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PSOD

PSOD. Lecture 2. MathCAD – vectors and matrix. MathCAD – vectors and matrix. Matrix operations Multiply by constant Matrix transpose [ctrl]+[1] Inverse [^][-][1] Matrix multiplying Determinant. MathCAD – vectors and matrix.

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  1. PSOD Lecture 2

  2. MathCAD – vectors and matrix

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

  4. 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)

  5. MathCAD – vectors and matrix • Calculations of dot product and cross product of vectors

  6. 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 Constrain: function arguments have to be integer

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

  8. 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]) Boundariescan be the real type numbers. (def. –5,5) Grids have to be integer type numbers (def. 20)

  9. MathCAD 3D graphs

  10. MathCAD 3D graphs - formating

  11. MathCAD 3D graphs – formatting: fill options

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

  13. MathCAD 3D graphs – formatting: line options

  14. MathCAD 3D graphs – formatting: Lighting

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

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

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

  18. 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)=

  19. MathCAD equation solving • Given-Find – solving methods • Linear (function of type c0x0 + c1x1 +...+ 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.

  20. 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

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

  22. 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

  23. 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-1Band show result: x= Or • Use the procedure LSOLVE: lsolve(A,B)=

  24. MathCAD, the system of equations solving

  25. 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,...)=

  26. MathCAD, the system of equations solving

  27. 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.

  28. 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 givenstarting 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

  29. 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

  30. 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

  31. MathCAD differential equations • Result is matrix (table) of two columns: first contain independent values second dependent ones • To show result as a plot: R<1>@R<0>

  32. MathCAD differential equations

  33. 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)

  34. 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<1>,R<2>@ R<0>

  35. MathCAD differential equations

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

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

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

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