Use Newton's method to find the real root function, accurate to five decimal places
f(x) = x^5+2x^2+3
How would I go about solving this?
I used x[n] +1 = x[n] -(f(x[n]))/(f '(x[n]))
So i got f(x)=x^5 +2x^2 +3
f '(x)= 5x^4 +4x
Then I plugged it in and got x[n]=(x^5+2x^2+3)/(5x^4+4x)
Now I'm stuck what do I do next?
Can someone walk me through the steps to the answer?
Thank youuu in advance!!!!
f(x) = x^5+2x^2+3
How would I go about solving this?
I used x[n] +1 = x[n] -(f(x[n]))/(f '(x[n]))
So i got f(x)=x^5 +2x^2 +3
f '(x)= 5x^4 +4x
Then I plugged it in and got x[n]=(x^5+2x^2+3)/(5x^4+4x)
Now I'm stuck what do I do next?
Can someone walk me through the steps to the answer?
Thank youuu in advance!!!!
-
You need to provide an initial estimate x[1].
Descartes' Rule of Signs tells us there are no positive real roots, so start looking at, say, negative integers (or graph the function and look for a zero)
f(-1) = 4
f(-2) = -13
The Intermediate Value Theorem tells us that there is at least one x-value between -2 and -1 for which f(x) = 0. Because |f(-1)| < |f(-2)|, let's choose x[1] = -1.3
x[2] = -1.3 - ((-1.3)^5 + 2(-1.3)² + 3) / (5(-1.3)^4 + 4(-1.3)) ≈ -1.593714
x[3] = -1.593714 - ((-1.593714)^5 + 2(-1.593714)² + 3) / (5(-1.593714)^4 + 4(-1.593714)) ≈ -1.5086506
x[4] ≈ -1.49540216
x[5] ≈ -1.495106542
x[6] ≈ -1.495106398
x[5] and x[6] agree to 6 decimal places (x[6] is in fact correct to all decimals shown) so to 5 decimal places, the root is -1.49511.
Graphing would have led me to choose x[1] = -1.5, which would have converged in fewer iterations.
Descartes' Rule of Signs tells us there are no positive real roots, so start looking at, say, negative integers (or graph the function and look for a zero)
f(-1) = 4
f(-2) = -13
The Intermediate Value Theorem tells us that there is at least one x-value between -2 and -1 for which f(x) = 0. Because |f(-1)| < |f(-2)|, let's choose x[1] = -1.3
x[2] = -1.3 - ((-1.3)^5 + 2(-1.3)² + 3) / (5(-1.3)^4 + 4(-1.3)) ≈ -1.593714
x[3] = -1.593714 - ((-1.593714)^5 + 2(-1.593714)² + 3) / (5(-1.593714)^4 + 4(-1.593714)) ≈ -1.5086506
x[4] ≈ -1.49540216
x[5] ≈ -1.495106542
x[6] ≈ -1.495106398
x[5] and x[6] agree to 6 decimal places (x[6] is in fact correct to all decimals shown) so to 5 decimal places, the root is -1.49511.
Graphing would have led me to choose x[1] = -1.5, which would have converged in fewer iterations.