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Introduction to Numerical Methods and MATLAB Programming for Engineers

Todd Young, Martin J. Mohlenkamp

Section 1.8 Review of Chapter 1

Subsection 1.8.1 Methods and Formulas

Subsubsection 1.8.1.1 Solving equations numerically:

\(f(x) = 0\)
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an equation we wish to solve.
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\(x^{*}\)
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a true solution.
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\(x_{0}\)
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starting approximation.
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\(x_{n}\)
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approximation after \(n\) steps.
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\(e_{n} = x_{n} - x^{*}\)
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error of \(n\)-th step.
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\(r_{n} = y_{n} = f(x_{n})\)
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residual at step \(n\text{.}\) Often \(|r_{n}|\) is sufficient.
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Subsubsection 1.8.1.2 Newton’s method:

\begin{equation*} x_{i+1}= x_{i} - \frac{f(x_{i})}{f'(x_{i})} \end{equation*}
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Subsubsection 1.8.1.3 Bisection method:

\(f(a)\) and \(f(b)\) must have different signs.
\begin{equation*} x = (a+b)/2 \end{equation*}
Choose \(a = x\) or \(b = x\text{,}\) depending on signs. \(x^{*}\) is always inside \([a,b]\text{.}\) \(e < (b-a)/2\text{,}\) current maximum error.
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Subsubsection 1.8.1.4 Secant method:

\begin{equation*} x_{i+1}= x_{i} - \frac{x_{i} - x_{i-1}}{y_{i} - y_{i-1}}y_{i} \end{equation*}
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Subsubsection 1.8.1.5 Regula Falsi

Is a hybrid between secant and bisection methods.
\begin{equation*} x = b - \frac{b - a}{f(b) - f(a)}f(b) \end{equation*}
Choose \(a = x\) or \(b = x\text{,}\) depending on signs.
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Subsubsection 1.8.1.6 Convergence:

Bisection is very slow. Newton is very fast. Secant methods are intermediate in speed. Bisection and Regula Falsi never fail to converge. Newton and Secant can fail if \(x_{0}\) is not close to \(x^{*}\text{.}\)
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Subsubsection 1.8.1.7 Locating roots:

Use knowledge of the problem to begin with a reasonable domain. Systematically search for sign changes of \(f(x)\text{.}\) Choose \(x_{0}\) between sign changes using bisection or secant.
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Subsubsection 1.8.1.8 Usage:

For Newton’s method one must have formulas for \(f(x)\) and \(f'(x)\text{.}\) Secant methods are better for experiments and simulations. Bisection and Regula Falsi are slower, but keep the root within the current bounds.
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Subsection 1.8.2 MATLAB

Subsubsection 1.8.2.1 Commands:

v = [0 1 2 3]
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Make a row vector.
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u = [0; 1; 2; 3]
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Make a column vector.
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w = v'
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Transpose: row vector \(\leftrightarrow\) column vector
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x = linspace(0,1,11)
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Make an evenly spaced vector of length 11.
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x = -1:.1:1
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Make an evenly spaced vector, with increments \(0.1\text{.}\)
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y = x.^2
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Square all entries.
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plot(x,y)
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plot y versus x.
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f = @(x) 2*x.^2 - 3*x + 1
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Make an anonymous function.
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y = f(x)
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A function can act on a vector.
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plot(x,y,'*','red')
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A plot with options.
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Control-c
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Stops a computation.
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Subsubsection 1.8.2.2 Program structures:

for ... end example: 
for i=1:20 
  S = S + i; 
end
if ... end example: 
if y == 0 
  disp('An exact solution has been found') 
end
while ... end example: 
while i <= 20 
  S = S + i; 
  i = i + 1; 
end
if ... else ... end example: 
if c*y>0 
  a = x; 
else 
  b = x; 
end
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Subsubsection 1.8.2.3 Function Programs:

  • Begin with the word function.
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  • There are inputs and outputs.
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  • The outputs, name of the function and the inputs must appear in the first line. i.e. function x = mynewton(f,x0,n)
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  • The body of the program must assign values to the outputs.
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  • Internal variables are not visible outside the function.
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  • A function program may not use variables in the current workspace unless they are inputs.
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Subsubsection 1.8.2.4 Script Programs:

  • There are no inputs and outputs.
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  • A script program may use, create, change and even delete variables in the current workspace.
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Subsubsection 1.8.2.5 Symbolic:

syms x y
f = 2*x^2 - sqrt(3*x)
subs(f,sym(pi))
double(ans)
g = log(abs(y)) MATLAB uses log for natural logarithm.
h(x) = compose(g,f)
k(x,y) = f*g
fplot(f)
fplot(g,-10,10)
ezsurf(k)
f1 = diff(f,'x')
F = int(f,'x') indefinite integral (antiderivative)
int(f,0,2*pi) definite integral
poly = x*(x - 3)*(x-2)*(x-1)*(x+1)
polyex = expand(poly)
polysi = simple(polyex)
solve(f)
solve(g)
solve(polyex)
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