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MathCAD Data

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

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Data in tables

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

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

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Inserting the text file

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Changes in the text file location Changes in the text file location

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

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Inserting the Excel sheets To insert worksheet: To insert worksheet: Menu: Insert/Component/Excel Menu: Insert/Component/Excel Browse file or create new 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 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) 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) In placeholder(s) type variable(s) Number of outputs/inputs and range of cells can be edited in properties of insertion Number of outputs/inputs and range of cells can be edited in properties of insertion

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

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Data analysis and optimisation Approximation

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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) 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)

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application Simplifying calculations when original function f(x) is defined by complicated expression 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. 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.

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

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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. 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.

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

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

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

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Function: 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. 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. function calculates the sum of square deviations as a function of parameters. p1, p2 – parameters of approximating function p1, p2 – parameters of approximating function Square-mean approximation in MathCAD

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Approximating algorithm: 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

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

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Other MathCAD tools for approximation

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

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regress Approximation by polynomial function Approximation by polynomial function Syntax: Z:= regress(X, Y, s) Syntax: Z:= regress(X, Y, s) X – vector of independent values from data set X – vector of independent values from data set Y - vector of dependent values from data set Y - vector of dependent values from data set s – polynomial degree s – polynomial degree Z – result: vector, s+1 last elements are parameters of polynomial (starting from x 0 parameter) Z – result: vector, s+1 last elements are parameters of polynomial (starting from x 0 parameter)

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Linear, cubic, polynomial spline interpolating approximation Approximation by linear (cubic etc.) spline function Approximation by linear (cubic etc.) spline function Syntax: Z:=lspline(X, Y) (cspline, pspline) Syntax: Z:=lspline(X, Y) (cspline, pspline) X – vector of independent values from data set X – vector of independent values from data set Y - vector of dependent 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 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 Z – result: vector of parameters of cubic spline function

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Can be derivate Can be integrate

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

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