```      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

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