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, A:code:`T` denotes the transpose of A , A:code:`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 ?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 . The order of the matrix A . n≥ 0
- 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) .