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dlatrs.f

      SUBROUTINE DLATRS( UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE,
     $                   CNORM, INFO )
*
*  -- LAPACK auxiliary routine (version 2.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1992
*
*     .. Scalar Arguments ..
      CHARACTER          DIAG, NORMIN, TRANS, UPLO
      INTEGER            INFO, LDA, N
      DOUBLE PRECISION   SCALE
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), CNORM( * ), X( * )
*     ..
*
*  Purpose
*  =======
*
*  DLATRS solves one of the triangular systems
*
*     A *x = s*b  or  A


'*x = s*b**  with scaling to prevent overflow.  Here A is an upper or lower*  triangular matrix, 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 DTRSV 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.
*
*  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
*          The triangular matrix A.  If UPLO = 'U', the leading n by n
*          upper triangular part of the array A contains the upper
*          triangular matrix, and the strictly lower triangular part of
*          A is not referenced.  If UPLO = 'L', the leading n by n lower
*          triangular part of the array A contains the lower triangular
*          matrix, and the strictly upper triangular part of A is not
*          referenced.  If DIAG = 'U', the diagonal elements of A are
*          also not referenced and are assumed to be 1.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max (1,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, DTRSV*  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 DTRSV 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 DTRSV 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, J, JFIRST, JINC, JLAST
      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, DTRSV, 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
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -7
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DLATRS', -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.
*
            DO 10 J = 1, N
               CNORM( J ) = DASUM( J-1, A( 1, J ), 1 )
   10       CONTINUE
         ELSE
*
*           A is lower triangular.
*
            DO 20 J = 1, N - 1
               CNORM( J ) = DASUM( N-J, A( J+1, 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 DTRSV 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
            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( A( J, J ) )
               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
   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            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( A( J, J ) )               IF( XJ.GT.TJJ )     $            XBND = XBND*( TJJ / XJ )   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 DTRSV( UPLO, TRANS, DIAG, N, A, LDA, 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*            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 = A( J, J )*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, A( 1, J ), 1, X,     $                           1 )                     I = IDAMAX( J-1, X, 1 )                     XMAX = ABS( X( I ) )                  END IF               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, A( J+1, J ), 1,     $                           X( J+1 ), 1 )                     I = J + IDAMAX( N-J, X( J+1 ), 1 )                     XMAX = ABS( X( I ) )                  END IF               END IF  110       CONTINUE*         ELSE**           Solve A' * x = b
*
            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 = A( J, J )*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, A( 1, J ), 1, X, 1 )
                  ELSE IF( J.LT.N ) THEN
                     SUMJ = DDOT( N-J, A( J+1, J ), 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 + ( A( I, J )*USCAL )*X( I )
  120                CONTINUE
                  ELSE IF( J.LT.N ) THEN
                     DO 130 I = J + 1, N
                        SUMJ = SUMJ + ( A( I, J )*USCAL )*X( 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
                     TJJS = A( J, J )*TSCAL
                  ELSE
                     TJJS = TSCAL
                     IF( TSCAL.EQ.ONE )
     $                  GO TO 150
                  END IF
*
*                    Compute x(j) = x(j) / A(j,j), scaling if necessary.
*
                  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


































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