# Curve Fitting of Type y=ab^x Algorithm

In this article we are going to develop an algorithm for fitting curve of type y = abx using least square regression method.

## Procedure for fitting y = abx

We have,

```
y = abx ----- (1)

```

Taking log on both side of equation (1), we get

```
log(y) = log(abx)

log(y) = log(a) + log(bx)

log(y) = log(a) + x*log(b) ----- (2)

Now let Y = log(y), A = log(a) and B = log(b)

then equation (2) becomes,

Y = A + Bx ----- (3),

Now we fit equation (3) using least square regression as:

1. Form normal equations:

∑Y = nA +  B ∑x

∑xY = A∑x + B∑x2

2. Solve normal equations as simulataneous
equations for A and B

3. We calculate afrom A and bfrom B as:
a = exp(A)
b = exp(B)

4. Substitute the value of a and b in
y= abx to find line of best fit.

```

## Algorithm for Fitting Curve y = abx

```1. Start

2. Read Number of Data (n)

3. For i=1 to n:
Next i

4. Initialize:
sumX = 0
sumX2 = 0
sumY = 0
sumXY = 0

5. Calculate Required Sum
For i=1 to n:
sumX = sumX + Xi
sumX2 = sumX2 + Xi * Xi
sumY = sumY + log(Yi)
sumXY = sumXY + Xi * log(Yi)
Next i

6. Calculate Required Constant A and b of Y = A + Bx:
B = (n * sumXY - sumX * sumY)/(n*sumX2 - sumX * sumX)
A = (sumY - B*sumX)/n

7. Transformation of A to a and B to b:
a = exp(A)
b = exp(B)

8. Display value of a and b

8. Stop
```