數值方法 NM 5 微分及積分(1)
5.1 Differentiation
5.1.1 Derivative and Integral
Calculus is an important tool for most modern professional practices. It is a branch of mathematical
sciences to study the “variation” in natural phenomena, of which includes two main components –
differentiation and integration.
Mathematically, derivative is the change or slope ∆y/∆x (in Cartesian coordinates) of an
imaginary tangent at a point on a curve.
y f (x i  x)  f (x i )

x
x
(5.1)
As ∆x approaches zero, ∆y/∆x becomes the so-called derivative, and can then be defined and
calculated numerically using the concept of “limit”:
dy
f (x  x)  f (x)
 f (x)  lim

x0
dx
x
(5.2)
The above derivative is calculated by FORWARD DIFFERENCE scheme.
*Note: The first-order derivative can be calculated in three ways:
1). Forward difference scheme: f (x)  lim
h0
f (x  h)  f (x)

h
f (x)  f (x  h)

h
(5.4)
f (x  h)  f (x  h) 2

2h
(5.5)
2). Backward difference scheme: f (x)  lim
h0
3). Central difference scheme: f (x)  lim
h 0
(5.3)
Integration is the “antiderivative” of a function in calculus, i.e.
I
b
 f (t)dt
a
(5.6)
For example, given y(t) as a function of time t
(or in term of position x), then its derivative v(t)
= dy(t)/dt and the integration of this derivative
yields the

t
y(t )  v(t )dt
0
NM 05 微分及積分(1)
(5.7)
1
5.1.2 Single variable in polynomial y = f(x)
5.1.2.1 Rounding error associated with very small increment.
Q1: Use Eq. (5.2) to evaluate the f   x  of f  x   x 9 , and assess the rounding error due to very
small increment h.
A1: S5121_Q1.m
fclose('all');
all; clc
x =
h =
fx =
fx1=
clear
variables;
close
1;
logspace(-20,0,21)';
@(x) x.^9;
@(x) 9*x.^8;
for ii = 1:length(h)
dfdx(ii,1) = ( fx(x
fx(x) )/h(ii);
end
+
h(ii))
-
loglog(h, abs( dfdx - fx1(1) ), '*b')
axis([1e-18, 1, 1e-8, 1e4]);
xlabel('h value'); ylabel('Error in
approximation');
Conclusion: As h goes smaller, error decreases before h  108 , but soars after h  1011 .
5.1.2.2 Differentiation using straight forward simple rule dx n / dx  nx n1
For this problems, the function polyder built in MATLAB can be used to compute the coefficients of
the derivative polynomial. Another built-in function diff will approximate the numerical derivative.
Q2: Find the first-order derivative of y(x) = 2x4 – 7x3 + 5x2 −1 using both MATLAB commands – f
polyder and diff with 99 points and 6 points, respectively. (Harman et al., p.574-576)
A2: (1) The first derivation of y(x) is y’(x) = 8x3 – 21x2 + 10x
(2) S5122_Q2.m
NM 05 微分及積分(1)
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fclose('all'); clear variables; close all;
clc
f = @(x) 2*x.^4-7*x.^3+5*x.^2-1; %
Polynomial
% ployder
p = [2, -7, 5, 0, -1];
% Coefficient of
the polynomial
pd = polyder(p);
% Take derivative
xi = linspace(0,3,100)'; % Assign x
coordinates
yd = polyval(pd,xi);
% Corresponding
f'(xi)
% diff >> 6-pt
x1 = linspace(0,3,6)';
y1 = diff( f(x1) )./diff(x1);
% diff >> 99-pt
x2 = linspace(0,3,99)';
y2 = diff( f(x2) )./diff(x2);
% Plot
plot(x1(1:end-1), y1, '^r')
hold on
plot(x2(1:end-1), y2, 'ok')
hold on
plot(xi, yd, '-b', 'LineWidth', 1.5)
xlabel('x'), ylabel('f^\prime(x)');
legend('diff@6pts','diff@99 pts',
'polyder','Location','NorthWest')
5.1.3 Multivariate function f(x,y)
For two variables f  f  x  x , y  y   f  x , y  , mathematically the derivative of f  x , y  is
df 
f
f
dx  dy
x
y
(use chain rule)
(5.8)
Other useful formulae for differentiation:
(1). y  u( x)n , then
dy
u
 nu n 1
dx
x
(2). y  u ( x)v( x) , then
dy
v
u
u v
dx
x
x
u
v
v u
u ( x)
dy
(3). y 
, then
 x 2 x
v( x)
dx
v
NM 05 微分及積分(1)
(5.9)
(5.10)
(5.11)
3
Q3: Find and plot first-order derivative of xcosnx, using forward difference scheme (Backstrom, 1997).
Note: Use n = 1 and h = 0.001.
A3: S513_Q3.m
fclose('all'); clear variables; close all;
clc
n = 1;
h = 1e-3;
f0 = @(x) x.*cos(x).^n;
f1 = @(x) cos(x).^n n*x.*sin(x).*cos(x).^(n-1);
x = linspace(0,10,51)';
% Exact solution
s1 = f1(x);
% Numerical result >> Finite difference
s2 = (f0(x+h) - f0(x))/h;
% Plot - Comparison
plot(x, s1, '-b')
hold on
plot(x, s2, 'or')
xlabel('x'); ylabel('f^\prime(x)');
legend('Exact','Numerical','Location','SouthW
est')
% Plot - Abs error
figure
plot(x, s2-s1, '-r')
xlabel('x'); ylabel('Absolute error');
Exercise: Try different n and h, and observe what would happen.
Q4: Plot the second-order derivative of xcos(x), using a central difference scheme. (Backstrom, 1997).
A4: S513_Q4.m
fclose('all'); clear variables; close all;
clc
h
f0
f1
x
=
=
=
=
1e-3;
@(x) x.*cos(x);
@(x) cos(x) - x.*sin(x);
linspace(0,10,51)';
% Exact solution
s1 = f1(x);
% Numerical result >> Finite difference
s2 = (f0(x+h) - f0(x-h))/2/h;
% Plot - Comparison
plot(x, s1, '-b')
NM 05 微分及積分(1)
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hold on
plot(x, s2, 'or')
xlabel('x'); ylabel('f^\prime(x)');
legend('Exact','Numerical','Location','SouthW
est')
% Plot - Abs error
figure
plot(x, s2-s1, '-r')
xlabel('x'); ylabel('Absolute error');
Exercise: Compare the errors between central difference scheme and forward difference scheme.
5.1.4 Mixed Problems in multivariate function
(1). Find first-order derivative of f (x)  x 2  x 2  1 and compare with exact solution
f (x) 

x 
 2x 

.
2
x 1 
2 x2  x2  1 
1
(2). Find first-order derivative of f (x) 
4  2x
x3
and compare with exact solution
f (x)  (x  6) / x 5/2 .
(3). Find first-order derivative of f (x)  tan(x 2  1) and compare with exact solution
f (x)  2x sec2 (x2  1) .
(4). Find first-order derivative of x 2  y 2  3 and compare with exact solution
f (x) 
d 2y
3
dy
x

and f (x)  2  ...   3
dx
y
dx
y
(5). Find first-order derivative of f ( )  ln
f ( )  1 
x

NM 05 微分及積分(1)
e   x
and compare with exact solution
x!
【 f ()  lne   lnx  lnx!   x ln  lnx! 】
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5.2 Extrema of real-values functions
5.2.1 Higher-order derivatives [Landfield and Penny 199, p.122-123]
Second-order derivatives are useful for the determination of local maximum and minimum. In central
difference scheme, the second-order derivative f (x) can be expressed as
f (x)  lim
h0
f (x  h)  f (x  h)
f (x  2h)  2 f (x)  f (x  2h)
 ...  lim
h0
2h
4h2
(5.12)
f (x  h)  2 f (x)  f (x  h)
h2
(5.13)
or alternatively,
f (x)  lim
h0
To evaluate the approximation error, derivation by Taylor series is suggested. Let us begin with a
Taylor series expanding f (x) from f (xi ) to f ( x i  h) ,
f (xi  h)  f (xi )  hf (xi )  (h2 / 2!) f (xi )  (h3 / 3!) f (xi )  (h4 / 4!) f (4) (xi )  ...
(5.14)
The above is equivalent to
fi 1  fi  hf (xi )  (h2 / 2!) f (xi )  (h3 / 3!) f (xi )  (h4 / 4!) f (4) (xi )  ...
(5.15)
Similarly, we can use the same form to immediately obtain:
fi 1  fi  hf (xi )  (h2 / 2!) f (xi )  (h3 / 3!) f (xi )  (h4 / 4!) f (4) (xi )  ...
(5.16)
Subtracting (5.16) from (5.15) gives
fi 1  fi 1  2hf (xi )  2(h3 / 3!) f (xi )  2(h5 / 5!) f (5) (xi )  ...
(5.17)
By neglecting terms with an order of h3 and higher, result will render a forward difference
approximation, expressed as:
f (xi ) 
fi 1  fi 1
 O(h2 )
2h
(5.18)
By adding (5.15) and (5.16), we obtain
fi1  fi1  2 fi  2(h2 / 2!) f (xi )  2(h4 / 4!) f (iv) (xi )  ...
(5.19)
And neglecting terms in h4 and higher, we have
f (xi ) 
fi 1  2 fi  fi 1
 O(h2 )
h2
(5.20)
By taking more terms in the Taylor series, together with those for f(x + 2h) and f(x − 2h) etc., and
performing similar manipulations, we can obtain higher derivatives, and perhaps more accurate
derivative approximations (see table below).
NM 05 微分及積分(1)
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Multipliers for fi-3 … fi+3
fi-3
fi-2
fi-1
fi
fi+1
fi+2
fi+3
Order of error
2hf (xi )
0
0
-1
0
1
0
0
h2
h2 f (xi )
0
0
1
-2
1
0
0
h2
2h3 f (xi )
0
-1
2
0
-2
1
0
h2
h4 f (4) (xi )
0
1
-4
6
-4
1
0
h2
12hf (xi )
0
1
-8
0
8
-1
0
h4
12h2 f (xi )
0
-1
16
-30
16
-1
0
h4
8h3 f (xi )
1
-8
13
0
-13
8
-1
h4
6h4 f (4) (xi )
-1
12
-39
56
-39
12
-1
h4
Exercise: S521.m Write a MATLAB program to determine the 2nd derivative of y = x7 when x = 1, using:
f (x) 
 f (x  2h)  16 f (x  h)  30 f (x)  16 f (x  h)  f (x  2h)
f (x  h)  2 f (x)  f (x  h)
and f (x) 
2
12h2
h
5.2.2 Local extrema (minimum or maximum) for single variable function
For a differentiable function y  f (x) in an open interval, a local or relative minimum or maximum
(including inflection point) exists at a point where f (x)  0 . This is called the necessary condition (充
分條件).
Relative minimum at x0: If f (x0 )  0 and f ( x0 )  0
(5.21)
Relative maximum at x0: If f (x0 )  0 and f ( x0 )  0
(5.22)
Eqs. (5.21) and (5.22) are called sufficient conditions (必要條件) for local extrema.
NM 05 微分及積分(1)
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5.2.3 Local maximum for multivariate function (Harman et al., 2000, p.581)
Necessary condition for the extrema of a given function f(x0) where its critical point is x0 = [x1, x2]:
 f f 
f
f
,
f’(x0)  

 0  find x0 = [x1, x2]

x1 x2
 x1 x2 
(5.23)
Sufficient conditions:
Let f (x , y) have continuous first-order and second-order partial derivatives in an open region R
containing the point ( x 0 , y 0 ) ,
If f (x0 , y0 ) x  0 , and f (x0 , y0 ) y  0 ,
(5.24)
and the product D
(1). D  fxx (x0 , y0 ) fyy (x0 , y0 )  fxy (x0 , y0 )  0
2
(5.25)
(a) f (x , y) has local minimum at 0  (x0 , y0 ) , if fxx (x0 , y0 )  0
(5.26)
(b) f (x , y) has local maximum at 0  (x0 , y0 ) , if fxx (x0 , y0 )  0
(5.27)
(2). If D < 0, f (x , y) is neither a minimum or maximum at 0  (x0 , y0 ) , and x 0 is called a
saddle point of f (x , y)
(3). If D = 0  No information about the critical points of f (x , y) . Higher order (n ≥ 3) is
required for judgement.
5.2.4 Useful MATLAB commands
fmin (or fminbnd)  to find minimum of a single variable function;
find  find specific value;
fzero  find zero crossing;
fmins  find minimum of a multivariate function.
NM 05 微分及積分(1)
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5.2.5 Examples for extrema of a function
Q5: Determine the minimum of the following given function: (Harman et al., 2000; p.583-)
z  f (x , y)  x 2  xy  y 2  3x  3y  1
(5.28)
A5: (1) Anayltical solution:
f
f
 2 x  y  3  0;
 x  2y  3  0 
x
y
 x0 , y0 
At (x0 , y0 )   3,3  fxx  2, fyy  2, fxy  1;
  3, 3 
D  fxx fyy  fxy2  3  0
>> Local minimum f(−3, 3)= −8 at the critical point.
(2) Plot the function S525_Q5a.m
fclose('all'); clear variables; close
all; clc
fn = 'x.^2+x.*y+y.^2+3*x-3*y+1'; %
Input >> Function
% fn = @(x,y) x.^2+x.*y+y.^2+3*x3*y+1; % Input >> Function
x0 = [-5, 5, -5, 5]; % Input >> Bounds
for x and y coordinate
X = x0(1):0.1:x0(2); X=X';
Y = x0(3):0.1:x0(4); Y=Y';
[x,y] = meshgrid(X,Y);
fgrid = eval(fn); % Get value of
f(x,y) by "eval"
% fgrid = fn(x,y); % Get value of
f(x,y) by @(x,y)
surfc(x, y, fgrid)
xlabel('x'); ylabel('y');
zlabel('f(x,y)')
(3) Find the minimum using command fmins S525_Q5b.m
fclose('all'); clear variables; close all; clc
fn = 'x(1)^2+x(1)*x(2)+x(2)^2+3*x(1)-3*x(2)+1';
xguess = [-1, 1];
xmin = fminsearch(fn, xguess);
x(1) = xmin(1); % In f(x,y) corresponds to x
x(2 )= xmin(2); % In f(x,y) corresponds to y
zmin = eval(fn);
>> xguess =
-1
1
>> xmin =
-3.0000
3.0000
>> zmin =
-8.0000
NM 05 微分及積分(1)
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5.3 Integral & Sums
Generally speaking, integral is a sum of a function over a specified range. In addition to do the summing
manually, this can be simply done by using the command sum in MATLAB, such as:
S(n) 
n
k
i
 1  k  k 2  k 3  ...  k n
(5.29)
i 0
In some cases, a sum tends to approach a value as i increases. Plot the integral for examination, if
required.
5.3.1 Convergent series
Q6: Sum of a geometric series S(n) 
n
k
i
 1  k  k 2  k 3  ...  k n , using k = 0.5, n =20.
i 0
A6: S531_Q6.m
k = 0.5; n = 20;
s = 0;
% Method#1
for i = 1:n
s = s + k^(i-1);
s1(i,1) = s;
end
% Method#2
s2 = cumsum( k.^(0:1:n-1)' );
plot((0:1:n-1)', s1, '-ob')
xlabel('i'); ylabel('Cumulative sum');
grid on; box off
axis([0, 20, 0.5, 2.5])
NM 05 微分及積分(1)
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How can we stop the calculations when a limit is reached?
Q7: Sum of a geometric series S(n) 
n
k
i
 1  k  k 2  k 3  ...  k n , using k = 0.5.
i 0
A7: S531_Q7.m
fclose('all'); clear variables; close all; clc
k = 0.5;
s = 0;
for i = 1:1e4
s = s + k^(i-1);
s1(i,1) = s;
if i > 1 && s1(i)-s1(i-1) < 1e-8; break; end
end
plot((0:1:i-1)', s1, '-ob')
xlabel('i'); ylabel('Cumulative sum'); grid on; box off
axis([0, i, 0.5, 2.5])
5.3.2 Trigonometric series
The sine and cosine functions expressed as an infinite series can be calculated numerically using the
summing technique:
sin( x) 

x 2i 1
x
x 2i
x2
x3
x5
 (1) (2i  1)!  1!  3!  5!  ...
i
(5.30)
i 0
cos( x) 

x4
 (1) (2i)!  1  2!  4!  ...
i
(5.31)
i 0
However, rather than summing up terms by terms directly, it may be useful to compare the ratio of
two consecutive terms. For example, for sin(x), the ratio gives
ti
(1)i x 2i 1 (2i  1)!
 x2


t i 1
(2i  1)! (1)i 1 x 2i 1 (2i  1)2i
(5.32)
Q8 & A8. Summing up the terms in series for sin(x)
S532_Q8.m
fclose('all'); clear variables; close
all; clc
k = 0.5;
s = 0;
for i = 1:1e4
s = s - k^2/(2*i+1)/2/i;
s1(i,1) = s;
if i > 1
rel_val = ( s1(i) - s1(i-1) )/s1(i1);
if abs(rel_val) < 1e-6; break; end
end
end
plot((0:1:i-1)', s1, '-ob')
xlabel('i'); ylabel('Cumulative sum');
grid on; box off
axis([0, i, -inf, inf])
NM 05 微分及積分(1)
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