1. MathCAD
Data

2. Data in tables Tables are analogous to matrix
The numbers of columns and rows can be dynamically changed (in contrast to matrix)
To enter table:
Menu: Insert/Data/Table
In placeholder type variable name which will be assigned to table
In cells type the values
Each rows must contains the same number of data. If data are missing the value ‘0’ will be assigned
Access to data in table are matrix like.

3. Data in tables

4. Data in tables
Row appears in matrix when only 1 data is inserted into the cell:
Matrix size = specified cell in the lowest row and in last column
Unfilled cells contains 0
Once specified cell can not be unspecified!
To overcome problem: create new matrix with correct number of rows i and columns j using

5. Data in files The most popular file formats accepted by MathCAD:
Text files
Excel worksheets
MATLAB
To insert text file containing data:
Menu: Insert/Data/File Input
Chose file format
Browse to the file location
In the appeared placeholder type variable name that will be assigned to the contents of file

6. Inserting the text file

7. Inserting the text file
Changes in the text file location

8. Inserting the Excel sheets
A range of Excel cells can be inserted into the MathCAD
There can be more then one range in single insertion
One variable is being assigned to one range
All variables forms a vector
Cells can contain numbers as well as text (in contrast to table and text files, ver. 2001)
Worksheets can be edited (double-click) using all Excel functions (object embedded). Excel has to be installed in system.

9. Inserting the Excel sheets
To insert worksheet:
Menu: Insert/Component/Excel
Browse file or create new
Choose number of ranges for input and output (relatively to Excel worksheet). If no data have to be inserted into the Excel worksheet type inputs number 0
Type ranges corresponding to outputs – e.g. A1:B10 (if sheet name is different from Sheet1 type sheet name – e.g. Sheet4!A1:B10)
In placeholder(s) type variable(s)
Number of outputs/inputs and range of cells can be edited in properties of insertion

11. MathCAD files as data source in MathCAD
MathCAD can use data included in other MathCAD files
Access to data is possible after embedding MathCAD file:
menu: Insert/References,
Brows file on disc or type file address
Below the insertion all data, definitions, assignment from inserted file are valid in the present document
Problem: matrix/vector variables.

13. Data analysis and optimisation
Approximation

14. definition
Approximation is a part of numerical analysis. It is concerned with how functions f(x) can be best approximated (fitted) with another functions F(x)

15. application
Simplifying calculations when original function f(x) is defined by complicated expression
Creation of continuous dependency when function f(x) is ascribed on discrete set of arguments. For known form of approximating function only values of function parameters giving the best approximation are to determine.

16. types of approximation
Interpolating approximation
Uniform approximation
Square-mean approximation

17. Interpolating approximation
Needs to satisfy condition: function given f(x) and approximating function F(x) have the same values on the set of nodes and (sometimes) the same values of derivatives (if given) too.

18. Uniform approximation
Function F(x) approximating function f(x) in the range [a,b], that maximal residuum reaches minimum

19. Square-mean approximation
Approximating function is determined by the use of condition :
Geometrically condition means: The area between curves representing functions have to reach minimum.

20. Square-mean approximation
Condition for discreet set of arguments:

21. Square-mean approximation in MathCAD
Function:
minimize(function, p1, p2,...)
can be used to determine parameters of approximating function minimizing the sum of square deviations between values given in the table and calculated from the function.
function calculates the sum of square deviations as a function of parameters.
p1, p2 – parameters of approximating function

22. Square-mean approximation in MathCAD
Approximating algorithm:
Insert data to be approximate
Build the approximating function
Create a counting variable with values from 0 to number of data minus 1
Build function that calculates sum of square of deviations with parameters of approximating function as variables
Assign starting values of parameters
Use the function minimize.

24. Advantageous of minimize function:
simple
explicit
suitable for any approximating function
can be used in optimisation problem solving

25. Other MathCAD tools for approximation

26. genfit Syntax: c:=genfit(X, Y, c0, F)
X – vector of independent values from data set
Y - vector of dependent values from data set
c0 – starting vector of searched parameters
F – vector function of independent variable and vector c, consists of approximating function and its derivatives on parameters
c - vector of searched parameters

28. regress Approximation by polynomial function
Syntax: Z:= regress(X, Y, s)
X – vector of independent values from data set
Y - vector of dependent values from data set
s – polynomial degree
Z – result: vector, s+1 last elements are parameters of polynomial (starting from x0 parameter)

30. Linear, cubic, polynomial spline interpolating approximation
Approximation by linear (cubic etc.) spline function
Syntax: Z:=lspline(X, Y) (cspline, pspline)
X – vector of independent values from data set
Y - vector of dependent values from data set
Data in set has to be sorted! Manually or by use function csort: W:=csort(W,i), W – matrix of data, i – nr of ordering column
Z – result: vector of parameters of cubic spline function

31. Can be integrate
Can be derivate

32. Interpreting function
Operates on vectors obtained from regress and l(c,p)spline functions
Building the continuous approximating function on the base of determined parameters
Syntax: F(x):=interp(Z, X, Y, x)
Z – vector given by approximating function
X – vector of independent values from data set
Y - vector of dependent values from data set
x – independent values
Interpreting function is implicit but can be derivated and integrated