1. PSOD
Lecture 2

3. MathCAD – vectors and matrix

4. MathCAD – vectors and matrix
Matrix operations
Multiply by constant
Matrix transpose [ctrl]+[1]
Inverse [^][-][1]
Matrix multiplying
Determinant

5. MathCAD – vectors and matrix
To read the matrix elements Ar, k: key [[] r- row nr, k – column nr
e.g. element A1,1 keys: [A][[][1][,][1][=]
To chose matrix column
First column A( A<0>): keys [A][ctrl]+[6][0]
Default first column number is 0, (to change : Math/Options/Array Origin)

6. MathCAD – vectors and matrix
Calculations of dot product and cross product of vectors

7. MathCAD – vectors and matrix
Special definition of matrix elements as a function of row-column number Mi,j=f(i,j)
E.g. Value of element is equal to product of column and row number
Argumenty funkcji są liczbami całkowitymi nieujemnymi od zera do ilości wierszy i/lub kolumn
Constrain: function arguments have to be integer

8. 3D graphs of function on the base of matrix : [ctrl]+[2] [M]
MathCAD 3D graphs
3D graphs of function on the base of matrix : [ctrl]+[2] [M]
M – matrix defined earlier

9. 3D Graphs of function of real type arguments
MathCAD 3D graphs
3D Graphs of function of real type arguments
Using procedure: CreateMesh(function, lb_v1, ub_v1, lb_v2, ub_v2, v1grid, v2grid)
Assign result to variable
Plot of the variable similarly to plot of matrix ([ctrl]+[2])
Boundaries can be the real type numbers. (def. –5,5)
Grids have to be integer type numbers (def. 20)

10. MathCAD 3D graphs

11. MathCAD 3D graphs - formating

12. MathCAD 3D graphs – formatting: fill options

13. MathCAD 3D graphs – formatting: fill options
Contours colour filled

14. MathCAD 3D graphs – formatting: line options

15. MathCAD 3D graphs – formatting: Lighting
Oświetlać - lighten

16. MathCAD 3D graphs – formatting: Fog and perspective

17. MathCAD 3D graphs – formatting: Backplane and Grids

18. Predefined constants e = 2,718 – natural logarithm base
g = 9,81 m/s2 – acceleration of gravity
 = 3,142 – circle perimeter/diameter ratio

19. MathCAD equation solving
Single equation (one unknown value)
Given-Find method
Input start point of variable
Type "Given"
Type equation with using [=] ([ctrl]+[=])
Type Find(variable)=
Variable that satisfy an equation

20. MathCAD equation solving
Given-Find – solving methods
Linear (function of type c0x0 + c1x cnxn) –starting point do not affects on results, it only defines size of matrix/vector of the solution.
Nonlinear – according to nonlinear equation. Obtained result could depend on starting point. Available methods:
Conjugate Gradient
Quasi – Newton
Levenberg-Marquardt
Quadratic
The choice of method is automatic by default. User can choose method from the pop-up menu over word Find.

21. MathCAD equation solving
Single equation (one unknown value)
Root procedure: Root(function, variable, low_limit, up_limit)=
Values of function at the bounds must have different signs or

22. MathCAD equation solving
Single equation (one unknown value)
Root procedure methods:
Secant method
Mueller method (2nd order polynomial) y1 x2 x3 x5 x4 x1 y3 y2

23. MathCAD equation solving
Single equation (one unknown value)
Special procedure: polyroots for the polynomials. Argument of procedure is a vector of polynomial coefficients (a0, a1...). The result is a vector too.
Methods:
Laguerre's method
companion matrix

24. MathCAD, the system of equations solving
The system of linear equations
Solving on the base of matrix toolbar:
Prepare square matrix of equations coefficients (A) and vector of free terms (B)
Do the operation x:=A-1B and show result: x= Or
Use the procedure LSOLVE: lsolve(A,B)=

25. MathCAD, the system of equations solving

26. MathCAD, the system of equations solving
The system of nonlinear equation
Can be solved using given-find method
Assign starting values to variables
Type Given
Type the equations using = sign (bold)
Type Find(var1, var2,...)=

27. MathCAD, the system of equations solving

28. Ordinary differential equations solving
Numerical methods:
Gives only values not function
Engineer usually needs values
There is no need to make complicated transformations (e.g. variables separation)
Basic method implemented in MathCAD is Runge-Kutta 4th order method.

29. Ordinary differential equations solving
Numerical methods principle
Calculation involve bounded range of independent variable only
Every point is being calculated on the base of one or few points calculated before or given starting points.
Independent variable is calculated using step:
xi+1 = x i + h = xi+Dx
Dependent value is calculated according to the method
yi+1 = y i +Dy= y i +Ki

30. Ordinary differential equations solving
Runge-Kutta 4th order method principles:
New point of the integral is calculated on the base of one point (given/calculated earlier) and 4 intermediate values

31. MathCAD differential equations
Single, first order differential equation
Assign the initial value of dependent variable (optionally)
Define the derivative function
Assign to the new variable the integrating function rkfixed:
R:=rkfixed(init_v, low_bound, up_bound, num_seg, function)
Initial condition

32. MathCAD differential equations
Result is matrix (table) of two columns: first contain independent values second dependent ones
To show result as a plot:

33. MathCAD differential equations

34. MathCAD differential equations
System of first order differential equations
Assign the vector of initial conditions of dependent variables (starting vector)
Define the vector function of derivatives (right-hand sides of equations)
Assign to the variable function rkfixed:
R:=rkfixed(init_vect, low_bound, up_bound, num_seg, function)

35. MathCAD differential equations
Result is matrix (table) of three columns: first contain independent values, 2nd first dependent values, third second ones :
Results as a plot: R<0>

36. MathCAD differential equations

37. MathCAD differential equations
Single second order equation
Transform the second order equation to the system of two first order equations:
Initial condition

38. MathCAD differential equations
Example:
Solve the second order differential equation (calculate: values of function and its first derivatives) given by equation:
While y=10 and y’=-1 for x=0
In the range of x=<0,1>

39. MathCAD differential equations
Starting vector
Vectoral function
System of equations