MATH 577fass1 3.2 The Secant Method Recall Newton’s method Main drawbacks: requires coding of the...
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Transcript of MATH 577fass1 3.2 The Secant Method Recall Newton’s method Main drawbacks: requires coding of the...
![Page 1: MATH 577fass1 3.2 The Secant Method Recall Newton’s method Main drawbacks: requires coding of the derivative requires evaluation.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d535503460f94a2ffd4/html5/thumbnails/1.jpg)
MATH 577
http://amadeus.math.iit.edu/
~fass 1
3.2 The Secant Method
Recall Newton’s method
Main drawbacks:•requires coding of the derivative•requires evaluation of and in every iteration
Work-aroundApproximate derivative with difference quotient:
![Page 2: MATH 577fass1 3.2 The Secant Method Recall Newton’s method Main drawbacks: requires coding of the derivative requires evaluation.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d535503460f94a2ffd4/html5/thumbnails/2.jpg)
MATH 577
http://amadeus.math.iit.edu/
~fass 2
Secant Method
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MATH 577
http://amadeus.math.iit.edu/
~fass 3
Graphical Interpretation
![Page 4: MATH 577fass1 3.2 The Secant Method Recall Newton’s method Main drawbacks: requires coding of the derivative requires evaluation.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d535503460f94a2ffd4/html5/thumbnails/4.jpg)
MATH 577
http://amadeus.math.iit.edu/
~fass 4
Graphical Interpretation
![Page 5: MATH 577fass1 3.2 The Secant Method Recall Newton’s method Main drawbacks: requires coding of the derivative requires evaluation.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d535503460f94a2ffd4/html5/thumbnails/5.jpg)
MATH 577
http://amadeus.math.iit.edu/
~fass 5
Convergence Analysis
![Page 6: MATH 577fass1 3.2 The Secant Method Recall Newton’s method Main drawbacks: requires coding of the derivative requires evaluation.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d535503460f94a2ffd4/html5/thumbnails/6.jpg)
MATH 577
http://amadeus.math.iit.edu/
~fass 6
Proof of Theorem 3.2
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MATH 577
http://amadeus.math.iit.edu/
~fass 7
Proof of Theorem 3.2 (cont.)
![Page 8: MATH 577fass1 3.2 The Secant Method Recall Newton’s method Main drawbacks: requires coding of the derivative requires evaluation.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d535503460f94a2ffd4/html5/thumbnails/8.jpg)
MATH 577
http://amadeus.math.iit.edu/
~fass 8
Proof of Theorem 3.2 (cont.)
![Page 9: MATH 577fass1 3.2 The Secant Method Recall Newton’s method Main drawbacks: requires coding of the derivative requires evaluation.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d535503460f94a2ffd4/html5/thumbnails/9.jpg)
MATH 577
http://amadeus.math.iit.edu/
~fass 9
Proof of Theorem 3.2 (cont.)
![Page 10: MATH 577fass1 3.2 The Secant Method Recall Newton’s method Main drawbacks: requires coding of the derivative requires evaluation.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d535503460f94a2ffd4/html5/thumbnails/10.jpg)
MATH 577
http://amadeus.math.iit.edu/
~fass 10
Proof of Theorem 3.2 (cont.)earlier formula
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MATH 577
http://amadeus.math.iit.edu/
~fass 11
Proof of Theorem 3.2 (Exact order)
![Page 12: MATH 577fass1 3.2 The Secant Method Recall Newton’s method Main drawbacks: requires coding of the derivative requires evaluation.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d535503460f94a2ffd4/html5/thumbnails/12.jpg)
MATH 577
http://amadeus.math.iit.edu/
~fass 12
Proof of Theorem 3.2 (Exact order)
(*):
![Page 13: MATH 577fass1 3.2 The Secant Method Recall Newton’s method Main drawbacks: requires coding of the derivative requires evaluation.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d535503460f94a2ffd4/html5/thumbnails/13.jpg)
MATH 577
http://amadeus.math.iit.edu/
~fass 13
Proof of Theorem 3.2 (Exact order)
![Page 14: MATH 577fass1 3.2 The Secant Method Recall Newton’s method Main drawbacks: requires coding of the derivative requires evaluation.](https://reader030.fdocuments.us/reader030/viewer/2022032704/56649d535503460f94a2ffd4/html5/thumbnails/14.jpg)
MATH 577
http://amadeus.math.iit.edu/
~fass 14
Proof of Theorem 3.2 (Exact order)
(cf. Theorem)
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MATH 577
http://amadeus.math.iit.edu/
~fass 15
Comparison of Root Finding Methods
Other facts:•bisection method always converges•Newton’s method requires coding of derivative
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MATH 577
http://amadeus.math.iit.edu/
~fass 16
Newton vs. Secant (“Fair” Comparison)
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MATH 577
http://amadeus.math.iit.edu/
~fass 17
Generalizations of the Secant Method
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MATH 577
http://amadeus.math.iit.edu/
~fass 18
Müller’s Method
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MATH 577
http://amadeus.math.iit.edu/
~fass 19
Müller’s Method (cont.)
Features:•Can locate complex roots (even with real initial guesses)•Convergence rate =1.84•Explicit formula rather lengthy (can be derived with more knowledge on interpolation – see Chapter 6)