An Introduction to MathCAD Finding Solutions and Symbolic Maths

2 MathCAD #4  dpl 1999 Finding Solutions u Finding roots of an equation u Finding roots of a polynomial u Solving systems of equations

3 MathCAD #4  dpl 1999 Finding roots of an equation u Equations of the form f(x) = 0 u One equation in one unknown u Define guess value for x u Use root(f(x),x) function to find root u plot function to get initial guess value(s)

4 MathCAD #4  dpl 1999 What if MathCAD can’t find roots u Expression has no roots u roots are far away from initial guess u local maxima or minima between guess and root u discontinuities between guess and root u complex root u Plot function to narrow in on solution

5 MathCAD #4  dpl 1999 Finding roots of a polynomial u Represent coefficients of polynomial at vector u Use polyroots to find roots u Doesn’t need initial guess u Finds all roots simultaneously u Finds real and complex roots

6 MathCAD #4  dpl 1999 Solving systems of equations #1 u Solve up to 50 simultaneous equations in 50 unknowns u Equations not restricted to linear equations u Allows equations and inequalities

7 MathCAD #4  dpl 1999 Solving systems of equations #2 u Assign guess values to all variables with := u Given starts solve block u Solve block contains system of equations u Equations in solve block use: – = – < < – > > –  – 

8 MathCAD #4  dpl 1999 Solving systems of equations #3 u Solve block terminates with find(a1,a2,a3,a4...) u returns scalar for one argument u returns vector of solutions for more than one argument u Only returns one solution u Use inequalities to force other solutions

9 MathCAD #4  dpl 1999 Symbolic algebra u Simplify expressions u Derive formulae u Solve equations symbolically u Get exact answers to integrals etc u Subset of Maple symbolic processor

10 MathCAD #4  dpl 1999 Use of symbolic processor u Type equations using = (boolean equals) u Select equation/variable to process u Options on symbolic menu change depending on what is selected u Derivation format changes result format

11 MathCAD #4  dpl 1999 Simplifying expressions #1 u Use up-arrow/space bar to select part of expression u Symbolic|Simplify u Result appears as defined in derivation format –Horizontal –Vertical –With or without comments

12 MathCAD #4  dpl 1999 Simplifying expressions #2 u Processor will simplify polynomials by collecting powers u Understands math & trig identities u Will simplify numeric operations & fractions

13 MathCAD #4  dpl 1999 Expansion / Factoring of expressions u Select expression u Symbolic|Expand or Symbolic|Factor u Factoring will also factor integer numbers

14 MathCAD #4  dpl 1999 Re-arranging equations to solve for variable u Use = to define equation u select variable you wish to solve for u Symbolic|Solve for Variable u Need to comment after result to remember which variable you have solved for !!

15 MathCAD #4  dpl 1999 Use of symbolic processor in problems u Define problem in terms of known equations u Use symbolic processor to solve for desired result u Copy resultant expression u Define variables u Paste in solution u Evaluate for numeric solution

16 MathCAD #4  dpl 1999 Why use symbolic maths ? u Gives general solution u Numeric solution only solves for one set of conditions u Symbolic solution shows how solution varies with inputs u Determine which terms are important

17 MathCAD #4  dpl 1999 Today’s practical session u Use of root() & polyroots() in conjunction with plotting to solve simple equations u Using a solve block to solve simple systems of equations u Using the symbol processor to: –simplify expressions –expand & factorise expressions –solve equations u Use symbolic processing for a real world problem

18 MathCAD #4  dpl 1999 Next week’s assesment u Morning lecture: revision session u Practical session 4-6 u Exam conditions u Graded series of exercises u Complete worksheet & u Marked on: –layout & commenting –understanding key points –thinking for yourself