MATLAB Information and Formulas OPERATOR PRECEDENCE 1
2
%f Fixed-Point
Notation
%e Exponential
Notation
./ / Right Division, Matrix Right Division,
%s String
Characters
.\ \ Left Division,
%c Single Character
.^ ^ Power,
~ Matrix Power
Logical Negation
.* * Multiplication,
5
6
Matrix Multiplication,
Matrix Left Division
of
+ Addition
\t Horizontal Tab
-­‐ Subtraction
\n New Line
: Colon Operator
%% Percent Character
Less Than,
< <= To,
7
Integer
%d .’ ‘ Transpose, Matrix Transpose,
3
4
fprintf SPECIFIER Parentheses
() Less Than Or Equal
\’
Single Quote Mark
’ > >= Greater Than, Greater Than Or Equal
To,
== ~= Equal To,
Not Equal To
8
& Element-wise AND
9
| Element-wise OR
1
0
&& Short-circuit AND
1
1
|| Short-circuit OR
COLOR SPECIFIER \\ Backslash
\b Backspace
Fixed-­‐Point Notation Syntax %<field_width>.<precision>f LINE STYLE SPECIFIER MARKER TYPE SPECIFIER r Red
-­‐ Solid Line
+ g Green
-­‐-­‐ Dashed Line
b Blue
: Dotted Line
c Cyan
-­‐. Dash-dot Line
m Magenta
y Yellow
k Black
w White
o * . x s d ^ v > < ENG1060 Computing for Engineers Plus Sign
Circle
Asterisk
Point
Cross
Square
Diamond
Triangle (Up)
Triangle (Down)
Triangle (Right)
Triangle (Left)
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Root Finding Bisection Method False Position Method xl + xu
2 xr =
xr = xu −
f ( xu )( xl − xu )
f ( xl ) − f ( xu )
Newton-­‐Raphson Method f ( xi )
xi +1 = xi −
f ' ( xi )
Secant Method f ( xi )( xi −1 − xi )
xi +1 = xi −
f ( xi −1 ) − f ( xi )
Modified Secant Method δxi f ( xi )
xi +1 = xi −
f ( xi + δxi ) − f ( xi )
Alternative Modified Secant Method δ f (xi )
xi+1 = xi −
f (xi + δ ) − f (xi )
Curve Fitting Linear Regression:
Coefficient of Determination y = ao + a1 x
a1 =
n∑ xi yi − ∑ xi ∑ yi
r2 =
2
n∑ xi2 − (∑ xi )
St − S r
St
Linearizing Nonlinear Models a0 = y − a1 x
Standard Deviation n
S t = ∑ ( yi − y ) 2
Standard Error of the Regression Estimate n
2
S r = ∑ ( y i − a 0 − a1 xi )
i =1
Nonlinear
Linearized
y = α1e β1x
ln y = lnα1 + β1x
y = α 2 x β2
log y = logα2 + β2 log x
y = α3
i =1
St
sy =
n −1
sy / x =
Sr
n−2
x
β3 + x
1
1 β3 1
=
+
y α3 α3 x
Page 2 of 3 Numerical Integration (n is the number of points) Trapezoidal Rule: I = (b − a )
f (b) + f (a )
2
1
3
Et = − f ʹ′ʹ′(ξ )(b − a )
12
Composite Trapezoidal Rule I=
n −1
h ⎡
⎤
(
)
f
x
+
2
f (xi ) + f (xn )⎥
∑
1
⎢
2 ⎣
i=2
⎦
where
h=
(b − a)
n −1
Composite Trapezoidal Rule with Unequal Segments I = (x2 − x1 )
f (x3 ) + f (x2 )
f (xn−1 ) + f (xn )
f (x2 ) + f (x1 )
+ (x3 − x2 )
+ ! + (xn − xn−1 )
2
2
2
Simpson’s 1/3 Rule h
I = [ f (x1 ) + 4 f (x2 ) + f (x3 )]
3
1
5
Et = −
f ( 4) (ξ )(b − a )
2880
Simpson’s 3/8 Rule 3h
[ f (x1 ) + 3 f (x2 ) + 3 f (x3 ) + f (x4 )]
8
1
Et = −
f ( 4) (ξ )(b − a)5
6480
I =
⎡
⎤
n −1
n−2
h ⎢
⎥
Composite Simpson’s 1/3 Rule: I = ⎢ f (x1 ) + 4 ∑ f (xi ) + 2 ∑ f (x j ) + f (xn )⎥
3
i = 2 , 4 , 6 ,...
j =3, 5, 7 ,...
⎢⎣
⎥⎦
i , even
j , odd
ODE: Initial Value Problems Euler’s Method yi +1 = yi + f (t i , yi )h
0
i +1
y
Heun’s Method = yi + f (ti , yi )h
Midpoint Method h
yi +1/ 2 = yi + f (ti , yi )
2
h
t i +1 / 2 = t i +
2
yi +1 = yi + f (ti +1/ 2 , yi +1/ 2 )h
f (ti , yi ) + f (ti +1 , yi0+1 )
yi +1 = yi +
h
2
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