Homework Help: Successive approximations

Posted by Devin on Saturday, October 20, 2012 at 9:00pm.

Find all root of ln(4-x^2)=x correct to 8 decimal places. List all successive approximations.

Successive approximations - Reiny, Thursday, April 4, 2013 at 12:35am
by definition:
4 - x^2 = e^x
by making a rough sketch of
y = 4-x^2 and y = e^x, I can see that there are roots at appr. 1 and -2

setting up for Newton's Method:
let y = 4 - x^2 - e^x
y' = -2x - e^x
His method says:

newx = x - y/y' = x - (4-x^2 - e^x)/(-2x - e^x)
= x + (4-x^2 - e^x)/(2x+e^x)
so start with x = 1

newx ---- x
???? ----- 1
1.059707788 ---1.059707788
1.058007813 --- 1.058007813
1.058006401 --- 1.058006401
1.058006401 YEahhhhh! correct to 9 decimal places

start with x = -2

------ -1.9649813...
-1.964635631 --- -1.964635631
-1.964635597 --- -1.964635597
-1.964635597 Yeah again,

x = 1.058006401 or x = -1.964635597
Successive approximations - Reiny, Thursday, April 4, 2013 at 12:37am
by definition:
4 - x^2 = e^x
by making a rough sketch of
y = 4-x^2 and y = e^x, I can see that there are roots at appr. 1 and -2

setting up for Newton's Method:
let y = 4 - x^2 - e^x
y' = -2x - e^x
His method says:

newx = x - y/y' = x - (4-x^2 - e^x)/(-2x - e^x)
= x + (4-x^2 - e^x)/(2x+e^x)
so start with x = 1

newx ---- x
???? ----- 1
1.059707788 ---1.059707788
1.058007813 --- 1.058007813
1.058006401 --- 1.058006401
1.058006401 YEahhhhh! correct to 9 decimal places

start with x = -2

------ -1.9649813...
-1.964635631 --- -1.964635631
-1.964635597 --- -1.964635597
-1.964635597 Yeah again,

x = 1.058006401 or x = -1.964635597

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