MathCAD Data

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.

Data in tables

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

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

Inserting the text file

Inserting the text file Changes in the text file location

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.

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

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.

Data analysis and optimisation Approximation

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)

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.

types of approximation Interpolating approximation Uniform approximation Square-mean approximation

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.

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

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.

Square-mean approximation Condition for discreet set of arguments:

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

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.

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

Other MathCAD tools for approximation

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

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)

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

Can be integrate Can be derivate

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