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Transcript of Www.le.ac.uk Numerical Methods: Finding Roots Department of Mathematics University of Leicester.
![Page 1: Www.le.ac.uk Numerical Methods: Finding Roots Department of Mathematics University of Leicester.](https://reader036.fdocuments.in/reader036/viewer/2022062620/551aa1dc550346e0158b595d/html5/thumbnails/1.jpg)
www.le.ac.uk
Numerical Methods: Finding Roots
Department of MathematicsUniversity of Leicester
![Page 2: Www.le.ac.uk Numerical Methods: Finding Roots Department of Mathematics University of Leicester.](https://reader036.fdocuments.in/reader036/viewer/2022062620/551aa1dc550346e0158b595d/html5/thumbnails/2.jpg)
Content
Motivation
Change of sign method
Iterative method
Newton-Raphson method
![Page 3: Www.le.ac.uk Numerical Methods: Finding Roots Department of Mathematics University of Leicester.](https://reader036.fdocuments.in/reader036/viewer/2022062620/551aa1dc550346e0158b595d/html5/thumbnails/3.jpg)
Reasons for Finding Roots by Numerical Methods• If the data is obtained from observations,
it often won’t have an equation which accurately models
• Some equations are not easy to solve
• Can program a computer to solve equations for us
Next
Iterative method
Newton-Raphson
Change of sign methodMotivation
![Page 4: Www.le.ac.uk Numerical Methods: Finding Roots Department of Mathematics University of Leicester.](https://reader036.fdocuments.in/reader036/viewer/2022062620/551aa1dc550346e0158b595d/html5/thumbnails/4.jpg)
Solving equations by change of sign
• This is also known as ‘Iteration by Bisection’
• It is done by bisecting an interval we know the solution lies in repeatedly
Next
Iterative method
Newton-Raphson
Change of sign methodMotivation
![Page 5: Www.le.ac.uk Numerical Methods: Finding Roots Department of Mathematics University of Leicester.](https://reader036.fdocuments.in/reader036/viewer/2022062620/551aa1dc550346e0158b595d/html5/thumbnails/5.jpg)
METHOD
• Find an interval in which the solution lies
• Split the interval into 2 equal parts
• Find the change of sign
• Repeat
Solving equations by change of sign
Next
Iterative method
Newton-Raphson
Change of sign methodMotivation
![Page 6: Www.le.ac.uk Numerical Methods: Finding Roots Department of Mathematics University of Leicester.](https://reader036.fdocuments.in/reader036/viewer/2022062620/551aa1dc550346e0158b595d/html5/thumbnails/6.jpg)
Solving equations by change of sign
Next
Iterative method
Newton-Raphson
Change of sign methodMotivation
![Page 7: Www.le.ac.uk Numerical Methods: Finding Roots Department of Mathematics University of Leicester.](https://reader036.fdocuments.in/reader036/viewer/2022062620/551aa1dc550346e0158b595d/html5/thumbnails/7.jpg)
Example: Find a root of the equation
given there is a solution close to x=-2
Step 1: Find the interval
So we know the solution lies between -2 and -1
Solving equations by change of sign
Next
Iterative method
Newton-Raphson
Change of sign methodMotivation
![Page 8: Www.le.ac.uk Numerical Methods: Finding Roots Department of Mathematics University of Leicester.](https://reader036.fdocuments.in/reader036/viewer/2022062620/551aa1dc550346e0158b595d/html5/thumbnails/8.jpg)
Step 2: We now half the interval and find the
value of f at the half way point
Now we know the solution lies between and
Solving equations by change of sign
Next
Iterative method
Newton-Raphson
Change of sign methodMotivation
𝑓 (−1.5 )=2 (−1.5 )3−2 (−1.5 )+7=6.625
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Step 3: Now we just keep repeating the process
Solving equations by change of sign
Next
Iterative method
Newton-Raphson
Change of sign methodMotivation
![Page 10: Www.le.ac.uk Numerical Methods: Finding Roots Department of Mathematics University of Leicester.](https://reader036.fdocuments.in/reader036/viewer/2022062620/551aa1dc550346e0158b595d/html5/thumbnails/10.jpg)
Solving equations by change of sign
Next
Iterative method
Newton-Raphson
Change of sign methodMotivation
![Page 11: Www.le.ac.uk Numerical Methods: Finding Roots Department of Mathematics University of Leicester.](https://reader036.fdocuments.in/reader036/viewer/2022062620/551aa1dc550346e0158b595d/html5/thumbnails/11.jpg)
So to 3 s.f. the solution is
Solving equations by change of sign
Next
Iterative method
Newton-Raphson
Change of sign methodMotivation
𝑥=−1.74
![Page 12: Www.le.ac.uk Numerical Methods: Finding Roots Department of Mathematics University of Leicester.](https://reader036.fdocuments.in/reader036/viewer/2022062620/551aa1dc550346e0158b595d/html5/thumbnails/12.jpg)
Solving equations by change of sign
Number of dp:
Next
Iterative method
Newton-Raphson
Change of sign methodMotivation
Find a solution
Clear information box
![Page 13: Www.le.ac.uk Numerical Methods: Finding Roots Department of Mathematics University of Leicester.](https://reader036.fdocuments.in/reader036/viewer/2022062620/551aa1dc550346e0158b595d/html5/thumbnails/13.jpg)
Solving using iterative method
• ‘Iteration’ is the process of repeatedly using a previous result to obtain a new result
Next
Iterative method
Newton-Raphson
Change of sign methodMotivation
![Page 14: Www.le.ac.uk Numerical Methods: Finding Roots Department of Mathematics University of Leicester.](https://reader036.fdocuments.in/reader036/viewer/2022062620/551aa1dc550346e0158b595d/html5/thumbnails/14.jpg)
METHOD
• Rearrange the equation to make the highest power the subject
• Use the power root to leave on its own on the LHS
• Make on the LHS
• Make on the RHS
Solving using iterative method
Next
Iterative method
Newton-Raphson
Change of sign methodMotivation
![Page 15: Www.le.ac.uk Numerical Methods: Finding Roots Department of Mathematics University of Leicester.](https://reader036.fdocuments.in/reader036/viewer/2022062620/551aa1dc550346e0158b595d/html5/thumbnails/15.jpg)
• Now that the function is in the form
we can use the value for to calculate , then we can use the value , and so on...
• When we eventually get a value repeating we have reached the solution
Solving using iterative method
Next
Iterative method
Newton-Raphson
Change of sign methodMotivation
𝑥𝑛+1= 𝑓 (𝑥𝑛)
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Solving using iterative method
Next
Iterative method
Newton-Raphson
Change of sign methodMotivation
![Page 17: Www.le.ac.uk Numerical Methods: Finding Roots Department of Mathematics University of Leicester.](https://reader036.fdocuments.in/reader036/viewer/2022062620/551aa1dc550346e0158b595d/html5/thumbnails/17.jpg)
Solving using iterative method
Next
Iterative method
Newton-Raphson
Change of sign methodMotivation
![Page 18: Www.le.ac.uk Numerical Methods: Finding Roots Department of Mathematics University of Leicester.](https://reader036.fdocuments.in/reader036/viewer/2022062620/551aa1dc550346e0158b595d/html5/thumbnails/18.jpg)
Solving using iterative method
Next
Iterative method
Newton-Raphson
Change of sign methodMotivation
![Page 19: Www.le.ac.uk Numerical Methods: Finding Roots Department of Mathematics University of Leicester.](https://reader036.fdocuments.in/reader036/viewer/2022062620/551aa1dc550346e0158b595d/html5/thumbnails/19.jpg)
Click on a seed value to see the cobweb:
start
here
start
here
start
here
start
here
start
here
start
here
𝑦=𝑥
𝑦= 𝑓 (𝑥 )
Clear Cobwebs
Next
Iterative method
Newton-Raphson
Change of sign methodMotivation
![Page 20: Www.le.ac.uk Numerical Methods: Finding Roots Department of Mathematics University of Leicester.](https://reader036.fdocuments.in/reader036/viewer/2022062620/551aa1dc550346e0158b595d/html5/thumbnails/20.jpg)
Example: Find a root of the equation
given that there is a solution close to
STEP 1: Rearrange the equation
Solving using iterative method
Next
Iterative method
Newton-Raphson
Change of sign methodMotivation
![Page 21: Www.le.ac.uk Numerical Methods: Finding Roots Department of Mathematics University of Leicester.](https://reader036.fdocuments.in/reader036/viewer/2022062620/551aa1dc550346e0158b595d/html5/thumbnails/21.jpg)
Step 2: We can now input (taken from the
question)
Solving using iterative method
Next
Iterative method
Newton-Raphson
Change of sign methodMotivation
![Page 22: Www.le.ac.uk Numerical Methods: Finding Roots Department of Mathematics University of Leicester.](https://reader036.fdocuments.in/reader036/viewer/2022062620/551aa1dc550346e0158b595d/html5/thumbnails/22.jpg)
This gives us the solution
to 3 d.p.
Solving using iterative method
Next
Iterative method
Newton-Raphson
Change of sign methodMotivation
𝑥=−1.893
![Page 23: Www.le.ac.uk Numerical Methods: Finding Roots Department of Mathematics University of Leicester.](https://reader036.fdocuments.in/reader036/viewer/2022062620/551aa1dc550346e0158b595d/html5/thumbnails/23.jpg)
Solving using iterative method
Starting value:
Number of d.p.:
Next
Iterative method
Newton-Raphson
Change of sign methodMotivation
Solve
Clear
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Newton-Raphson Method
• Sometimes known as the Newton Method
• Named after Issac Newton and Joseph Raphson
• Iteratively finds successively better approximations to the roots
Next
Iterative method
Newton-Raphson
Change of sign methodMotivation
![Page 25: Www.le.ac.uk Numerical Methods: Finding Roots Department of Mathematics University of Leicester.](https://reader036.fdocuments.in/reader036/viewer/2022062620/551aa1dc550346e0158b595d/html5/thumbnails/25.jpg)
Newton-Raphson Method
The formula is
We start with an arbitrary and wait for the
iteration to converge
Next
Iterative method
Newton-Raphson
Change of sign methodMotivation
𝑥𝑛+1=𝑥𝑛−𝑓 (𝑥𝑛)𝑓 ′ (𝑥𝑛)
![Page 26: Www.le.ac.uk Numerical Methods: Finding Roots Department of Mathematics University of Leicester.](https://reader036.fdocuments.in/reader036/viewer/2022062620/551aa1dc550346e0158b595d/html5/thumbnails/26.jpg)
Newton-Raphson Method
𝑥0𝑥1𝑥3
Next
Iterative method
Newton-Raphson
Change of sign methodMotivation
![Page 27: Www.le.ac.uk Numerical Methods: Finding Roots Department of Mathematics University of Leicester.](https://reader036.fdocuments.in/reader036/viewer/2022062620/551aa1dc550346e0158b595d/html5/thumbnails/27.jpg)
Newton-Raphson Method
Example: Use the Newton-Raphson Method to
approximate the cube root of 37
The equation we use is
Now we need to evaluate
Next
Iterative method
Newton-Raphson
Change of sign methodMotivation
![Page 28: Www.le.ac.uk Numerical Methods: Finding Roots Department of Mathematics University of Leicester.](https://reader036.fdocuments.in/reader036/viewer/2022062620/551aa1dc550346e0158b595d/html5/thumbnails/28.jpg)
Newton-Raphson Method
We then obtain the formula
Choose
Next
Iterative method
Newton-Raphson
Change of sign methodMotivation
𝑓 (𝑥𝑛+1 )=𝑥𝑛−(𝑥𝑛)3−373 (𝑥𝑛 )2
![Page 29: Www.le.ac.uk Numerical Methods: Finding Roots Department of Mathematics University of Leicester.](https://reader036.fdocuments.in/reader036/viewer/2022062620/551aa1dc550346e0158b595d/html5/thumbnails/29.jpg)
Newton-Raphson Method
So this means that the cube root of 37 is approximately 3.3322 Next
Iterative method
Newton-Raphson
Change of sign methodMotivation
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