Numerical methods and programming
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This document provides an algorithm and flowchart for fitting an exponential curve to data using the equation y = aebx. It involves the following steps: 1. Convert the equation to linear form by taking the log of both sides as ln(y) = ln(a) + bx 2. Create a table with the x, y, ln(y), x2, and xy values 3. Derive equations to solve for the coefficients a' and b' 4. Calculate the necessary deltas and ratios to determine the values of a' and b' 5. Substitute the coefficient values back into the original exponential equation to obtain the best-fit curve.
Numerical methods and programming
- 1. EXPONENTIAL EQUATION (y = aebx) FITTING :- ALGORITHM, FLOWCHART AND NUMERICAL Presented by, VAIBHAV SANJAY PATIL TEME – A – 13 (CIA – 3) Guided by, Prof. D.R. Satpute
- 3. (B)
- 4. * Fill the exponential curve y = aebx to the following data • Step 1 : - Convert the equation into y = ax + b form To find out the values of a & b we require here to prepare a table as below. The equation being y = aebx Taking loge both sides i.e. In y = In a + bx In (e) since In (e) = 1 We get In y = In a + bx Y = a’x + b’ Where Y = In y a’ = b b ’ = In a X 2 4 6 8 Y 25 38 56 84
- 5. • Step 2 : - Prepare the table indicating the value of ΣX, ΣY, ΣX2, ΣXY • Step 3 : - Write the desired equation for straight line Σy = A Σx + nB 15.312631 = 20(A) + 4(B) ……………………………………………………(1) Σxy = A Σx2 + B Σx 80.58673 = 120(A) + 20(B) ………………………….………………………….(2) x y Y xy x2 2 4 6 8 25 38 56 84 3.218876 3.637586 4.025352 4.430817 3.43775 14.55034 24.15211 35.44653 4 16 36 64 Σx = 20 ΣY=15.312631 Σxy=80.58673 Σx2 = 120
- 6. • Step 4 : - Find the value of Δ Δ = |20 4 | |120 20| Δ = 80 • Step 5 : - Find the value of Δ a’ Δ a’ = | 15.312631 4 | | 80.58673 20| Δ a’ = 16.0943 • Step 6 : - Find the value of Δ b’ Δ b’ = | 20 15.312631| | 120 80.58673 | Δ b’ = 225.78112
- 7. • Step 7 : - Then the bet curve is given by the equation ∴ a‘ = Δ a’ ÷ Δ = 16.0743 ÷ 80 a’ = 0.201178 b’ = Δ b’ ÷ Δ = 225.78112 ÷ 80 b’ = 2.822264 ∴ Δ b’ = Δ a’ = 0.201178 a = Antilog b’ = 16.81488 ∴ The equation becomes y = 16.81488 e0.201178 x
- 8. THANK YOU…..