excel for scientists and engineers phần 10 potx

-30 L X Figure A4-1. Polynomial of order 3. The curve follows equation A42 with a = 5, b = -1, c = -5 and d = 1. The Trendline type is Polynomial. The highest-order polynomial that Trendline can use as a fitting function is a regular polynomial of order six, i.e., y = ax6 + bx5 +cx4 + ak3 + ex2 +fx + g. LINEST is not limited to order six, and LINEST can also fit data using other polynomials such as y = ax2 + bx3'2 + cx + + e. Exponential Decrease. 0.1 0 0.08 *, 0.06 0.04 0.02 0.00 0 2 4 6 8 10 X Figure A4-2. Exponential decrease to zero. The curve follows equation A43 with a = 0.1 and b = -0.5. The Trendline equation is shown on the chart. Data with the behavior shown in Figure A4-2 can be fitted by the exponential equation APPENDIX 4 EOUATIONS FOR CURVE FITTING 41 1 y = aebx (A4-3) The sign of b is often negative (as in radioactive decay), giving rise to the The linearized form of the equation is In y = bx + In a; the Trendline type is decreasing behavior shown in Figure A4-2. Exponential. Exponential Growth. curvature is upwards, as in Figure A4-3. If the sign of b in equation A4-3 is positive, the r, 0 2 4 6 8 10 X Figure A4-3. Exponential increase. The curve follows equation A4-3 with a = 0.1 and b = 0.5. The Trendline equation is shown on the chart. Exponential Decrease or Increase Between Limits. If the curve decreases exponentially to a nonzero limit, or rises exponentially to a limiting value as in Figure A4-4, the form of the equation is y = aebx + c Excel's Trendline cannot handle data of this type. (A4-4) 412 EXCEL: NUMERICAL METHODS 1 0.8 0.6 0.4 x 0.2 0 0 2 4 6 8 10 X Figure A4-4. Exponential increase to a limit. The curve follows equation A4-4 with a = -1, b = -0.5 and c = 1. The linearized form of the equation is In 0, - c) = bx + In a. Double Exponential Decay to Zero. The sum of two exponentials (equation A4-5) gives rise to behavior similar to that shown in Figure A4-5. This type of behavior is observed, for example, in the radioactive decay of a mixture of two nuclides with different half-lives, one short-lived and the other relatively longer-lived. y = ae-bt + ce-dl (A4-5) >r 0 21 1.5 >r o.q\ 0 , , 0 2 4 6 8 10 X Figure A4-5. Double exponential decay. The curve follows equation A4-5 with a = 1, b = -2, c = 1 and d = -0.2. If the second term is subtracted rather than added, a variety of curve shapes are possible. Figures A4-6 and A4-7 illustrate two of the possible behaviors. APPENDIX 4 EQUATIONS FOR CURVE FITTING 413 I- - I I I I I I 2 4 6 8 10 -1 L X Figure A4-6. Double exponential decay. The curve follows equation A4-5 with a = 1, b = 4.2, c = -2 and d = -2, 0 2 4 6 8 10 X -0.8 Figure A4-7. Double exponential decay. The curve follows equation A4-5 with a = 1, b = -2, c = -1 and d = -0.2. Equation A4-5 is intrinsically nonlinear (cannot be converted into a linear form). Power. Data with the behavior shown in Figure A4-8 can be fitted by equation A4-6. (A4-6) b y=aX 4 14 EXCEL: NUMERICAL METHODS y= 1.1x-O.~ 0 2 4 6 8 10 X Figure A4-8. Power curve. The curve follows equation A4-6 with a = 1.1, b = -0.5. The Trendline equation is shown on the chart. The linearized form of equation A4-6 is In y = b In x + In a; the Trendline form is Power. Logarithmic. 4 2 -0 -2 y = 2Ln(x) + 1 I 10 “t -6 X Figure A4-9. Logarithmic function. The curve follows equation A4-7 with a = 2, b = 1. Data with the behavior shown in Figure A4-9 can be fitted by the logarithmic equation A4-7. y = a lnx + b (A4-7) APPENDIX 4 EQUATIONS FOR CURVE FITTING 415 The Trendline type is Logarithmic. "Plateau" Curve. A relationship of the form ax y=- b+x exhibits the behavior shown in Figure A4-10. 1 >r 0.5 0 (A4-8) 0 2 4 6 8 10 X Figure A4-10. Plateau curve. The curve follows equation A4-8 with a = 1, b = 1. In biochemistry, this type of curve is encountered in a plot of reaction rate of an enzyme-catalyzed reaction of a substrate as a function of the concentration of the substrate, as in Figure A4-10. The behavior is described by the Michaelis- Menten equation, (A4-9) where V is the reaction velocity (typical units mmol/s), K,,, is the Michaelis- Menten constant (typical units mM), V,, is the maximum reaction velocity and [S] is the substrate concentration. Some typical results are shown in Figure A4- 11. 416 EXCEL: NUMERICAL METHODS 50 40 30 20 10 0 Figure A4-11. Michaelis-Menten enzyme kinetics. The curve follows equation A4-9 with V,, = 50, K,, = 0.5. Double Reciprocal Plot. The Michaelis-Menten equation can be converted to a straight line equation by taking the reciprocals of each side. This treatment is called a Lineweaver-Burk plot, a plot of the reciprocal of the enzymatic reaction velocity (UV) versus the reciprocal of the substrate concentration (l/[SI). 1 K,1 1 - +- V Vmax S Vmax (A4- 10) A double-reciprocal plot of the data of Figure A4-11 is shown in Figure A4- 12. The parameters V,,, and K,,, can be obtained from the slope and intercept of the straight line (Vmm = Uintercept, K,,, = interceptlslope). However, relationships dealing with the propagation of error must be used to calculate the standard deviations of V,,, and K, from the standard deviations of slope and intercept. By contrast, when the Solver is used the expression does not need to be rearranged, ycalc is calculated directly from equation A4-19, the Solver returns the coefficients V,,, and K,,,, and SolvStat.xls returns the standard deviations of V,,, and K,. APPENDIX 4 EQUATIONS FOR CURVE FITTING 417 0.00 ' 0 5 10 WSI Figure A4-12. Double-reciprocal plot of enzyme kinetics. The curve follows equation A4-10 with V,,, = 50, K,,, = 0.5. Logistic Function. The logistic equation or dose-response curve (A4-11) 1 y=- 1 + e-" produces an S-shaped curve like the one shown in Figure A4-13. Y -5 0 5 10 -10 X Figure A4-13. Simple logistic curve. The curve follows equation A4- 1 1 with a = 1. 418 EXCEL: NUMERICAL METHODS In the dose-response form of the equation, the y-axis (the response) is normalized to 100% and the x-axis (usually logarithmic) is normalized so that the midpoint (the half-maximum response or ECSo) occurs at x = 0. Logistic Curve with Variable Slope. In equation A4-11, the coefficient a determines the slope of the rising part of the curve; in biochemistry a is referred to as the Hill slope. Figure A4-14 illustrates the effect of varying Hill slope. At the midpoint the slope is a/4. -5 0 5 10 X -10 Figure A4-14. Variable slopes of logistic curve. The three curves have a = 0.5, 1 and 2, respectively. Logistic Curve with Additional Parameters. Equation A4-12 is the logistic equation with addition parameters that determine the height of the "plateau" and the offset of the mid-point from x = 0. b c + e-ax The height of the plateau is equal to b/c. Y= (A4- 12) APPENDIX 4 EQUATIONS FOR CURVE FITTING 419 Figure A4-15. Logistic curve with additional variables. The curve follows equation A4-12 with a = 1, b = 0.5 and c = 5. Logistic Curve with Offset on the y-Axis. The logistic equation -10 -5 0 5 10 15 20 A Figure A4-16. Logistic curve with offset on the y-axis. The curve follows equation A4-13 with a = 1, b = -2, c = 1 and d = -0.2. (A4- 13) This equation takes into account the value of the plateau maximum and minimum (coefficients a and d, respectively), the offset on the x-axis, and the Hill slope. [...]... 90 91 92 93 94 95 6 2 A B C D E F G H I 3 K L M N 0 P Q R S T U v W X Y Z [ \ 1 A - 427 27 escape 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 1 1 1 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 ' a b c d e f g h i j k I m n 0 p q r s t u v w x y z { I } N (bksp) EXCEL: NUMERICAL METHODS 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148... JntegrateT 136 custom lookup formula 80 CVErr keyword 50 D Data Analysis 303, 347 debug toolbar (VBA) 57 Debug 55 debugging 54 deck of cards (problem) 362 decrease, exponential 4 12 definite integral 127 derivative calculating first and second 99, 104 cubic fitting function for calculating 105 first 99, 100 formulas for computing 104 INDEX of a function 109 of a worksheet formula 110, 1 11, 112 partial... Numerical Methodsfor Engineers and Scientists, McGraw-Hill, 1992 Johnson, K Jeffrey, Numerical Methods in Chemistry, Marcel Dekker, 1980 Kuo, Shan S., Numerical Methods and Computers, Addison-Wesley, 1965 Press, William H., et al., Numerical Recipes in FORTRAN,2nd ed., Cambridge University Press, 1992 Rao, S S., Applied Numerical Methodsfor Engineers and Scientists, Prentice-Hall, 2002 Rusling, J F and Kumosinski,... ANSWERS AND COMMENTS FOR PROBLEMS 44 1 Random Numbers and Monte Carlo 1 The answer spreadsheet contains two examples The first uses 32 points, and is intended mainly to illustrate the method Random number formulas are used to generate a pair of x, y coordinates in columns A and B The formula in column C uses an IF statement to determine whether the point is inside the circle; if inside, the formula returns... Bilal M and Richard H McCuen, Numerical Methodsfor Engineers, Prentice-Hall, 1996 Bourg, David M., Excel ScientiJc and Engineering Cookbook,OReilly, 2006 Chapra, Steven C and Raymond P Canale, Numerical Methodsfor Engineers, 4'h ed., McGraw-Hill, 2002 Cheney, Ward and David Kincaid, Numerical Mathematics and Computing, Brooks/Cole, 1985 Gerald, Curtis F and Patrick 0 Wheatley, Applied Numerical Analysis,... cm-’), and in is the value of x at Amax.The parameters is related to the bandwidth at half-height 10 8 6 4 2 0 0 2 4 X 6 8 Figure A4-17 Gaussian curve The curve follows equation A4-15 with A,, = 10, m Log vs Reciprocal The function ( y=exp a 3 10 =5 and s = 1.5 (A4-16) is often seen in the relationship of physical properties to temperature The linearized form is In y = -b/x + a This equation form is... derivatives, formulas for 104 constraints, in Solver model 324 Convergence (Solver options) 325 convergence, slow 153 convergent series 69 ConvertFormula method 1 17, 1 18 correlation coefficient, R 288 Cramer's rule 169, 190 Crank-Nicholson 274,280 create an Add-In macro 53 critical points 100 cubic equation 147 EXCEL: NUMERICALMETHODS fitting data to 295 fitting function for calculating derivatives 105 interpolating... The spreadsheet answer also incorporates the formula for the initial estimate (problem 9) 9 Here is one possible formula The number is in cell C2; the initial estimate formula is =LEFT(C2,0.!5*(LEN(C2)+1)) 10 The series is described in Edward Kasner and James R Newman, Mathematics and the Imagination, Simon & Schuster, 1940; Harper & Row, 1989 The sum (10 terms) is 'I = 3.14159265359 (9 x I % error)... Throttle and Power vs Speed and experimented; quadratic or APPENDIX 8 ANSWERS AND COMMENTS FOR PROBLEMS 439 cubic (polynomials of 2nd or 3rd order) fitted the data quite well Using that information I used LINEST to find regression coefficients that fitted Power to ,? Speed (5') for each value of Throttle (0 (the fitting function was a ! + b.5') I then fitted the regression coefficients a and b individually... tried The preceding fitting function can be written in the following form: c.p-9+ d F p + e - 9 + f F S + g S + h Chapter 14 Nonlinear Regression and the Solver 1 Enter formula for Acdc (you'll need a cell for k, the changing cell) Enter formula for (residual)* and sum the squares of residuals (this is the target cell) Use the Solver to minimize the target Answer: k = 0.3290 2 Follow the same procedure . W X Y Z [ 1 A - 96 ' 97 a 98 b 99 c 100 d 101 e 102 f 103 g 104 h 105 i 106 j 107 k 108 I 109 m 110 n 111 0 112 p 113 q 114 r 115 s 116 t 117. the coefficients V,,, and K,,,, and SolvStat.xls returns the standard deviations of V,,, and K,. APPENDIX 4 EQUATIONS FOR CURVE FITTING 417 0.00 ' 0 5 10 WSI Figure A4-12 Bibliography Ayyub, Bilal M. and Richard H. McCuen, Numerical Methods for Engineers, Bourg, David M., Excel ScientiJc and Engineering Cookbook, OReilly, 2006. Chapra, Steven C. and Raymond P. Canale,