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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 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 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.
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.
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
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
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
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