PH15010

1. PH15010 Laboratory Techniques - An Introduction to MATHCAD

2. Introduction • Review of last week • Vector and Matrix Maths • Statistics • Solving simultaneous equations with matrices

3. Review of Last Week • Processing experimental data with arrays • Adding Error bars to graphs • Pre-Processing Data • Convert l to f • Logarithms for power law • Extract slope & intercept from data

4. Functions Reviewed • Take one or more inputs (parameters) and return a value or values • From f(x) toolbar if you don’t know the name • Afunction(parameters)= • Avariable:=Afunction(p1,p2,p3…) • Use the help system/f(x) dialog

5. Matrix & Vector Maths • Used for: • Processing experimental data • Statistics • Resolving forces etc in 2D, 3D • Co-ordinate transformations • Simultaneous Equations • Applying transforms to data

6. Creating Arrays in MathCAD • 2 ways of entering arrays • Input Table (already encountered) • Spreadsheet type entry • Expand to fit data available • Insert Matrix Dialog box • Array of placeholders • Good if you know how many elements in matrix/vector

7. Creating an array with Insert|Matrix… • Same for defining ordinary variable • Follow : with • [:::] from toolbar • <ctrl-m> • Insert|Matrix • Dialog box for rows & columns

8. Empty 3 – element vector Placeholders to fill in Can use units Creating arrays with Insert|Matrix…

9. Matrix Maths #1

10. Matrix Maths #2(Symbolically) • Matrix Multiplication • Determinant • Addition of constant

11. Matrix & Vector Maths • Add, Multiply & Divide matrices/vectors • Must have correct numbers of rows & columns

12. Addressing Arrays with Indices • Index addressing allows access to individual elements of array • Examine & Set elements • Indices start at 0 • Indices use [ key • 2D matrices use two indices for row, column selection

13. Indices #1 Vector Example • Simple 2 element vector • Examine elements • Set element 0 • Result of setting

14. Indices #2Matrix Example • Sample matrix • Examine elements • Subscript too big • Set element • Result

15. Accessing Parts Of Arrays • Index addressing – single element  • Column Extract operator M<>  • Submatrix() function • Transposing matrices

16. Matrix to operate on Rows Columns The submatrix() function Define a matrix Use submatrix() to extract part of the array

17. Transpose Rows & Columns • Uses MT operator • From matrix toolbar

18. Other matrix functions1. Information • Return information about: • Size of matrix • rows(M), cols(M), last(V), length(V) • Contents of matrix • min(A), max(A) • V

19. Other matrix functions2. Manipulation • Create new arrays • Put 2 arrays together • stack(A1,A2) • augment(A1,A2) • Sorting • sort(V), reverse(A) • rsort(M,r), csort(M,c)

20. Statistical Functionson Vectors #1 • From f(x) dialog box under “Statistics” • Averaging • mean() ‘Average value’ • median() ‘Half way item’ • mode() ‘Most common value’

21. Statistical Functionson Vectors #2 • Variance & Standard Deviation • 2 forms of each • Analysis of sample • Stdev(), Var() • Analysis of whole population • stdev(), var() • Note capitalisation

22. Vectorise Operator #1 • From matrix toolbar • Forces evaluation on element by element basis • Overrides normal matrix maths • ‘dot’ product • Determinant |M| • Powers Mx

23. Dot product With vectorise operator Vectorise Operator #2

24. Range Variables • Take on a series of values • Define using semicolon [;]

25. Range Variables #2 • Restrictions on where used • ‘illegal context’ error message • Useful for filling arrays

26. Filling Arrays using RVs • Define RV • Use RV in array index & expression

27. Processing Arrays using RVs • Use RV in expressions to create output array from input array

28. Range Variables with different step sizes • Can create RVs with different step sizes. • Enter first 2 values separated by comma [,] before typing [;]

29. Summation #1 • Summation operator on matrix palette • Sum of all elements in a vector

30. Summation #2 • Operators on Calculus palette • Do summation on any expression • 2 forms • Plain => defines local RV • Range Variable => uses existing RV

31. Sum of first 5 integers Sum of elements 20-30 of vector Counts Summation #3

32. 2 terms 4 terms 9 terms Summation #4Reciprocals of factorials

33. Data Files • Used Input table to put data into MathCAD by hand. • Can use Import on input table to get from a file (makes a copy) • Select table & right-click • Also can link to data file with File Read Component (preferred)

34. Reading Data Files #1 • File Read Component • Reads from datafile  array • Text files • Numbers separated by comma or tab • Other formats supported

35. Reading Data Files #2 • Insert|Component…|File Read or Write • Wizard: • (Select file type) • Browse for file • Finish • Give name for array

36. Reading Data Files #3 • Example

37. This weeks worksheet • Arrays & Indices • Columns (Revision) • Submatrix() function • Other array functions • Reading data from file