MathCAD

Data

Data in tables

Tables are analogous to matrixThe numbers of columns and rows can be dynamically changed (in contrast to matrix)To enter table:Menu: Insert/Component/Input Table In placeholder type variable name which will be

assigned to table In cells type the valuesEach 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 files

The most popular file formats accepted by MathCAD:Text filesExcel worksheetsMATLAB

To insert text file containing data: Menu: Insert/Component/File Read or WriteChose file formatBrowse 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 fileChanges 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)

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 prosperities of insertion

MathCAD files as data source in MathCAD

MathCAD can use data included in other MathCAD filesAccess to data is possible after embedding MathCAD file:menu: Insert/References, Brows file on disc or type file addressBelow the insertion all data, definitions,

assignment from inserted file are valid in the present document

Problem: indexed 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)

aplicationSimplifying 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 they are 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.

min2 dxxfxFEb

a

Condition for discreet set of arguments:

min2

iii xfxFE

Square-mean approximation

Function:minimize(function, p1, p2,...)

can be used to determine parameters of approximating function minimizing the sum of mean square deviations between values given in the table and calculated from the function. function calculates the sum of mean square

deviations as a function of parameters.p1, p2 – parameters of approximating function

Square-mean approximation in MathCAD

Approximating algorithm:1. Insert data to be approximate

2. Build the approximating function

3. Create a counting variable with values from 0 to number of data minus 1

4. Build function that calculates sum of square of deviations with parameters of approximating function as variables

5. Assign starting values of parameters

6. Use the function minimize.

Square-mean approximation in MathCAD

Advantageous of minimize function:simpleexplicitsuitable for any approximating functioncan 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 setY - vector of dependent values from data setc0 – starting vector of searched parametersF – vector function of independent variable and

vector c, consists of approximating function and its derivatives on parameters

c - vector of searched parameters

regressApproximation 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

Linear, cubic Spline

Approximation by linear (cubic etc.) spline function Syntax: Z:=lspline(X, Y) (cspline)

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 derivate

Can be integrate

Interpreting functionOperates on vectors obtained from regress and l(c)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