Developer Reference for Intel® oneAPI Math Kernel Library for Fortran
?latrs
Solves a triangular system of equations with the scale factor set to prevent overflow.
Syntax
call slatrs(uplo, trans, diag, normin, n, a, lda, x, scale, cnorm, info)
call dlatrs(uplo, trans, diag, normin, n, a, lda, x, scale, cnorm, info)
call clatrs(uplo, trans, diag, normin, n, a, lda, x, scale, cnorm, info)
call zlatrs(uplo, trans, diag, normin, n, a, lda, x, scale, cnorm, info)Include Files
mkl.fi
Description
The routine solves one of the triangular systems
A*x = s*b , or A^{T}*x = s*b, or A^{H}*x = s*b (for complex flavors)
with scaling to prevent overflow. Here A is an upper or lower triangular matrix, AT denotes the transpose of A , AH 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 ?trsv 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.
Input Parameters
- uplo
-
CHARACTER*1 . Specifies whether the matrix A is upper or lower triangular. = ‘U’ : Upper triangular = ‘L’ : Lower triangular
- trans
-
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
-
CHARACTER*1 . Specifies whether or not the matrix A is unit triangular. = ‘N’ : non–unit triangular = ‘U’ : unit triangular
- normin
-
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. O n exit, the norms will be computed and stored in cnorm .
- n
-
INTEGER. Specifies the order of the matrix A. The value of n must be at least zero.
- a
-
REAL for slatrs DOUBLE PRECISION for dlatrs COMPLEX for clatrs DOUBLE COMPLEX for zlatrs . Array, DIMENSION ( lda , n ). Contains 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
-
INTEGER . The leading dimension of the array a . lda≥ max(1, n) .
- x
-
REAL for slatrs DOUBLE PRECISION for dlatrs COMPLEX for clatrs DOUBLE COMPLEX for zlatrs . Array, DIMENSION ( n ). On entry, the right hand side b of the triangular system.
- cnorm
-
REAL for slatrs / clatrs DOUBLE PRECISION for dlatrs / zlatrs . 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.
Output Parameters
- x
-
On exit, x is overwritten by the solution vector x .
- scale
-
REAL for slatrs / clatrs DOUBLE PRECISION for dlatrs / zlatrs . Array, DIMENSION ( lda , n ). The scaling factor s for the triangular system as described above. 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
-
If normin = 'N' , cnorm is an output argument and cnorm(j) returns the 1-norm of the off-diagonal part of the j -th column of A .
info
INTEGER .
= 0: successful exit
< 0: if info = -k , the k -th argument had an illegal value
Application Notes
A rough bound on x is computed; if that is less than overflow, ?trsv 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 \(Ax = b\) . 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
and
Since |x(j)| ≤ M(j) , we use the Level 2 BLAS routine ?trsv 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 \(Ax = 0\) 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)|
and we can safely call ?trsv if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow) .