Linear preconditioner matrix

The matrix A is known as the preconditioner and has to be chosen such that the condition number of the transformed linear system is smaller than that of the original system. 

In cases where iterative methods are employed to solve large, sparse linear systems, both the efficiency and robustness of these methods can be significantly improved by use of preconditioners. A preconditioner of a matrix A is a matrix such that has a smaller condition number than A. The... 

This version solves a linear system or the equivalent minimization of a quadratic function using the PCG method where the preconditioner is the band matrix with the band dimension provided by the integer in the argument. 

Solve the previous linear system by using the Solve function and the band matrix with bandwidth 1 as the preconditioner. 

This result is the motivation behind the use of a preconditioner, a transformation of the linear system into a related one whose condition number is closer to 1. This reduces the number of iterations necessary for iterative methods to converge to a solution, and is common practice in the numerical solution of BVPs. Let us choose some nonsingular upper-triangular matrix Mi defining a coordinate transformation,... 

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