SLATRS(l) — LAPACK auxiliary routine (version 2.0)

NAME

SLATRS - solve one of the triangular systems A ∗x = s∗b or A’∗x = s∗b with scaling to prevent overflow

SYNOPSIS

SUBROUTINE SLATRS(
UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE, CNORM, INFO )

CHARACTER DIAG, NORMIN, TRANS, UPLO

INTEGER INFO, LDA, N

REAL SCALE

REAL A( LDA, ∗ ), CNORM( ∗ ), X( ∗ )

PURPOSE

SLATRS solves one of the triangular systems 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 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) REAL 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) REAL 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) REAL
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) REAL 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, STRSV 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

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 STRSV if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow).