content: text: Numerical Methods for Ordinary Differential Equations

content: text: also plays an important role in error analysis (investigating the difference between the numerical
approximation and the solution).

content: text: Calculating with only a finite subset of the rational numbers has many consequences. For exam-
ple: a computer cannot distinguish between two polynomials of sufficiently high degree. Conse-
quently, methods based on the main theorem of algebra (i.e. that an nth degree polynomial has
exactly n complex zeros) cannot be trusted. Errors that follow from the use of finitely many digits
are called rounding errors (Section 1.4).

content: text: An important aspect of numerical mathematics is the emphasis on efficiency. Contrary to or-
dinary mathematics, numerical mathematics considers an increase in efficiency, i.e. a decrease
of the number of operations and / or amount of storage required, as an essential improvement.
Progress in this aspect is of great practical importance and the end of this development has not
been reached yet. Here, the creative mind will meet many challenges. On top of that, revolutions
in computer architecture will overturn much conventional wisdom.

content: text: 1.3 Why numerical mathematics?

content: text: A big advantage of numerical mathematics is that it can provide answers to problems that do not
admit closed-form solutions. Consider for example the integral

content: text: \int_0^\pi\sqrt{1+\cos^2x}dx\text{.}

content: text: This is an expression for the arc length of one arc of the curve y(x) = sin x, which does not have
a solution in closed form. A numerical method, however, can approximate this integral in a very
simple way (Chapter 5). An additional advantage is that a numerical method only uses stan-
dard function evaluations and the operations addition, subtraction, multiplication and division.
Because these are exactly the operations a computer can perform, numerical mathematics and
computers form a perfect combination.

content: text: An advantage of analytical methods is that the solution is given by a mathematical formula.
From this, insight in the behavior and the properties of the solution can be gained. For numerical
approximations, however, this is not the case. In that case, visualization tools may be used to gain
insight in the behavior of the solution. Using a numerical method to draw a graph of a function
is usually a more useful tool than evaluating the solution at a large number of points.

content: text: 1.4 Rounding errors

content: text: A computer uses a finite representation of the all numbers in R. These are stored in a computer
in the form

content: text: \pm0.d_1d_2\ldotsd_n\cdot\beta^e\text{,}

content: text: (1.1)

content: text: in which, by definition, d1 > 0 and 0 ≤ di < �. The normalization is needed in order to prevent a
waste of digits and to make the representation unambiguous. We call the value in equation (1.1)
a floating point number (representation) in which 0.d1d2 . . . dn is called the mantissa, � the base and
e (integer) the exponent, where L < e < U. Characteristic values for |L| and U are in the range
[100,1000], often, � = 2 (binary representation) and n = 24 (single precision) or n = 53 (double
precision). Most computers and software packages (Matlab) satisfy the IEEE-754 standard, and
hence provide single-1 and double-precision2 computations.

content: text: Let for x ∈ R

content: text: 0.d_1\ldotsd_n\cdot\beta^e\leqx<0.d_1d_2\ldots\left(d_n+1\right)\cdot\beta^e\text{,}

content: text: 1http://en.wikipedia.org/wiki/Single-precision_floating-point_format
2http://en.wikipedia.org/wiki/Double-precision_floating-point_format