CHEE 222: PROCESS DYNAMICS AND NUMERICAL METHODS

Winter 2017

Implementation of Numerical Methods via MATLAB

Instructor: Xiang Li

Outline 1. Introduction - Command, script and function - MATLAB function for mathematical functions

2. Programming Flow Control - IF-ELSE statement: Conditional commands - FOR statement: Repetitive commands - WHILE statement: Repetitive commands - Logical expressions: Description of conditions

3. Example Implementations - Direct substitution - Newton’s method - Explicit Euler method

1

Why MATLAB?

1. Introduction •  MATLAB has become a standard tool for almost all

scientific and engineering disciplines.

•  MATLAB has become more and more popular in the R&D in industry.

•  For CHEE 222: You don’t have to perform tedious calculation by hand, and you can solve complicated problems with the principles learned in the course.

•  Will be used again in: CHEE 311, CHEE 319, CHEE 321, CHEE 434, CHEE 412, CHEE 418 …

2

References

1. Introduction •  Moler, C. Numerical Computing with MATLAB, The MathWorks, Inc.,

Natick, MA (2004). (Electronic version: http://www.mathworks.com/moler) •  MATLAB help files •  Hudon, N, DeHaan, D., and Guay, M. Introduction to MATLAB.

Department of Chemical Engineering, Queen’s University (2006). - This MATLAB guide is available on D2L as HDG2006.pdf - Part of this presentation is prepared based on this guide.

3

Prompt line command – Straightforward way to implement an operation

1. Introduction

Prompt line – Where your type your command

Current folder – Place the MATLAB file you want to call in the current folder.

Workspace – Existing variables are stored here.

4

One to several commands in a single prompt line 1. Introduction

5

Script file – A batch of commands 1. Introduction

6

Function file – A batch of commands with inputs and outputs 1. Introduction

Function name should be the file name!

7

Function file – A batch of commands with inputs and outputs 1. Introduction

8

Function file – A batch of commands with inputs and outputs 1. Introduction

9

Row Vector, Colum Vector, Matrix 1. Introduction

10

Math MATLAB

a = ( 1 3 6 )

b =258

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

A = 1 2 3

4 5 6⎛

⎝⎜⎞

⎠⎟

a = [ 1 3 6 ];

b = [ 1; 3; 6 ];

a(1) = 1; a(2) = 3; a(3) = 6;

b(1) = 2; b(2) = 5; b(3) = 8; b = ′b ;

A = [ 1 2 3; 4 5 6 ];

A(1,:) = [ 1 2 3 ]; A(2,:) = [ 4 5 6 ];

Approach 1

Approach 2

Approach 1

Approach 2

Approach 1

Approach 2

MATLAB function as “f(x)” in algebraic equations (1) 1. Introduction

11

absin(x1)+ cos(x2 ) = 0,

a ln(x1)+ eb2 (x1+x2 ) = 0,

a = 1,b = 0.

where x =x1

x2

⎛

⎝⎜

⎞

⎠⎟ , f(x) =

absin(x1)+ cos(x2 )

a ln(x1)+ eb2 (x1+x2 )

⎛

⎝⎜⎜

⎞

⎠⎟⎟

,

a = 1, b = 0.

f(x) = 0

MATLAB function as “f(x)” in algebraic equations (2) 1. Introduction f(x) =

absin(x1)+ cos(x2 )

a ln(x1)+ eb2 (x1+x2 )

⎛

⎝⎜⎜

⎞

⎠⎟⎟

where a = 1, b = 0.5.

To solve f(x) = 0 with x0 = (1.2, 1.4), type in MATLAB prompt (command) line:

12

or

MATLAB function as “f(x)” in algebraic equations (3) 1. Introduction f(x) =

absin(x1)+ cos(x2 )

a ln(x1)+ eb2 (x1+x2 )

⎛

⎝⎜⎜

⎞

⎠⎟⎟

where a = 1, b = 0.5.

To solve f(x) = 0 with x0 = (1.2, 1.4), type in MATLAB prompt (command) line:

13

MATLAB function for “f(t,x)” in ODEs 1. Introduction x = f(t,x) =

absin(x1)+ cos(x2 )

a ln(x1)+ eb2 (x1+x2 )

⎛

⎝⎜⎜

⎞

⎠⎟⎟

where a = 1, b = 0.5.

if MATLAB built-in ode solvers (e.g., ode45) are to be used for solution, the ODE needs to be represented in this form no matter whether time “t” is present in the right-hand-sides of the equations.

To solve x=f(t,x) for time period [0, 0.5] with x(0) = (3, 4), type in prompt line:

14

or

Some MATLAB built-in functions for math functions

1. Introduction

Function Description sin(x) sine of x; x in radians cos(x) cosine of x; x in radians tan(x) tangent of x; x in radians sec(x) secant of x; x in radian csc(x) cosecant of x; x in radian cot(x) cotangent of x; x in radian asin(x) arc sine of x sinh(x) hyperbolic sine of x asinh(x) inverse hyperbolic sine of x exp(x) exponential of x log(x) natural logarithm of x log10(x) base 10 logarithm of x sqrt(x) square root of x power(x,a) x to the power of a (equivalent to x^a)abs(x) absolute value of x round(x) round x towards nearest integer floor(x) round x towards minus infinity ceil(x) round x towards infinity fix(x) round x towards zero

15

IF-ElSE statements – Conditional commands 2. Programming Flow Control for Conditional and Repetitive Commands “CONDITION” is expressed via logical operations.

“ACTION” may involve one or several commands.

The math function |x| can be computed via the following MATLAB function ‘myabs’ that involves conditional commands:

16

FOR statements – Repetitive commands (1)

After the execution of the FOR loop, the value of y becomes 5.

After the execution of the FOR loop, the value of vector y becomes [1, 0, 0, 0, 25, 0, 0, 0, 81].

2. Programming Flow Control for Conditional and Repetitive Commands

17

FOR statements – Repetitive commands (2)

Using FOR statements for computing s = xii=1

20

∑

2. Programming Flow Control for Conditional and Repetitive Commands

18

WHILE statements – Repetitive commands (1) 2. Programming Flow Control for Conditional and Repetitive Commands

19

Using WHILE statements for computing s = xii=1

20

∑

WHILE statements – Repetitive commands (2) 2. Programming Flow Control for Conditional and Repetitive Commands

20

Using WHILE statements for computing s = xii=1

20

∑

Logical expressions – Description of conditions

Logical operators assess conditions and return Boolean values (1/0 or True/False).

2. Programming Flow Control for Conditional and Repetitive Commands

21

Direct Substitution Method (Fixed-Point Method)

3. Nonlinear Equation System (1) Initialization:

Reformulate equation f (x) = 0 as x=g(x).Give an initial guess x0 and convergence tolerance ε , set k = 0.

(2) Iteration k+1: Compute the next candidate solution xk+1 = g(xk ).

(3) Termination Check: If | xk+1 − xk |≤ ε ,

return xk+1 as the numerical solution; Otherwise,

k = k +1, and go to (2).

22

Direct substitution – Implementation via script or function

To start the solution procedure, type in prompt line:

To start the solution procedure, type in prompt line:

23 3. Example Implementations

Newton’s method 3. Example Implementations (1) Initialization: Give a guess x0 and tolerances ε,εa,εr, set k = 0.

(2) Iteration k+1: Compute f(xk ), Jk, and xk+1 = xk − Jk

−1f(xk ).

(3) Termination Check: If || f(xk ) ||≤ ε or || xk+1 − xk ||≤|| xk ||εr +εa, return xk+1 as the numerical solution; Otherwise, k = k +1, and go to (2).

24

Algorithm initial conditions

Solution information

Newton’s method – Example of function and Jacobian evaluation

f(x) =absin(x1)+ cos(x2 )

a ln(x1)+ eb2 (x1+x2 )

⎛

⎝⎜⎜

⎞

⎠⎟⎟,

where a = 1, b = 0.5.

J =abcos(x1) −sin(x2 )

ax1

+ b2eb2 (x1+x2 ) b2eb

2 (x1+x2 )

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

, where a = 1, b = 0.5.

3. Example Implementations

25

Newton’s method – MATLAB command for execution

Note: MATLAB m-files “Newton.m”, “fun.m”, “Jfun.m” need to be included in the current folder for execution of the command!

If initial guess x0 =x1,0

x2,0

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 0.2

2⎛⎝⎜

⎞⎠⎟

, tolerances: ε = 10−6, εa = 10−6, ε r = 10−6,

then the MATLAB command is:

3. Example Implementations

26

Newton’s method – Alternative implementation (1) Initialization: Give a guess x0 and tolerances ε,εa,εr, set k = 0.

(2) Iteration k+1: Compute f(xk ), Jk, and xk+1 = xk − Jk

−1f(xk ).

(3) Termination Check: If || f(xk ) ||≤ ε or || xk+1 − xk ||≤|| xk ||εr +εa, return xk+1 as the numerical solution; Otherwise, k = k +1, and go to (2).

3. Example Implementations

27

Explicit Euler - From algorithm to a MATLAB function

(1) Initialization: Give t0 (= 0), t f , Δt, set k = 0.

(2) Step k +1: m(k) = f(x(k)), x(k +1) = x(k)+m(k)Δt

(3) Termination Check : If k +1= n(= t f / Δt), terminate and return [x(1) ... x(n)]T ; Otherwise, set k = k +1, go to (2).

x = f(x), x(0) = x0

3. Example Implementations

28

Explicit Euler - Example of search direction evaluation

x1 = absin(x1)+ cos(x2 ),

x2 = a ln(x1)+ eb2 (x1+x2 ),

where a = 1, b = 0.5.

3. Example Implementations

29

Explicit Euler - MATLAB command for execution

Note: MATLAB m-files “Euler.m”, “odefun.m” need to be included in the current folder for execution of the command!

If time period [t0, t f ]= [0,1], time step Δt = 0.1,

initial condition x(0) =x1(0)x2 (0)

⎛

⎝⎜

⎞

⎠⎟ =

11

⎛⎝⎜

⎞⎠⎟

,

then the MATLAB command is:

3. Example Implementations

30

Explicit Euler - Alternative implementation

(1) Initialization: Give t0 (= 0), t f , Δt, set k = 0.

(2) Step k +1: m(k +1) = f(x(k)), x(k +1) = x(k)+m(k)Δt

(3) Termination Check : If k +1= n(= t f / Δt), terminate and return [x(1) ... x(n)]T ; Otherwise, set k = k +1, go to (2).

x = f(x), x(0) = x0

3. Example Implementations

31