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/TableIn placeholder type variable name which will be assigned to tableIn cells type the valuesEach rows must contains the same number of data. If data are missing the value ‘0’ will be assignedAccess to data in table are matrix like.
4 Data in tablesRow appears in matrix when only 1 data is inserted into the cell:Matrix size = specified cell in the lowest row and in last columnUnfilled cells contains 0Once 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 filesExcel worksheetsMATLABTo insert text file containing data:Menu: Insert/Data/File InputChose file formatBrowse to the file locationIn the appeared placeholder type variable name that will be assigned to the contents of 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 MathCADThere can be more then one range in single insertionOne variable is being assigned to one rangeAll variables forms a vectorCells 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/ExcelBrowse file or create newChoose 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 0Type 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 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 documentProblem: matrix/vector variables.
14 definitionApproximation 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 applicationSimplifying calculations when original function f(x) is defined by complicated expressionCreation 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 approximationUniform approximationSquare-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 approximateBuild the approximating functionCreate a counting variable with values from 0 to number of data minus 1Build function that calculates sum of square of deviations with parameters of approximating function as variablesAssign starting values of parametersUse the function minimize.
26 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 parametersc - vector of searched parameters
28 regress Approximation by polynomial function Syntax: Z:= regress(X, Y, s)X – vector of independent values from data setY - vector of dependent values from data sets – polynomial degreeZ – 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 functionSyntax: Z:=lspline(X, Y) (cspline, pspline)X – vector of independent values from data setY - vector of dependent values from data setData in set has to be sorted! Manually or by use function csort: W:=csort(W,i), W – matrix of data, i – nr of ordering columnZ – result: vector of parameters of cubic spline function
32 Interpreting function Operates on vectors obtained from regress and l(c,p)spline functionsBuilding the continuous approximating function on the base of determined parametersSyntax: F(x):=interp(Z, X, Y, x)Z – vector given by approximating functionX – vector of independent values from data setY - vector of dependent values from data setx – independent valuesInterpreting function is implicit but can be derivated and integrated