clatrs-f(3)

clatrs.f –

Synopsis

Functions/Subroutines

subroutine clatrs (UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE, CNORM, INFO)
CLATRS solves a triangular system of equations with the scale factor set to prevent overflow.

Function/Subroutine Documentation

subroutine clatrs (characterUPLO, characterTRANS, characterDIAG, characterNORMIN, integerN, complex, dimension( lda, * )A, integerLDA, complex, dimension( * )X, realSCALE, real, dimension( * )CNORM, integerINFO)

CLATRS solves a triangular system of equations with the scale factor set to prevent overflow.

Purpose:

 CLATRS solves one of the triangular systems
 A * x = s*b, A**T * x = s*b, or A**H * x = s*b,
 with scaling to prevent overflow. Here A is an upper or lower
 triangular matrix, A**T denotes the transpose of A, A**H denotes the
 conjugate 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
 CTRSV 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.

Parameters:

UPLO

 UPLO is CHARACTER*1
 Specifies whether the matrix A is upper or lower triangular.
 = 'U': Upper triangular
 = 'L': Lower triangular

TRANS

 TRANS is CHARACTER*1
 Specifies the operation applied to A.
 = 'N': Solve A * x = s*b (No transpose)
 = 'T': Solve A**T * x = s*b (Transpose)
 = 'C': Solve A**H * x = s*b (Conjugate transpose)

DIAG

 DIAG is CHARACTER*1
 Specifies whether or not the matrix A is unit triangular.
 = 'N': Non-unit triangular
 = 'U': Unit triangular

NORMIN

 NORMIN is 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 is INTEGER
 The order of the matrix A. N >= 0.

 A is COMPLEX 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

 LDA is INTEGER
 The leading dimension of the array A. LDA >= max (1,N).

 X is COMPLEX 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

 SCALE is REAL
 The scaling factor s for the triangular system
 A * x = s*b, A**T * x = s*b, or A**H * 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

 CNORM is 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

 INFO is INTEGER
 = 0: successful exit
 < 0: if INFO = -k, the k-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

September 2012

Further Details:

 A rough bound on x is computed; if that is less than overflow, CTRSV
 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 CTRSV 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**T *x = b or
 A**H *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 CTRSV if 1/M(n) and 1/G(n) are both greater
 than max(underflow, 1/overflow).

Definition at line 239 of file clatrs.f.

Author

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