PmU!. @"Data.app1@ .12 37@F;@O&Y @9 AWell behaved exampleSecant convergenceWegstein AccelerationNewton convergenceRegula falsi convergenceRidder's convergenceSteinman convergenceDifficult exampleSecant non-convergenceNewton non-convergenceRegula falsi convergenceRidder's convergenceCD)^  1. Numerical AnalysisSolve for Log Reciprocal Equationlu@1. Numerical AnalysisSolve for Power Reciprical Square eqnyp1. Numerical AnalysisSolve for simple linear equationyP73. Chemical PropertiesNormal Boiling Point, Z & omegaMC3. Chemical PropertiesThermal Conductivity, Gas, kGSPG3. Chemical PropertiesVapour Pressure definitionP73. Chemical PropertiesNormal Boiling Point, Z & omegaMC3. Chemical PropertiesThermal Conductivity, Gas, kGSPG3. Chemical PropertiesVapour Pressure definitionP73. Chemical PropertiesNormal Boiling Point, Z & omegaMC3. Chemical PropertiesThermal Conductivity, Gas, kGSPG3. Chemical PropertiesVapour Pressure definitionP73. Chemical PropertiesNormal Boiling Point, Z & omegaMC3. Chemical PropertiesThermal Conductivity, Gas, kGSPG3. Chemical PropertiesVapour Pressure definitionP73. Chemical PropertiesNormal Boiling Point, Z & omegaMC3. Chemical PropertiesThermal Conductivity, Gas, kGSPG3. Chemical PropertiesVapour Pressure definitionP73. Chemical PropertiesNormal Boiling Point, Z & omegaMC3. Chemical PropertiesThermal Conductivity, Gas, kGSPG3. Chemical PropertiesVapour Pressure definitionP73. Chemical PropertiesNormal Boiling Point, Z & omegaMC3. Chemical PropertiesThermal Conductivity, Gas, kGSPG3. Chemical PropertiesVapour Pressure definitionP73. Chemical PropertiesNormal Boiling Point, Z & omegaMC3. Chemical PropertiesThermal Conductivity, Gas, kGSPG3. Chemical PropertiesVapour Pressure definitionP73. Chemical PropertiesNormal Boiling Point, Z & omegaMC3. Chemical PropertiesThermal Conductivity, Gas, kGSPG3. Chemical PropertiesVapour Pressure definitionP73. Chemical PropertiesNormal Boiling Point, Z & omegaMC3. Chemical PropertiesThermal Conductivity, Gas, kGSPG3. Chemical PropertiesVapour Pressure definitionP73. Chemical PropertiesNormal Boiling Point, Z & omegaMC3. Chemical PropertiesThermal Conductivity, Gas, kGSPG3. Chemical PropertiesVapour Pressure definitionP73. Chemical PropertiesNormal Boiling Point, Z & omegaMC3. Chemical PropertiesThermal Conductivity, Gas, kGSPG3. Chemical PropertiesVapour Pressure definitionP73. Chemical PropertiesNormal Boiling Point, Z & omegaMC3. Chemical PropertiesThermal Conductivity, Gas, kGSPG3. Chemical PropertiesVapour Pressure definitionP73. Chemical PropertiesNormal Boiling Point, Z & omegaMC3. Chemical PropertiesThermal Conductivity, Gas, kGSPG3. Chemical PropertiesVapour Pressure definitionA9"Times New Roman$"Times New Roman%D%M%Y%/0%1%/1%2%/2%3%/3 _}`"Times New Roman77\cef dtDesign Tools for Chemical Engineers1. Numerical Analysis\cefd&1999/10/21, Page 5.AAd(cxn)/dx=ncxn-1d(uv)/dx=udv/dx+vdu/dxd(uvw)/dx=uvdwdx+uwdv/dx+ vwdu/dxd((u/v)/dx=[vdu/dx-udv/dx]/v2dy/dx=(dy/du)(du/dx)d(sin u)/dx=(cos u)du,dxd(cos u)//dx=(-sin u)du/dxd(tan u)/dx=(secu)du/dxd(ln u)/dx=1/udu/dxd(eu)/dx=eudu/dxA    B7!-1.NA.DTCE Overview 0harmedj@freeuk.comPsion series 5 & 7 palmtopsN,cGeneral description of this module, its use, scope and applicability.Read meharmedj@freeuk.com-H1.NA.06"Solve General Non-linear Equationsconvergence methodsN,^Newton approximation secant regula falsi Steinman solving equations Ridder's method bracketedlDetermining the solution of an implicit equation with one variable by convergent, iterative means.Read meC _T 4 @& 'HC  'HC @<&aWE Sketch!!(!?????? 0<????????< 0??0???????????0???? 3???????????? ??<0? ????? ??  ???? ? ?? ??? ???? ?0 ?  ? ?? ?<? ?? ?????<???00  }&Paint.app}F*D Q@iTable1"ColA9 (ColB9ColA10 (ColB10ColA12 2ColB12ColA13 2ColB13ColA14 2ColB14ColA16 ColA18ColB18ColA20ColB20ColA21ColB21 Index1ColA9(ColA10 B|@1.NA.062Roots of Quadra}@1.NA.063Roots of Cubic ~@1.NA.064Roots of Quarti1.NA.07Optimisation{@1.NA.08Solve Linear Eq@1.NA.09Regression, Mul1.NA.092Regression, All1.NA.0ANumerical ApproB1.NA.01Solve for coeff@1.NA.06Solve General N B 1.NA.02Interpolation, 1.NA.02Interpolation, 1.NA.03Differentiation1.NA.03Integration, ge1.NA.04Numerical Integ1.NA.05Matrix arithmet1.NA.06Solve General NqA  Table1&Category: ( dTitle: (d*Reference: 2dInput: 2dLimits: 2dDate d&Keywords:d2Description:d"modules:dB@1.NA.DTCE Overview1.NA.01Solve for coeff1.NA.01Solve for coeff1.NA.01Solve for coeff1.NA.02Interpolation, 1.NA.02Interpolation, 1.NA.04Numerical Integ1.NA.05Matrix arithmetVAy=cfA+cfB*x+cfC*x+cfD*xln(y)=cfA-cfB/x+cfCln(x)+cfDy/(x) (Harlacker)ln(y)=cfA+cfB/x+cfCx+cfDx(1-x)ln(P/Pc)=cfAx+cfBx1,5+cfCx)+cfDx6 (inconsistant)(1-x)ln(P/Pc)=cfAx+cfBx1,5+cfCx+cfDx6 (consistant)ln(y)=cfA+cfB/x+cfCx+cfDxy=cfA+cfBx+cfCxcfDy=cfA(xcfB)/(1+cfC/x+cfD/x)y=cfA+cfB/(cfC+xcfD)C       OOOOOOOOO       O1! "Times New Roman$#O1! "Times New Roman$#O1! "Times New Roman$#O1! "Times New Roman$#O1! "Times New Roman$#O1! "Times New Roman$#O1! "Times New Roman$#O1! "Times New Roman$#O1! "Times New Roman$#:Ay=cfA+cfB*x+cfC*x+cfD*x e.g. liquid and gas heat capacity, gas thermal conductivtyln(y)=cfA-cfB/x+cfCln(x)+cfDy/(x) e.g. vapour pressureln(y)=cfA+cfB/x+cfCx+cfDx e.g. liquid viscosity(1-x)ln(P/Pc)=cfAx+cfBx1,5+cfCx)+cfDx6 e.g. vapour pressurey=cfA(xcfB)/(1+cfC/x+cfD/x) e.g. gas viscosity@jWord "Times New RomanN123LHP&Heading 1L"Times New RomanJ k&Heading 2L "Times New Roman x x k&Heading 3L "Times New Roman < <k.Bullet listO"Times New Roman Swiss2Home AddressO"Times New Roman:Postal AddressOsh!"Times New Roman! h! "Times New Roman%D%M%Y%/0%1%/1%2%/2%3%/3 _`a,h!"Times New Roman"Times New Roman77\c efd1. Numerical AnalysisSolve for Simple linear equation y=cfA+cfBx\cefdX1999/11/06, Page 1 of 0, harmedj@freeuk.com.ASolve for y=cfA+cfBxFor y1=cfA+cfBx 1 (1)y2=cfA+cfBx2 (2)Subtract (2) from (1) to get:-(y1-y2)=cfB(x1-x2)Thus:-cfB=(y1-y2)/(x1-x2)cfA=y1-cfBx1  !  "Word.app C}rCF*D3 QQJ\ Word   7"Times New RomanN123LHP0&Heading 1L"Times New RomanJ k&Heading 2L "Times New Roman x x k&Heading 3L "Times New Roman < <k.Bullet listO"Times New Roman Swiss2Home AddressO"Times New Roman:Postal AddressO>Numbered bulletORXRDh!"Times New Roman/h!."Times New Roman%D%M%Y%/0%1%/1%2%/2%3%/3 _`a,h!"Times New Roman"Times New Roman77n\cIefQd1. Numerical AnalysisGauss Elimination Solution of Linear Equations\cef:dX1999/11/07, Page 1 of 0, harmedj@freeuk.com.A=Analytical SolutionThe Gaussian Elimination technique uses equation multiplication and subtraction to solve for n variables with n unique linear equations. The matrix is reduced to singular form in a manner very similar to that taught in schools and is best ilustrated with a simple example.Suppose we have the following equations:-2x+3y=83x-y=1The first step is to deal with the solution of x in the first column.1 Choose 2x (top left) as the pivot element and reduce pivot point to unity by dividing the whole first equation by 2 .x+3/2y=4 (1) 3x-y=1 (2)2 Eliminate surrounding x-column elements to zero by proportional subtraction (subtract 3(1) from (2)) x+3/2y=4 0-11/2y=-113 Choose 2nd element in column 2 (11/2y) as the pivot point and reduce it to unity x+3/2y=4 0+y=24 Now repeat step 2 in the y column x+0=1 0+y=2Clearly the solutions of the equation are, x=1 and y=2. In matrix form the initial and final equations are written thus:-2 3 83 -1 1and1 0 10 1 2harmedj@freeuk.com(*Fvh T $z"Word.app CC F*D@ QC ;"Word.app CxvC F*D3 QfWq xSketch'bb(b,?????????   <????<0???03???<??????<?<< ?1111,,??+?+?+?????<??..?????? ?0%?"%?? ? ?? ? -0,03<=x<=+0,03"Times New Roman>-0,01<=x<=+0,01"Times New Roman @6?"Times New RomanF1/(1+x) = 1-x+x"Times New Roman6-0,1<=x<=+0,1"Times New Roman>-0,21<=x<=+0,21"Times New Roman ư"Times New Roman JSQRT(1+x) = 1+x/2"Times New Roman >-0,08<=x<=+0,10"Times New Roman >-0,24<=x<=+0,32"Times New Roman +uf"Times New Roman^SQRT(1+x) = 1+x/2-x/8"Times New Roman>-0,22<=x<=+0,27"Times New Roman>-0,44<=x<=+0,66"Times New Roman G p>"Times New Roman R1/SQRT(1+x) = 1-x/2"Times New Roman>-0,04<=x<=+0,05"Times New Roman>-0,15<=x<=+0,16"Times New Roman 8{?"Times New RomanfSQRT(1+x) = 1-x/2-3*x/8"Times New Roman>-0,14<=x<=+0,15"Times New Roman>-0,30<=x<=+0,32"Times New Roman 3H䔾"Times New Roman>(1+x) = 1+a*x"Times New Roman"Times New Roman"Times New Roman uO, =Y?"Times New Roman for a="Times New Roman 0"Times New Roman "Times New Roman "Times New Roman0? 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Numerical AnalysisSolving for 3 Coefficient equation, y=cfA+fBx1+cfCx2\cefdX1999/11/06, Page 1 of 0, harmedj@freeuk.com.A Analystical SolutionFor y1=cfA+cfBxb1+cfCxc1 (1)y2=cfA+cfBxb2+cfCxc2 (2)y3=cfA+cfBxb3+cfCxc3 (3)Subtract (2) from (1) to get:-(y1-y2)=cfB(xb1-xb2)+cfC(xc1-xc2)let zz2=xb1-xb2 zc2=(xc1-xc2)/zz2 zy2=(y1-y2)/zz2Thus:-zy2=cfB+cfCzc2 (4)and with the same operations on equations (1) and (3) we get:-zy3=cfB+cfCzc3 (5)Subtract (5) from (4) and summarising we get:-cfC=(zy2-zy3)/(zc2-zc3)cfB=zy2-cfCzc2cfA=y1-cfBxb1-cfCxc1wherezz2=xb1-xb2zz3=xb1-xb3zc2=(xc1-xc2)/zz2zc3=(xc1-xc3)/zz3zy2=(y1-y2)/zz2zy3=(y1-y3)/zz3If there are any problems please contact harmedj@freeuk.com'$   ?/      <       ;"Word.app CxvC F*D3 Qf@TWordS "Times New RomanN123LHP&Heading 1L"Times New RomanJ k&Heading 2L "Times New Roman x x k&Heading 3L "Times New Roman < <k.Bullet listO"Times New Roman Swiss2Home AddressO"Times New Roman:Postal AddressODh!"Times New Romanh!"Times New Roman%D%M%Y%/0%1%/1%2%/2%3%/3 _`a*h!"Times New Roman"Times New Roman77n\c efd`1. Numerical AnalysisSolve Nonlinear Equations\cefdT1999/11/06, Page 1 of 0harmedj@freeuk.com.Ay/Secant SearchTo solve for function(x)=0, we need use linear extrapolation between points (x1,y1) and (x2, y2) to the x axis. The similar triangles drawn above allow us to assign the simple ratios below:-(y1-y2)/(x1-x2)=y1/(x1-xnew)Rearranging we getx1-xnew=y1(x1-x2)/(y1-y2)orxnew=x1-y1(x1-x2)/(y1-y2)A variation of the secant method is the regula falsi method where the values of y1 and y2 are always chosen with opposite signs to ensure convergence.Newton's approximationIf the two points in the secant search method are very close to each other then:-x/y=(x1-x2)/(y1-y2)and thus the secant equation becomes:-xnew=x1-y1x/y.For very small x and y, y1=f(x) and y/x=dy/dx=f'(x) to getxnew=x-f(x)/f'(x)Newton's method is only applicable to situations where the function f(x) can be differentiated, and even then its use is only superior to the secant search when convergence is difficult and the second order differential can be determined to warn of divergence. Should divergence be imminent, a quadratic approximation could help via:-xnew=(f'(x)-f''(x)x)/f''(x)where discriminant=f'(x)-2f'(x)f''(x)+f''(x)xWegstein Accelerated SubstitutionWegstein used a modified secant to solve non-linear equations where direct substitution is commonly employed, such as with recycle streams.If xk=f(xk-1) and xk+1=f(xk) then a value q is calculated such:-q=[f(xk-1)-f(xk)]/[f(xk-1)-f(xk)-xk+1+xk]and the new iteration would be:-xk+1=qxk+(1-q)f(xk)Ridder's MethodThis method determines a new estimate x4, based on the results of three previous guesses x1, x2, & x3 such that x4 is bracketed between x1 and x2. The formula is:-x4=x3+(x3-x1)Thus x4 is guaranteed to lie between x1 and x2 and the convergence is quadratic.Steinman's approximationFirst locate an approximate root d1 and write the equation in the forma0xn=a1xn-1+a2xn-2+...+anThen, with x=d1,d2=(a1+2a2/d1+3a3/d1+4a4/d1+...)/(a0+a1/d1+2a3/d1+3a4/d14...)Choosing a techniqueThe iterative solutions on the Psion spreadsheet are not very efficient because calculations are slower than in a compiled computer program and because the function equation has to be retyped for each iteration. Furthermore, the number of iterations has to be set in the spreadsheet, whereas the compiled program can continue with the iterations until the desired accuracy is achieved. For that reason, where possible use an analytical solution rather than an iterative solution.If an iterative convergence technique is inescapable then it is recommended that the function is closely examined over the entire operating range and a few search techniques are tested. If the function is a polynomial, use the Steinman approximation. If the function is well-behaved (nearly linear) and the derivative can be evaluated, then use the Newton approximation method. 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"Times New Roman%D%M%Y%/0%1%/1%2%/2%3%/3 _`a,h!"Times New Roman"Times New Roman77\c efd1. Numerical AnalysisSolve for Power x function,y=cfA(cfB^x)\cefdX1999/11/06, Page 1 of 0, harmedj@freeuk.com.ASolve for y=cfA(cfB^x) & y=cfC*(cfD^(-(1-Tr)^2/7))See solution of 2 coefficient equations. Use this equation for the density equationrho=A(B^(-(1-Tr)2/7))4T"Times New Roman"Word.app C}nCF40*A0DN0Sheet,) }h!"Times New Roman  "Times New Roman"Times New Roman"Times New Roman%D%M%Y%/0%1%/1%2%/2%3%/3 _` '"Times New Roman"Times New Roman77\cef!d~1. Numerical AnalysisSolve for Power x function,y=cfA(cfBx)\c ef3dN1999/11/06, Page 1, harmedj@freeuk.com.Ax0' 'Z' @@' @@'  '@'@  '@ '@    '@ '@%% '%       %% %%       N%%%  v% t%t%% qv% t%t%% qN% %%  N% %%    '@ '@   '@ '@  b ?'    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"Times New Roman0"Times New Roman43333|@"Times New Roman":{P"Times New Roman "Times New Roman0"Times New Roman~@"Times New RomanݐQ"Times New Roman "Times New Roman0"Times New Romanr@"Times New Roman9IR"Times New Roman "Times New Roman0"Times New Roman}@"Times New Roman;S"Times New Roman "Times New Roman0"Times New RomanDDDDD@"Times New Roman ~аT"Times New Roman "Times New Roman0"Times New Roman@"Times New Roman ( U"Times New Roman "Times New Roman0"Times New Roman@"Times New Roman$f`&W"Times New Roman "Times New Roman0"Times New RomanEDDDD@"Times New RomanC؈X"Times New Roman "Times New Roman0 "Times New Roman@"Times New Roman&86IZ"Times New Roman "Times New Roman0 "Times New Roman@"Times New Roman(9"["Times New Roman "Times New Roman0? &6"Univers.06$2$Sheet1cfDcfCTestxpower2power1 TccfA cfB yy2yy1 xx2xx1*%9%1%kk,Z6y=cfA*(cfB^x)0xI@ |@@4ySJ[SJ[ |@!6y=cfA*(cfB^x)!/&&"CG Times"CG Times6y=cfA*(cfB^x)kk,Yny=cfC*(cfD^(-(1-x/Tc)^2/7))0xI@ |@@ 4ySJ[SJ[ |@%. ny=cfC*(cfD^(-(1-x/Tc)^2/7))%.'e( ny=cfC*(cfD^(-(1-x/Tc)^2/7))A'(&Sheet.app &!))F-6*:6DH6Word_ "Times New RomanN123LHP&Heading 1L"Times New RomanJ k&Heading 2L "Times New Roman x x k&Heading 3L "Times New Roman < <k.Bullet listO"Times New Roman Swiss2Home AddressO"Times New Roman:Postal AddressO"Times New Romanh! "Times New Roman%D%M%Y%/0%1%/1%2%/2%3%/3 _`a,h!"Times New Roman"Times New Roman77\c efd1. Numerical AnalysisSolve for Simple Power Curvey=cfA(x^cfB)\cefdX1999/11/06, Page 1 of 0, harmedj@freeuk.com.ASolve for y=cfA(x^cfB)Take logarithms and solve via the solution of 2 coefficient equations.G"Word.app Cl^CF_*_D_Sheet ?"Times New Roman "Times New Roman"Times New Roman"Times New Roman%D%M%Y%/0%1%/1%2%/2%3%/3 _`'"Times New Roman"Times New Roman77\cef!d1. Numerical AnalysisSolve for Simple Power Curvey=cfA(xcfB)\cef"dN1999/11/06, Page 1, harmedj@freeuk.com.AF,'  ' ''    ' V' @@'' @@ ' N&Test '&=J&Test %&=*%&=%%R%'%  V% t%%t qz% t%t%t%t   ' '   ' '    '  '   '  '   '  '    ' '  B43333|@ ,*43333|@r433333?'? vSolve for Simple Power Curve  "Times New Roman"Times New Roman"Times New Roman "Times New RomanGeneral solution of y=cfA*(x^cfB)"Times New Roman"Times New Roman"Times New Roman  "Times New Roman"Times New RomanDATA"Times New Roman"Times New Roman "Times New Roman Title:"Times New Roman ^viscosity=0,6*(Tk^-0,4)"Times New Roman "Times New Roman "Times New RomanData "Times New RomanT "Times New Roman&viscosity "Times New Roman "Times New Roman1"Times New Roman43333|@"Times New RomanfJs?"Times New Roman "Times New Roman2"Times New Roman@"Times New Roman1?"Times New Roman "Times New Roman"Times New Roman"Times New Roman"Times New Roman "Times New Roman Test T="Times New Roman  43333|@"Times New Roman "Times New Roman "Times New Roman$"Times New Roman$2CALCULATIONS"Times New Roman$"Times New Roman $"Times New Roman(cfB="Times New Roman(ٿ"Times New Roman('=(LN(yy1)-LN(yy2))/(LN(xx1)-LN(xx2))"Times New Roman ("Times New Roman,cfA="Times New Roman, 33333?"Times New Roman,j'=EXP(LN(yy1)-cfB*LN(xx1))"Times New Roman ,"Times New Roman0>Test viscosity="Times New Roman0}?"Times New Roman0"Times New Roman 0"Times New Roman4"Times New Roman4*GRAPH DATA"Times New Roman4"Times New Roman 4"Times New Roman8 "Times New Roman8T "Times New Roman8&viscosity "Times New Roman 8"Times New Roman<0"Times New Roman<43333|@"Times New Roman"<dJs?"Times New Roman$ <"Times New Roman@0"Times New Roman@~@"Times New Roman@dP?"Times New Roman$ @"Times New RomanD0"Times New RomanDr@"Times New RomanDD@"??"Times New Roman$ D"Times New RomanH0"Times New RomanH}@"Times New RomanHu{tr?"Times New Roman$ H"Times New RomanL0"Times New RomanLDDDDD@"Times New Roman Ll 8w?"Times New Roman$ L"Times New RomanP0"Times New RomanP@"Times New Roman P)y%_?"Times New Roman$ P"Times New RomanT0"Times New RomanT@"Times New Roman&T >V?"Times New Roman$ T"Times New RomanX0"Times New RomanXEDDDD@"Times New RomanXu[q?"Times New Roman$ X"Times New Roman\0 "Times New Roman\@"Times New Roman*\Y?"Times New Roman$ \"Times New Roman`0 "Times New Roman`@"Times New Roman(`1?"Times New Roman$ `"Times New Roman0?0,,,,,,,, , , , , ,,,,,,,,,,,,  &6"UniversSheet1TitleyNamexNamecfA cfB yy2yy1 xx2xx1y|4C;PP,ZbViscosity vs temperature0.TemperatureI@ |@@/$?4&ViscosityMbP?/$?:v? |@Range 1Range 2+"CG Times"CG Times&Viscosityt&Sheet.app !Fe*'eD5e, QQ0Q6Q_*QeQ:@Wordt  "Times New RomanN123LHP&Heading 1L"Times New RomanJ k&Heading 2L "Times New Roman x x k&Heading 3L "Times New Roman < <k.Bullet listO"Times New Roman Swiss2Home AddressO"Times New Roman:Postal AddressOh!"Times New RomanGh!F"Times New Roman%D%M%Y%/0%1%/1%2%/2%3%/3 _`a,h!"Times New Roman"Times New Roman77\cefd1. Numerical AnalysisSolving for General 4 Coefficient Equations, y=cfA+cfBx+cfCx2+cfDx3\cefdX1999/11/06, Page 1 of 0, harmedj@freeuk.com.AeSolve for y=cfA+cfBx1+cfCx2+cfDx3y1=cfA+cfBxb1+cfCxc1+cfDxd1 (1)y2=cfA+cfBxb2+cfCxc2+cfDxd2 (2)y3=cfA+cfBxb3+cfCxc3+cfDxd3 (3)y4=cfA+cfBxb4+cfCxc4+cfDxd4 (4)Subtract (2) from (1) to get:-(y1-y2)=cfB(xb1-xb2)+cfC(xc1-xc2)+cfD(xd1-xd2)let zz2=xb1-xb2 zc2=(xc1-xc2)/zz2 zd2=(xd1-xd2)/zz2 zy2=(y1-y2)/zz2Thus:-zy2=cfB+cfCzc2+cfDzd2 (5)and with the same operations on equations (1) and (3) we get:-zy3=cfB+cfCzc3+cfDzd3 (6)Subtract (6) from (5) to get:-zy2-zy3=cfC(zc2-zc3)+cfD(zd22-zd3) (7)Let gy3=(zy2-zy3)/(zc2-zc3) gd3=(zd2-zd3)/(zc2-zc3)Then equation (7) becomes:-gy3=cfC+cfDgd3 (8)And operations involving equatio (4) produce:-gy4=cfC+cfDgd4 (9)Subtract (9) from (8) and summarising:-cfD=(gy3-gy4)=(gd3-gd4)cfC=gy3-cfDgd3cfB=zy2-cfCzc2-cfDzd2cfA=y1-cfBxb1-cfCxc1-cfDxd1wheregy3=(zy2-zy3)/(zc2-zc3)gy4=(zy2-zy4)/(zc2-zc4)gd3=(zd2-zd3)/(zc2-zc3)gd4=(zd2-zd4)/(zc2-zc4)zz2=xb1-xb2zz3=xb1-xb3zz4=xb1-xb4zc2=(xc1-xc2)/zz2zc3=(xc1-xc3)/zz3zc4=(xc1-xc4)/zz4zd2=(xd1-xd2)/zz2zd3=(xd1-xd3)/zz3zd4=(xd1-xd4)/zz4zy2=(y1-y2)/zz2zy3=(y1-y3)/zz3zy4=(y1-y4)/zz4harmedj@freeuk.comA%$ $ $ $ 2   ?*   /(                C