2.7 Floating point numbers and round-off errors.
Round-off errors are due to the fact that people, calculators, and computers usually do not keep track of or
store numbers exactly in the course of a series of calculations. Scientific and engineering compu
1.5 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
1.7 Applications to Electrical Networks.
In section 1.1 we talked about applications of systems of equations to getting approximate solutions of
boundary value problems. In this section we look at applications of linear equations to electrical networks
1.6 How Long Does it Take to Solve Linear Equations.
1.6.1 Basic Priciples.
We saw in section 1.4.1 that for large n the amount of time T it takes to solve a linear system of n equations
and n unknowns from scratch is approximately proportional to n3, i.e
In this chapter we look at how errors propagate when we solve linear equations. Even though Gaussian
elimination is an exact algorithm in theory, errors are an important consideration in this subject. Two
reasons for this are the following.
1.7.2 Loop Analysis.
In this section we discuss another approach to electric circuits called loop analysis. This time we construct a
system of linear equations that the currents in the lines satisfy. We illustrate the concepts by means of the
2.4 Norms of Matrices.
In the previous section we looked at norms of vectors and saw how they could be used to estimate the
propagation of errors in a linear formula. In this section we look at norms of matrices and see how they
extend the results of the
2.2 Propagation of errors in linear formulas componentwise analysis.
In this section we look at the propagation of errors in linear formulas. Here we treat the errors
componentwise. In the following sections we will treat the errors vectorwise which is le
2.3 Norms of Vectors.
In the previous section we looked at the propagation of errors in linear formulas by describing the errors in
the variables individually. This gives the most precise result, but becomes more difficult in larger problems.
In this and
1.4 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