3.4.2 Norms of Matrices.
As with vectors, a norm of a matrix A is a numerical measure of its size. However, now we are also
interested in the effect of the matrix A on a vector x when we form y = Ax. The norm on the matrix depends
on the norms we use for
3
Systems of Equations
3.1 Systems of equations
3.1.1 Notation and Examples
In this chapter we look at algorithms to solve systems of equations. A system of n linear equations with n
unknowns can be written in the form Ax = b, where A is the n matrix of c
3.2 Gaussian Elimination Without Pivoting.
Gaussian elimination is one popular procedure to solve linear equations. As we shall see, it leads to a
decomposition of the coefficient matrix A as the product A = LU of a lower triangular matrix L and an upper
3.3 Gaussian Elimination With Partial Pivoting.
In the previous section we discussed Gaussian elimination. In that discussion we used equation 1 to
eliminate x1 from equations 2 through n. Then we used equation 2 to eliminate x2 from equations 2 through
n
4.2 Divided differences.
In section 4.1 we considered two methods for constructing an nth degree polynomial
(1)
y = a0 + a1x + a2x2 + a3x3 +
+ anxn
that passes through n + 1 points
(2)
(x0, y0), (x1, y1), , (xn, yn)
This is called the interpolating polyn
2
Nonlinear Equations
This chapter is concerned with approximating the solution of a single nonlinear equation. We begin with
general principles involved in the solution of nonlinear equations such as existence and uniqueness and then
look at various appr
3.4.4 Propagation of errors in linear formulas componentwise analysis.
In this section we look at the propagation of errors in linear formulas by treating the errors component-wise.
This is more precise than the analysis using vector and matrix norms in t
4
Approximation
Approximation is concerned with finding functions y = f(x) that either approximate other functions or
experimental data. In the first case we want to approximate a given function by a simpler function. In the
second case we want to find a
3.4.1 Norms of Vectors.
In this and the next two sections we look at the propagation of errors in linear formulas using vector and
matrix norms. Although this does not give as precise a result as when works with the errors of each
component separately, it
4.3 The Error in the Interpolating Polynomial.
In this section we derive a formula for the error in the interpolating polynomial when it is used to
approximate a given function. In this section we assume
(1)
y = f(x) is a function with n + 1 continuous de
Exam 1
Math 472/572
Fall 2010
Name: _ This is a closed book exam. You may use a calculator and the
formulas handed out along with the exam. Show your work so I can see how you arrived at your
answers. (This is particularly important if I am to be able to