Numerical methods
Jiří Zelinka
Autumn 2017
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Numerical methods
Literature
Literature
• Mathews, J.H., Fink, K.D.: Numerical methods using MATLAB, Pearson Prentice Hall, 2003
• Stoer, J., Bulirsch R.: Introduction to Numerical Analysis, Spriger, 1992
Assumptions
a Linear algebra
• Differential calculus
• Integral calculus
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Numerical methods
x: exact value, x: approximation x
x — x: absolute error x error
x — x
< a: estimate of the absolute
x—x x
: relative error
x—x x
< S: estimate of the relative error
Approximation x of x to s digits:
x — x
x
< 5.10"5.
Jiří Zelinka
□
Numerical methods
Error for the computer representation of the numbers
Matlab code:
» c=0.1;
» x=c*ones(l,100); » sum (x)-10 ans =
-1.953992523340276e-14
°/o rearrangement of the computation
» A=reshape(x,10,10); » sum(sum(A))-10 ans =
-1.776356839400250e-15
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Numerical methods
SP
Conditionality
Condition number:
Ay
y
Ax
X
output relative error I input relative error
1 - the task is well-conditioned
Cp > 100 - the task is ill-conditioned
Condition number for matrix
A - non-singular matrix
CA= A
A
-i
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□ i3" Numerical methods
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Example of ill-conditioned matrix: Hilbert matrix H = (/j/j), hjj = -r^j
Matlab code:
» H=hilb(8) H =
1	1/2	1/3	1/4	1/5	1/6	1/7	1/8
1/2	1/3	1/4	1/5	1/6	1/7	1/8	1/9
1/3	1/4	1/5	1/6	1/7	1/8	1/9	1/10
1/4	1/5	1/6	1/7	1/8	1/9	1/10	1/11
1/5	1/6	1/7	1/8	1/9	1/10	1/11	1/12
1/6	1/7	1/8	1/9	1/10	1/11	1/12	1/13
1/7	1/8	1/9	1/10	1/11	1/12	1/13	1/14
1/8	1/9	1/10	1/11	1/12	1/13	1/14	1/15
» cond(H)							
ans =
15257575253
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Typical ill-conditioned tasks
• dividing by a small number
• subtracting almost the same numbers
• cumulation of errors in iterative calculation
An = n- An_ľ
Example:
ln = J xnex-1dx o
li = \f itegration by parts: ln = 1 — nln_i
The initial error is multiplied by n in every step, i.e.
n = 10 is the error multiplied by 10! = 3, 628, 800.
Jiří Zelinka
□ i3" Numerical methods
Symbols Oy o
f,g- functions defined in the neighbourhood of a point a (it is possible a = ±oc)
f (x) = 0(g(x)) for x —>► a      there exists a constant C > 0
|f(x)| < C-\g(x)\
in the neigbourhood of a point a. Meaning:
function f is similar to g in the neigbourhood of a point a.
r(x) = o(g(x)) for x ^ a
limM = 0.
Meaning:
function f converges to 0 faster then g in the point a.
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For sequences {an)™=0, {bn)™=0
an = 0(bn) or an = o(bn) pro n —>► oc
Sometimes the point a is clear - we can omit it:
an = 0{1/n),      f(h) = o(h3)
Example: Taylor series
f (x + h) = f (x) + f'(x)h +
f(x+h) = f(x)+f'(x)h+0{h2),    f(x+h) = f{x)+f'{x)h+o{h)
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