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

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( * )

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.

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

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).