( IP+1 ), 1,
         $                           X( J+1 ), 1 )
                         I = J + IDAMAX( N-J, X( J+1 ), 1 )
                         XMAX = ABS( X( I ) )
                      END IF
                      IP = IP + N - J + 1
                   END IF
      110       CONTINUE
    *
             ELSE
    *
    *           Solve A' * x = b
    *
                IP = JFIRST*( JFIRST+1 ) / 2
                JLEN = 1
                DO 160 J = JFIRST, JLAST, JINC
    *
    *              Compute x(j) = b(j) - sum A(k,j)*x(k).
    *                                    k<>j
    *
                   XJ = ABS( X( J ) )
                   USCAL = TSCAL
                   REC = ONE / MAX( XMAX, ONE )
                   IF( CNORM( J ).GT.( BIGNUM-XJ )*REC ) THEN
    *
    *                 If x(j) could overflow, scale x by 1/(2*XMAX).
    *
                      REC = REC*HALF
                      IF( NOUNIT ) THEN
                         TJJS = AP( IP )*TSCAL
                      ELSE
                         TJJS = TSCAL
                      END IF
                      TJJ = ABS( TJJS )
                      IF( TJJ.GT.ONE ) THEN
    *
    *                       Divide by A(j,j) when scaling x if A(j,j) > 1.
    *
                         REC = MIN( ONE, REC*TJJ )
                         USCAL = USCAL / TJJS
                      END IF
                      IF( REC.LT.ONE ) THEN
                         CALL DSCAL( N, REC, X, 1 )
                         SCALE = SCALE*REC
                         XMAX = XMAX*REC
                      END IF
                   END IF
    *
                   SUMJ = ZERO
                   IF( USCAL.EQ.ONE ) THEN
    *
    *                 If the scaling needed for A in the dot product is 1,
    *                 call DDOT to perform the dot product.
    *
                      IF( UPPER ) THEN
                         SUMJ = DDOT( J-1, AP( IP-J+1 ), 1, X, 1 )
                      ELSE IF( J.LT.N ) THEN
                         SUMJ = DDOT( N-J, AP( IP+1 ), 1, X( J+1 ), 1 )
                      END IF
                   ELSE
    *
    *                 Otherwise, use in-line code for the dot product.
    *
                      IF( UPPER ) THEN
                         DO 120 I = 1, J - 1
                            SUMJ = SUMJ + ( AP( IP-J+I )*USCAL )*X( I )
      120                CONTINUE
                      ELSE IF( J.LT.N ) THEN
                         DO 130 I = 1, N - J
                            SUMJ = SUMJ + ( AP( IP+I )*USCAL )*X( J+I )
      130                CONTINUE
                      END IF
                   END IF
    *
                   IF( USCAL.EQ.TSCAL ) THEN
    *
    *                 Compute x(j) := ( x(j) - sumj ) / A(j,j) if 1/A(j,j)
    *                 was not used to scale the dotproduct.
    *
                      X( J ) = X( J ) - SUMJ
                      XJ = ABS( X( J ) )
                      IF( NOUNIT ) THEN
    *
    *                    Compute x(j) = x(j) / A(j,j), scaling if necessary.
    *
                         TJJS = AP( IP )*TSCAL
                      ELSE
                         TJJS = TSCAL
                         IF( TSCAL.EQ.ONE )
         $                  GO TO 150
                      END IF
                      TJJ = ABS( TJJS )
                      IF( TJJ.GT.SMLNUM ) THEN
    *
    *                       abs(A(j,j)) > SMLNUM:
    *
                         IF( TJJ.LT.ONE ) THEN
                            IF( XJ.GT.TJJ*BIGNUM ) THEN
    *
    *                             Scale X by 1/abs(x(j)).
    *
                               REC = ONE / XJ
                               CALL DSCAL( N, REC, X, 1 )
                               SCALE = SCALE*REC
                               XMAX = XMAX*REC
                            END IF
                         END IF
                         X( J ) = X( J ) / TJJS
                      ELSE IF( TJJ.GT.ZERO ) THEN
    *
    *                       0 < abs(A(j,j)) <= SMLNUM:
    *
                         IF( XJ.GT.TJJ*BIGNUM ) THEN
    *
    *                          Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM.
    *
                            REC = ( TJJ*BIGNUM ) / XJ
                            CALL DSCAL( N, REC, X, 1 )
                            SCALE = SCALE*REC
                            XMAX = XMAX*REC
                         END IF
                         X( J ) = X( J ) / TJJS
                      ELSE
    *
    *                       A(j,j) = 0:  Set x(1:n) = 0, x(j) = 1, and
    *                       scale = 0, and compute a solution to A'*x = 0.
    *
                         DO 140 I = 1, N
                            X( I ) = ZERO
      140                CONTINUE
                         X( J ) = ONE
                         SCALE = ZERO
                         XMAX = ZERO
                      END IF
      150             CONTINUE
                   ELSE
    *
    *                 Compute x(j) := x(j) / A(j,j)  - sumj if the dot
    *                 product has already been divided by 1/A(j,j).
    *
                      X( J ) = X( J ) / TJJS - SUMJ
                   END IF
                   XMAX = MAX( XMAX, ABS( X( J ) ) )
                   JLEN = JLEN + 1
                   IP = IP + JINC*JLEN
      160       CONTINUE
             END IF
             SCALE = SCALE / TSCAL
          END IF
    *
    *     Scale the column norms by 1/TSCAL for return.
    *
          IF( TSCAL.NE.ONE ) THEN
             CALL DSCAL( N, ONE / TSCAL, CNORM, 1 )
          END IF
    *
          RETURN
    *
    *     End of DLATPS
    *
          END