dlatrs.f

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      SUBROUTINE dlatrs( UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE,
     $                   cnorm, info )
*
*  -- LAPACK auxiliary routine (version 3.2) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     .. 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'*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 ) ) )
  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 DLATRS
*
      END