Chapter 4: Numerical Solution Of Systems Of Equations Flashcards
Numerical solutions to linear and nonlinear systems of equations.
Name the two branches of methods to get a numerical solution to the linear system Ax=b
Direct and indirect. Direct uses a set number of pre-determined operations, indirect uses iterations.
Name the two direct methods of solving linear systems.
Gauss elimination and LU factorization.
Name the two indirect methods of solving linear systems.
The Jacobi method and the Gauss-Seidel method
What is the difference between a homogeneous and nonhomogeneous system?
The system is homogeneous if every bk=0 in a11x1+a12x2+…..a1nxn = b1
What is the trivial solution for Ax=b
When all x componenets in the x vector are set to 0
Describe Gauss Elimination.
Gauss elimination uses elementary row operations to transform the linear system into an upper triangular matrix. This means a11 id the pivot, so a11 cannot =0.
What are the two methods of LU factorization
The Cholesky and Doolittle methods
Describe the process of permutation matrices
In transforming the matrix(A) to and upper triangular matrix(U) there must be an nxn matrix P such that multiplying Ax=b by P yields the new system Ux=b’
How many operations are required for gauss elimination?
Roughly (2/3)*n^3
Describe the method for solving Tridiagonal systems.
The Thomas Method scales all elements in its matrixso the diagonal entries are 1. Back substitution is then employed to solve the system.
Describe LU Factorization
LU factorization breaks a matrix into an upper and lower triangular matrix. The upper triangular matrix (U) is solved via back substitution. The loewr triangular matrix (L) is solve via forward substitution. Two other methods may be employed known as Doolittle and Crout factorization.
Describe Doolittle factorization.
This is an LU factorization method where the L matrix has 1’s along the diagonal. Requires (1/3)n^3 operations, so it is easier than gauss elimination
Describe Cholesky factorization
Ggfgf
