DLATPS man page on IRIX

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DLATPS(3F)							    DLATPS(3F)

NAME
     DLATPS - solve 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

SYNOPSIS
     SUBROUTINE DLATPS( UPLO, TRANS, DIAG, NORMIN, N, AP, X, SCALE, CNORM,
			INFO )

	 CHARACTER	DIAG, NORMIN, TRANS, UPLO

	 INTEGER	INFO, N

	 DOUBLE		PRECISION SCALE

	 DOUBLE		PRECISION AP( * ), CNORM( * ), X( * )

PURPOSE
     DLATPS solves one of the triangular systems 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.

									Page 1

DLATPS(3F)							    DLATPS(3F)

     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}.

									Page 2

DLATPS(3F)							    DLATPS(3F)

     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).

									Page 3

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