The secant method is a root-finding algorithm that uses successive secant lines to converge on a root of an equation. It begins with two initial points and finds where the secant line between those points intersects the x-axis. It then uses the intersection point as the next estimate and draws a new secant line. This process repeats until the estimate converges within a specified tolerance of the root. The secant method requires only function evaluations, unlike other methods that also require derivative evaluations. However, it may not always converge and provides no error bounds for the estimates.

1. Numerical Computing
Secant Method
Lecture # 9
By Nasima Akhtar

2. Secant Method Working Rule
• The secant method begins by finding two points on the curve of f(x),
hopefully near to the root we seek.
• A graph or a few applications of bisection might be used to determine the
approximate location of the root.
• we draw the line through these two points and find where it intersects the x-
axis.
• The line through two points on the curve is called the secant line.

3. Graphical Representation of Secant Method 1

4. Derivation Approach 1
A
C
B
D
E

5. Derivation Approach 1
• From the similar Triangle
• ABC = ADE
•
𝐷𝐸
𝐵𝐶
=
𝐴𝐷
𝐴𝐵
•
𝑓(𝑥0)
𝑓(𝑥1)
=
𝑥0−𝑥2
𝑥1−𝑥2
By re-arrangement we get (I have done to whole derivation on page
See the next slide.
𝑥2 = 𝑥1- f(𝑥1)*
𝑥0−𝑥1
𝑓 𝑥0 −𝑓(𝑥1)
𝑥𝑛+1 = 𝑥𝑛- f(𝑥𝑛)*
𝑥𝑛−1−𝑥𝑛
𝑓 𝑥𝑛−1 −𝑓(𝑥𝑛) A
C
B
D
E

7. Derivation Approach 2
From Newton Raphson Method we have,
𝑥𝑖+1 = 𝑥𝑖-
𝑓(𝑥𝑖)
𝑓′(𝑥𝑖)
-----------------------1
From the equation of line we have slope,
𝑓′
(𝑥𝑖)=
𝑓 𝑥𝑖−1 −𝑓 𝑥𝑖
𝑥𝑖−1−𝑥 𝑖
------------------------2
Putting equation 2 in 1
𝑥𝑖+1 = 𝑥𝑖-
𝑓(𝑥𝑖)
𝑓 𝑥𝑖−1 −𝑓 𝑥𝑖
𝑥𝑖−1−𝑥 𝑖
𝑥𝑖+1 = 𝑥𝑖- f(𝑥𝑖)*
𝑥𝑖−1−𝑥𝑖
𝑓 𝑥𝑖−1 −𝑓(𝑥𝑖)

8. Algorithm for Secant Method
• To determine a root of f(x) = 0, given two values, 𝑥0 and 𝑥1 , that are near the root,
If If (𝑥0 ) l < I f (𝑥1)l then
Swap 𝑥0 with 𝑥1
Repeat
Set 𝑥2 = 𝑥1-f(𝑥1)*
𝑥0−𝑥1
𝑓 𝑥0 −𝑓(𝑥1)
Set 𝑥0 = 𝑥1 .
Set 𝑥1 = 𝑥2 .
Until If(𝑥2)l < tolerance value.

9. Graphical Representation of Secant Method 2

10. Matlab Code
f=@(x)(x^3-5*x+1);
x1=2;
x2 = 2.5;
i = 0;
while i <= 5
val = f(x2);
val1 = f(x1);
temp = x2 - x1;
temp1 = val - val1;
nVal = temp/temp1;
nVal = nVal * val;
nVal = x2 - nVal;
x1 = x2;
x2 = nVal;
i = i+1;
end
fprintf('Point is %fn',x2)

11. Example Numerical
• F(x)=𝒙𝟑
− 𝟓𝒙 + 𝟏
• So we consider 𝒙𝟎 =2, 𝒙𝟏 = 𝟐. 𝟓
• F(2)=-1
• F(2.5)=4.125
• 𝒙𝒏+𝟏 = 𝒙𝒏-f(𝒙𝒏)*
𝒙𝒏−𝒙𝒏−𝟏
𝒇 𝒙𝒏 −𝒇 𝒙𝒏−𝟏
• 𝒙𝟐 = 𝒙𝟏-f(𝒙𝟏)*
𝒙𝟏−𝒙𝟎
𝒇 𝒙𝟏 −𝒇(𝒙𝟎)
• 𝒙𝟐 = 𝟐. 𝟓 – 4.125*
𝟐.𝟓−𝟐
(𝟒.𝟏𝟐𝟓)−(−𝟏)
=2.0975
• 𝒙𝟑 = 𝒙𝟐 - f(𝒙𝟐)*
𝒙𝟐−𝒙𝟏
𝒇 𝒙𝟐 −𝒇(𝒙𝟏)
=2.1213
• 𝒙𝟒 =2.1285
• 𝒙𝟓 =2.1284
X F(x)
1 -3
2 -1
2.5 4.125
3 13

12. Secant Method Pros and Cons
• Advantages
1. No computations of derivatives
2. Only f(x) computation each step
3. It converges faster than linear rate.
• Disadvantages
1. It may not converge
2. There is no guaranteed error bound for the computed iterates

13. Convergence Issue

14. Thank You 