A sequence of numbers {xn}n = 1¥ converges to a number x if there is an e > 0 and a number N(e) such that n > N(e) implies |xn - x| < e.
The derivative of a function at a point, f¢(x) is
| ||||||||
Functions with n continuous derivatives are f(x) Î Cn[a,b]. Note that differentiable functions are continuous but not viceversa.
Theorem 1.7 (p. 4) Rolle's. If f(x) is continuous and differentiable in [a,b] and f(a) = f(b) = 0 then, for some a < c < b f¢(c) = 0.
Theorem 1.8 (p. 4) Mean Value. If f(x) is continuous and differentiable in [a,b] then, for some a < c < b f¢(c) = [(f(b) - f(a))/(b-a)].
Theorem 1.9 (p. 5) Extreme Value. If f(x) Î C[a,b] there are two points c1, c2 such that f(c1) < f(x) < f(c2) for each x Î [a,b].
Definition 1.10 (p. 7) Riemmann integral. The integral of f(x) from a to b is
| |||||||||||
Theorem 1.11 (p. 8) Average. The average value of a function in [a,b] is f(c) = [1/(b-a)] òab f(x) dx.
Theorem 1.12 (p. 9) Generalized Rolle. If f(x) Î Cn[a,b] and is equal to zero at n+1 points, then there is a c for which f(n)(c) = 0.
Theorem 1.13 (p. 9) Intermediate Value. If f(x) Î C[a,b] and K Î [f(a),f(b)], then there is a c for which f(c) = K.
Theorem 1.14 (p. 10) Taylor polynomial. If f(x) Î Cn[a,b] and x0 Î [a,b], for every x Î [a,b] there is a x(x) between x0 and x such that f(x) can be represented in terms of the nth-order Taylor polynomial Pn(x) and the remainder or truncation error Rn(x), i.e.
| ||||
| ||||||||
| ||||||
If x0 = 0 one obtains the Maclauring polynomial.
Example 1.1.3 (p. 10). Determine the Taylor polynomial of f(x) = cos(x) about x0 = 0. See Fig. 1.10, p. 11 and Maple code on p. 13.
Only a small subset of real numbers can be represented in computers. Roundoff errors always occur in computer calculation since only approximate representations of real numbers are used. Roundoff errors can be reduced by using high-order arithmetic (double precision) but they can not be eliminated. Computers handle numbers using a floating point representation consisting of a fractional part (mantissa) and an exponential part (characteristic).
The maximum and minimum numbers the computer can represent are fixed. When an attempt to exceed the maximum number is made, an overflow takes place. When an attempt to go below the minimum number is made, an underflow takes place.
If a number is within the numerical range of the computer, its floating point representation can be obtained either by chopping or rounding. As a result, when the floating point numbers are used in computation, roundoff error always occurs.
Example 1.2.2 (p. 19) Represent the number p to five decimals using choping and rounding.
Definition 1.15 Errors. Let p* approximate p, the absolute error is |p-p*| and the relative error is |p-p*|/|p| (p ¹ 0).
Definition 1.16 (p. 20) Significant digits. p* approximates p to t significant digits if
| ||||||
A numerical method is an approach for the determination of approximate solutions of mathematical problems. An algorithm is a procedure describing clearly the steps of a computation. A high level computer languaje is used to code the algorithm for machine computation.
Numerical methods are stable if small changes in data produce correspondingly small changes in output. A methos in unstable if that is not the case. Linear growth of errors occur in stable methods while exponential growth is common in unstable methods.
Example 1.3.3 (p. 34) Determine the rate of error growth when computing with the recursive equation given.
Definition 1.18 (p.36) Rate of Covergence. If {bn}n = 1¥ is a sequence converging to zero while {an}n = 1¥ converges to a and there is a K > 0 such that
| ||||
| ||||
The software included with the text is sufficient for the purposes of this class. Coded programs of the numerical methods described are available in FORTRAN, C, Pascal, Maple, Mathematica and Matlab. More sophisticated software, essential for the solution of more complicated problems is readily available.