The LA_TRIMPROVE function improves the solution to a system of linear equations with a tridiagonal array, AX = B, and provides optional error bounds and backward error estimates.
The LA_TRIMPROVE function may also be used to improve the solutions for multiple systems of linear equations, with each column of B representing a different set of equations. In this case, the result is a k-by-n array where each of the k columns represents the improved solution vector for that set of equations.
LA_TRIMPROVE is based on the following LAPACK routines:
Given the tridiagonal system of equations:
-4t + u = 6
2t - 4u + v = -8
2u - 4v + w = -5
2v -4w = 8
the solution can be found and improved by using the following program:
aupper = [1, 1, 1]
adiag = [-4, -4, -4, -4]
alower = [2, 2, 2]
b = [6, -8, -5, 8]
dlower = alower
darray = adiag
dupper = aupper
LA_TRIDC, dlower, darray, dupper, u2, index
x = LA_TRISOL(dlower, darray, dupper, u2, index, b)
xnew = LA_TRIMPROVE(Alower, Adiag, Aupper, $
dlower, darray, dupper, u2, index, b, x)
PRINT, 'LA_TRISOL improved solution:'
LA_TRISOL improved solution:
-1.00000 2.00000 2.00000 -1.00000
Result = LA_TRIMPROVE( AL, A, AU, DAL, DA, DAU, DU2, Index, B, X [, BACKWARD_ERROR=variable] [, /DOUBLE] [, FORWARD_ERROR=variable] )
The result is an n-element vector or k-by-n array.
A vector of length (n - 1) containing the subdiagonal elements of the original array.
A vector of length n containing the main diagonal elements of the original array.
A vector of length (n - 1) containing the superdiagonal elements of the original array.
The (n - 1) elements of the lower bidiagonal array, created by the LA_TRIDC procedure.
The n diagonal elements of the upper triangular array, created by the LA_TRIDC procedure.
The (n - 1) superdiagonal elements of the upper triangular array, created by the LA_TRIDC procedure.
The (n - 2) elements of the second superdiagonal of the upper triangular array, created by the LA_TRIDC procedure.
An input vector, created by the LA_TRIDC procedure, containing the row permutations which occurred as a result of partial pivoting.
An n-element input vector containing the right-hand side of the linear system, or a k-by-n array, where each of the k columns represents a different linear system.
An n-element input vector, or a k-by-n array, containing the approximate solutions to the linear system, created by the LA_TRISOL function.
Set this keyword to a named variable that will contain the relative backward error estimate for each linear system. If B is a vector containing a single linear system, then BACKWARD_ERROR will be a scalar. If B is an array containing k linear systems, then BACKWARD_ERROR will be a k-element vector. The backward error is the smallest relative change in any element of A or B that makes X an exact solution.
Set this keyword to use double-precision for computations and to return a double-precision (real or complex) result. Set DOUBLE = 0 to use single-precision for computations and to return a single-precision (real or complex) result. The default is /DOUBLE if AL is double precision, otherwise the default is DOUBLE = 0.
Set this keyword to a named variable that will contain the estimated forward error bound for each linear system. If B is a vector containing a single linear system, then FORWARD_ERROR will be a scalar. If B is an array containing k linear systems, then FORWARD_ERROR will be a k-element vector. For each linear system, if Xtrue is the true solution corresponding to X, then the forward error is an estimated upper bound for the magnitude of the largest element in (X - Xtrue) divided by the magnitude of the largest element in X.
Resources and References
For details see Anderson et al., LAPACK Users' Guide, 3rd ed., SIAM, 1999.