Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations...
Transcript of Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations...
![Page 1: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/1.jpg)
Solutions of Equations in One Variable
Fixed-Point Iteration II
Numerical Analysis (9th Edition)R L Burden & J D Faires
Beamer Presentation Slidesprepared byJohn Carroll
Dublin City University
c© 2011 Brooks/Cole, Cengage Learning
![Page 2: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/2.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Outline
1 Functional (Fixed-Point) Iteration
2 Convergence Criteria for the Fixed-Point Method
3 Sample Problem: f (x) = x3 + 4x2 − 10 = 0
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 2 / 54
![Page 3: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/3.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Outline
1 Functional (Fixed-Point) Iteration
2 Convergence Criteria for the Fixed-Point Method
3 Sample Problem: f (x) = x3 + 4x2 − 10 = 0
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 2 / 54
![Page 4: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/4.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Outline
1 Functional (Fixed-Point) Iteration
2 Convergence Criteria for the Fixed-Point Method
3 Sample Problem: f (x) = x3 + 4x2 − 10 = 0
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 2 / 54
![Page 5: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/5.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Outline
1 Functional (Fixed-Point) Iteration
2 Convergence Criteria for the Fixed-Point Method
3 Sample Problem: f (x) = x3 + 4x2 − 10 = 0
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 3 / 54
![Page 6: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/6.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Now that we have established a condition for which g(x) has a uniquefixed point in I, there remains the problem of how to find it. Thetechnique employed is known as fixed-point iteration.
Basic ApproachTo approximate the fixed point of a function g, we choose an initialapproximation p0 and generate the sequence {pn}∞n=0 by lettingpn = g(pn−1), for each n ≥ 1.If the sequence converges to p and g is continuous, then
p = limn→∞
pn = limn→∞
g(pn−1) = g(
limn→∞
pn−1
)= g(p),
and a solution to x = g(x) is obtained.This technique is called fixed-point, or functional iteration.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 4 / 54
![Page 7: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/7.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Now that we have established a condition for which g(x) has a uniquefixed point in I, there remains the problem of how to find it. Thetechnique employed is known as fixed-point iteration.
Basic ApproachTo approximate the fixed point of a function g, we choose an initialapproximation p0 and generate the sequence {pn}∞n=0 by lettingpn = g(pn−1), for each n ≥ 1.
If the sequence converges to p and g is continuous, then
p = limn→∞
pn = limn→∞
g(pn−1) = g(
limn→∞
pn−1
)= g(p),
and a solution to x = g(x) is obtained.This technique is called fixed-point, or functional iteration.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 4 / 54
![Page 8: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/8.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Now that we have established a condition for which g(x) has a uniquefixed point in I, there remains the problem of how to find it. Thetechnique employed is known as fixed-point iteration.
Basic ApproachTo approximate the fixed point of a function g, we choose an initialapproximation p0 and generate the sequence {pn}∞n=0 by lettingpn = g(pn−1), for each n ≥ 1.If the sequence converges to p and g is continuous, then
p = limn→∞
pn = limn→∞
g(pn−1) = g(
limn→∞
pn−1
)= g(p),
and a solution to x = g(x) is obtained.
This technique is called fixed-point, or functional iteration.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 4 / 54
![Page 9: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/9.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Now that we have established a condition for which g(x) has a uniquefixed point in I, there remains the problem of how to find it. Thetechnique employed is known as fixed-point iteration.
Basic ApproachTo approximate the fixed point of a function g, we choose an initialapproximation p0 and generate the sequence {pn}∞n=0 by lettingpn = g(pn−1), for each n ≥ 1.If the sequence converges to p and g is continuous, then
p = limn→∞
pn = limn→∞
g(pn−1) = g(
limn→∞
pn−1
)= g(p),
and a solution to x = g(x) is obtained.This technique is called fixed-point, or functional iteration.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 4 / 54
![Page 10: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/10.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
x x
yyy 5 x
p2 5 g(p1)
p3 5 g(p2)
p1 5 g(p0)
(p1, p2)(p2, p2)
(p0, p1)
y 5 g(x)
(p1, p1)
p1 p3 p2 p0
(a) (b)
p0 p1 p2
y 5 g(x)
(p2, p2)
(p0, p1)
(p2, p3)
p1 5 g(p0)
p3 5 g(p2)
y 5 x
p2 5 g(p1)
(p1, p1)
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 5 / 54
![Page 11: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/11.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Fixed-Point AlgorithmTo find the fixed point of g in an interval [a, b], given the equationx = g(x) with an initial guess p0 ∈ [a, b]:
1. n = 1;2. pn = g(pn−1);3. If |pn − pn−1| < ε then 5;4. n → n + 1; go to 2.5. End of Procedure.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 6 / 54
![Page 12: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/12.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Fixed-Point AlgorithmTo find the fixed point of g in an interval [a, b], given the equationx = g(x) with an initial guess p0 ∈ [a, b]:
1. n = 1;2. pn = g(pn−1);3. If |pn − pn−1| < ε then 5;4. n → n + 1; go to 2.5. End of Procedure.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 6 / 54
![Page 13: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/13.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Fixed-Point AlgorithmTo find the fixed point of g in an interval [a, b], given the equationx = g(x) with an initial guess p0 ∈ [a, b]:
1. n = 1;
2. pn = g(pn−1);3. If |pn − pn−1| < ε then 5;4. n → n + 1; go to 2.5. End of Procedure.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 6 / 54
![Page 14: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/14.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Fixed-Point AlgorithmTo find the fixed point of g in an interval [a, b], given the equationx = g(x) with an initial guess p0 ∈ [a, b]:
1. n = 1;2. pn = g(pn−1);
3. If |pn − pn−1| < ε then 5;4. n → n + 1; go to 2.5. End of Procedure.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 6 / 54
![Page 15: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/15.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Fixed-Point AlgorithmTo find the fixed point of g in an interval [a, b], given the equationx = g(x) with an initial guess p0 ∈ [a, b]:
1. n = 1;2. pn = g(pn−1);3. If |pn − pn−1| < ε then 5;
4. n → n + 1; go to 2.5. End of Procedure.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 6 / 54
![Page 16: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/16.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Fixed-Point AlgorithmTo find the fixed point of g in an interval [a, b], given the equationx = g(x) with an initial guess p0 ∈ [a, b]:
1. n = 1;2. pn = g(pn−1);3. If |pn − pn−1| < ε then 5;4. n → n + 1; go to 2.
5. End of Procedure.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 6 / 54
![Page 17: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/17.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Fixed-Point AlgorithmTo find the fixed point of g in an interval [a, b], given the equationx = g(x) with an initial guess p0 ∈ [a, b]:
1. n = 1;2. pn = g(pn−1);3. If |pn − pn−1| < ε then 5;4. n → n + 1; go to 2.5. End of Procedure.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 6 / 54
![Page 18: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/18.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
A Single Nonlinear Equation
Example 1The equation
x3 + 4x2 − 10 = 0
has a unique root in [1, 2]. Its value is approximately 1.365230013.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 7 / 54
![Page 19: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/19.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
f (x) = x3 + 4x2 − 10 = 0 on [1, 2]
y
x1 2
14
−5
y = f(x) = x3 + 4x2 −10
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 8 / 54
![Page 20: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/20.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
f (x) = x3 + 4x2 − 10 = 0 on [1, 2]
Possible Choices for g(x)
There are many ways to change the equation to the fixed-pointform x = g(x) using simple algebraic manipulation.For example, to obtain the function g described in part (c), we canmanipulate the equation x3 + 4x2 − 10 = 0 as follows:
4x2 = 10−x3, so x2 =14(10−x3), and x = ±1
2(10−x3)1/2.
We will consider 5 such rearrangements and, later in this section,provide a brief analysis as to why some do and some notconverge to p = 1.365230013.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 9 / 54
![Page 21: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/21.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
f (x) = x3 + 4x2 − 10 = 0 on [1, 2]
Possible Choices for g(x)
There are many ways to change the equation to the fixed-pointform x = g(x) using simple algebraic manipulation.
For example, to obtain the function g described in part (c), we canmanipulate the equation x3 + 4x2 − 10 = 0 as follows:
4x2 = 10−x3, so x2 =14(10−x3), and x = ±1
2(10−x3)1/2.
We will consider 5 such rearrangements and, later in this section,provide a brief analysis as to why some do and some notconverge to p = 1.365230013.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 9 / 54
![Page 22: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/22.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
f (x) = x3 + 4x2 − 10 = 0 on [1, 2]
Possible Choices for g(x)
There are many ways to change the equation to the fixed-pointform x = g(x) using simple algebraic manipulation.For example, to obtain the function g described in part (c), we canmanipulate the equation x3 + 4x2 − 10 = 0 as follows:
4x2 = 10−x3, so x2 =14(10−x3), and x = ±1
2(10−x3)1/2.
We will consider 5 such rearrangements and, later in this section,provide a brief analysis as to why some do and some notconverge to p = 1.365230013.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 9 / 54
![Page 23: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/23.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
f (x) = x3 + 4x2 − 10 = 0 on [1, 2]
Possible Choices for g(x)
There are many ways to change the equation to the fixed-pointform x = g(x) using simple algebraic manipulation.For example, to obtain the function g described in part (c), we canmanipulate the equation x3 + 4x2 − 10 = 0 as follows:
4x2 = 10−x3, so x2 =14(10−x3), and x = ±1
2(10−x3)1/2.
We will consider 5 such rearrangements and, later in this section,provide a brief analysis as to why some do and some notconverge to p = 1.365230013.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 9 / 54
![Page 24: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/24.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Solving f (x) = x3 + 4x2 − 10 = 0
5 Possible Transpositions to x = g(x)
x = g1(x) = x − x3 − 4x2 + 10
x = g2(x) =
√10x− 4x
x = g3(x) =12
√10− x3
x = g4(x) =
√10
4 + x
x = g5(x) = x − x3 + 4x2 − 103x2 + 8x
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 10 / 54
![Page 25: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/25.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Numerical Results for f (x) = x3 + 4x2 − 10 = 0
n g1 g2 g3 g4 g5
0 1.5 1.5 1.5 1.5 1.51 −0.875 0.8165 1.286953768 1.348399725 1.3733333332 6.732 2.9969 1.402540804 1.367376372 1.3652620153 −469.7 (−8.65)1/2 1.345458374 1.364957015 1.3652300144 1.03× 108 1.375170253 1.365264748 1.3652300135 1.360094193 1.365225594
10 1.365410062 1.36523001415 1.365223680 1.36523001320 1.36523023625 1.36523000630 1.365230013
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 11 / 54
![Page 26: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/26.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Solving f (x) = x3 + 4x2 − 10 = 0
x = g(x) with x0 = 1.5
x = g1(x) = x − x3 − 4x2 + 10 Does not Converge
x = g2(x) =
√10x− 4x Does not Converge
x = g3(x) =12
√10− x3 Converges after 31 Iterations
x = g4(x) =
√10
4 + xConverges after 12 Iterations
x = g5(x) = x − x3 + 4x2 − 103x2 + 8x
Converges after 5 Iterations
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 12 / 54
![Page 27: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/27.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Outline
1 Functional (Fixed-Point) Iteration
2 Convergence Criteria for the Fixed-Point Method
3 Sample Problem: f (x) = x3 + 4x2 − 10 = 0
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 13 / 54
![Page 28: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/28.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
A Crucial QuestionHow can we find a fixed-point problem that produces a sequencethat reliably and rapidly converges to a solution to a givenroot-finding problem?
The following theorem and its corollary give us some cluesconcerning the paths we should pursue and, perhaps moreimportantly, some we should reject.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 14 / 54
![Page 29: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/29.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
A Crucial QuestionHow can we find a fixed-point problem that produces a sequencethat reliably and rapidly converges to a solution to a givenroot-finding problem?The following theorem and its corollary give us some cluesconcerning the paths we should pursue and, perhaps moreimportantly, some we should reject.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 14 / 54
![Page 30: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/30.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Convergence ResultLet g ∈ C[a, b] with g(x) ∈ [a, b] for all x ∈ [a, b]. Let g′(x) exist on(a, b) with
|g′(x)| ≤ k < 1, ∀ x ∈ [a, b].
If p0 is any point in [a, b] then the sequence defined by
pn = g(pn−1), n ≥ 1,
will converge to the unique fixed point p in [a, b].
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 15 / 54
![Page 31: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/31.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Convergence ResultLet g ∈ C[a, b] with g(x) ∈ [a, b] for all x ∈ [a, b]. Let g′(x) exist on(a, b) with
|g′(x)| ≤ k < 1, ∀ x ∈ [a, b].
If p0 is any point in [a, b] then the sequence defined by
pn = g(pn−1), n ≥ 1,
will converge to the unique fixed point p in [a, b].
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 15 / 54
![Page 32: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/32.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Proof of the Convergence Result
By the Uniquenes Theorem, a unique fixed point exists in [a, b].Since g maps [a, b] into itself, the sequence {pn}∞n=0 is defined forall n ≥ 0 and pn ∈ [a, b] for all n.Using the Mean Value Theorem MVT and the assumption that|g′(x)| ≤ k < 1, ∀ x ∈ [a, b], we write
|pn − p| = |g(pn−1)− g(p)|≤
∣∣g′(ξ)∣∣ |pn−1 − p|≤ k |pn−1 − p|
where ξ ∈ (a, b).
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 16 / 54
![Page 33: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/33.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Proof of the Convergence ResultBy the Uniquenes Theorem, a unique fixed point exists in [a, b].
Since g maps [a, b] into itself, the sequence {pn}∞n=0 is defined forall n ≥ 0 and pn ∈ [a, b] for all n.Using the Mean Value Theorem MVT and the assumption that|g′(x)| ≤ k < 1, ∀ x ∈ [a, b], we write
|pn − p| = |g(pn−1)− g(p)|≤
∣∣g′(ξ)∣∣ |pn−1 − p|≤ k |pn−1 − p|
where ξ ∈ (a, b).
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 16 / 54
![Page 34: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/34.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Proof of the Convergence ResultBy the Uniquenes Theorem, a unique fixed point exists in [a, b].Since g maps [a, b] into itself, the sequence {pn}∞n=0 is defined forall n ≥ 0 and pn ∈ [a, b] for all n.
Using the Mean Value Theorem MVT and the assumption that|g′(x)| ≤ k < 1, ∀ x ∈ [a, b], we write
|pn − p| = |g(pn−1)− g(p)|≤
∣∣g′(ξ)∣∣ |pn−1 − p|≤ k |pn−1 − p|
where ξ ∈ (a, b).
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 16 / 54
![Page 35: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/35.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Proof of the Convergence ResultBy the Uniquenes Theorem, a unique fixed point exists in [a, b].Since g maps [a, b] into itself, the sequence {pn}∞n=0 is defined forall n ≥ 0 and pn ∈ [a, b] for all n.Using the Mean Value Theorem MVT and the assumption that|g′(x)| ≤ k < 1, ∀ x ∈ [a, b], we write
|pn − p| = |g(pn−1)− g(p)|
≤∣∣g′(ξ)∣∣ |pn−1 − p|
≤ k |pn−1 − p|
where ξ ∈ (a, b).
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 16 / 54
![Page 36: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/36.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Proof of the Convergence ResultBy the Uniquenes Theorem, a unique fixed point exists in [a, b].Since g maps [a, b] into itself, the sequence {pn}∞n=0 is defined forall n ≥ 0 and pn ∈ [a, b] for all n.Using the Mean Value Theorem MVT and the assumption that|g′(x)| ≤ k < 1, ∀ x ∈ [a, b], we write
|pn − p| = |g(pn−1)− g(p)|≤
∣∣g′(ξ)∣∣ |pn−1 − p|
≤ k |pn−1 − p|
where ξ ∈ (a, b).
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 16 / 54
![Page 37: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/37.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Proof of the Convergence ResultBy the Uniquenes Theorem, a unique fixed point exists in [a, b].Since g maps [a, b] into itself, the sequence {pn}∞n=0 is defined forall n ≥ 0 and pn ∈ [a, b] for all n.Using the Mean Value Theorem MVT and the assumption that|g′(x)| ≤ k < 1, ∀ x ∈ [a, b], we write
|pn − p| = |g(pn−1)− g(p)|≤
∣∣g′(ξ)∣∣ |pn−1 − p|≤ k |pn−1 − p|
where ξ ∈ (a, b).
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 16 / 54
![Page 38: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/38.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Proof of the Convergence Result (Cont’d)Applying the inequality of the hypothesis inductively gives
|pn − p| ≤ k |pn−1 − p|≤ k2 |pn−2 − p|≤ kn |p0 − p|
Since k < 1,lim
n→∞|pn − p| ≤ lim
n→∞kn |p0 − p| = 0,
and {pn}∞n=0 converges to p.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 17 / 54
![Page 39: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/39.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Proof of the Convergence Result (Cont’d)Applying the inequality of the hypothesis inductively gives
|pn − p| ≤ k |pn−1 − p|
≤ k2 |pn−2 − p|≤ kn |p0 − p|
Since k < 1,lim
n→∞|pn − p| ≤ lim
n→∞kn |p0 − p| = 0,
and {pn}∞n=0 converges to p.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 17 / 54
![Page 40: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/40.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Proof of the Convergence Result (Cont’d)Applying the inequality of the hypothesis inductively gives
|pn − p| ≤ k |pn−1 − p|≤ k2 |pn−2 − p|
≤ kn |p0 − p|
Since k < 1,lim
n→∞|pn − p| ≤ lim
n→∞kn |p0 − p| = 0,
and {pn}∞n=0 converges to p.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 17 / 54
![Page 41: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/41.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Proof of the Convergence Result (Cont’d)Applying the inequality of the hypothesis inductively gives
|pn − p| ≤ k |pn−1 − p|≤ k2 |pn−2 − p|≤ kn |p0 − p|
Since k < 1,lim
n→∞|pn − p| ≤ lim
n→∞kn |p0 − p| = 0,
and {pn}∞n=0 converges to p.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 17 / 54
![Page 42: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/42.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Proof of the Convergence Result (Cont’d)Applying the inequality of the hypothesis inductively gives
|pn − p| ≤ k |pn−1 − p|≤ k2 |pn−2 − p|≤ kn |p0 − p|
Since k < 1,lim
n→∞|pn − p| ≤ lim
n→∞kn |p0 − p| = 0,
and {pn}∞n=0 converges to p.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 17 / 54
![Page 43: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/43.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Corrollary to the Convergence ResultIf g satisfies the hypothesis of the Theorem, then
|pn − p| ≤ kn
1− k|p1 − p0|.
Proof of Corollary (1 of 3)For n ≥ 1, the procedure used in the proof of the theorem implies that
|pn+1 − pn| = |g(pn)− g(pn−1)|≤ k |pn − pn−1|≤ · · ·≤ kn |p1 − p0|
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 18 / 54
![Page 44: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/44.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Corrollary to the Convergence ResultIf g satisfies the hypothesis of the Theorem, then
|pn − p| ≤ kn
1− k|p1 − p0|.
Proof of Corollary (1 of 3)For n ≥ 1, the procedure used in the proof of the theorem implies that
|pn+1 − pn| = |g(pn)− g(pn−1)|
≤ k |pn − pn−1|≤ · · ·≤ kn |p1 − p0|
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 18 / 54
![Page 45: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/45.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Corrollary to the Convergence ResultIf g satisfies the hypothesis of the Theorem, then
|pn − p| ≤ kn
1− k|p1 − p0|.
Proof of Corollary (1 of 3)For n ≥ 1, the procedure used in the proof of the theorem implies that
|pn+1 − pn| = |g(pn)− g(pn−1)|≤ k |pn − pn−1|
≤ · · ·≤ kn |p1 − p0|
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 18 / 54
![Page 46: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/46.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Corrollary to the Convergence ResultIf g satisfies the hypothesis of the Theorem, then
|pn − p| ≤ kn
1− k|p1 − p0|.
Proof of Corollary (1 of 3)For n ≥ 1, the procedure used in the proof of the theorem implies that
|pn+1 − pn| = |g(pn)− g(pn−1)|≤ k |pn − pn−1|≤ · · ·
≤ kn |p1 − p0|
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 18 / 54
![Page 47: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/47.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Corrollary to the Convergence ResultIf g satisfies the hypothesis of the Theorem, then
|pn − p| ≤ kn
1− k|p1 − p0|.
Proof of Corollary (1 of 3)For n ≥ 1, the procedure used in the proof of the theorem implies that
|pn+1 − pn| = |g(pn)− g(pn−1)|≤ k |pn − pn−1|≤ · · ·≤ kn |p1 − p0|
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 18 / 54
![Page 48: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/48.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
|pn+1 − pn| ≤ kn |p1 − p0|
Proof of Corollary (2 of 3)
Thus, for m > n ≥ 1,
|pm − pn| = |pm − pm−1 + pm−1 − pm−2 + · · ·+ pn+1 − pn|≤ |pm − pm−1|+ |pm−1 − pm−2|+ · · ·+ |pn+1 − pn|≤ km−1 |p1 − p0|+ km−2 |p1 − p0|+ · · ·+ kn |p1 − p0|
≤ kn(
1 + k + k2 + · · ·+ km−n−1)|p1 − p0| .
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 19 / 54
![Page 49: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/49.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
|pn+1 − pn| ≤ kn |p1 − p0|
Proof of Corollary (2 of 3)Thus, for m > n ≥ 1,
|pm − pn| = |pm − pm−1 + pm−1 − pm−2 + · · ·+ pn+1 − pn|
≤ |pm − pm−1|+ |pm−1 − pm−2|+ · · ·+ |pn+1 − pn|≤ km−1 |p1 − p0|+ km−2 |p1 − p0|+ · · ·+ kn |p1 − p0|
≤ kn(
1 + k + k2 + · · ·+ km−n−1)|p1 − p0| .
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 19 / 54
![Page 50: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/50.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
|pn+1 − pn| ≤ kn |p1 − p0|
Proof of Corollary (2 of 3)Thus, for m > n ≥ 1,
|pm − pn| = |pm − pm−1 + pm−1 − pm−2 + · · ·+ pn+1 − pn|≤ |pm − pm−1|+ |pm−1 − pm−2|+ · · ·+ |pn+1 − pn|
≤ km−1 |p1 − p0|+ km−2 |p1 − p0|+ · · ·+ kn |p1 − p0|
≤ kn(
1 + k + k2 + · · ·+ km−n−1)|p1 − p0| .
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 19 / 54
![Page 51: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/51.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
|pn+1 − pn| ≤ kn |p1 − p0|
Proof of Corollary (2 of 3)Thus, for m > n ≥ 1,
|pm − pn| = |pm − pm−1 + pm−1 − pm−2 + · · ·+ pn+1 − pn|≤ |pm − pm−1|+ |pm−1 − pm−2|+ · · ·+ |pn+1 − pn|≤ km−1 |p1 − p0|+ km−2 |p1 − p0|+ · · ·+ kn |p1 − p0|
≤ kn(
1 + k + k2 + · · ·+ km−n−1)|p1 − p0| .
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 19 / 54
![Page 52: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/52.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
|pn+1 − pn| ≤ kn |p1 − p0|
Proof of Corollary (2 of 3)Thus, for m > n ≥ 1,
|pm − pn| = |pm − pm−1 + pm−1 − pm−2 + · · ·+ pn+1 − pn|≤ |pm − pm−1|+ |pm−1 − pm−2|+ · · ·+ |pn+1 − pn|≤ km−1 |p1 − p0|+ km−2 |p1 − p0|+ · · ·+ kn |p1 − p0|
≤ kn(
1 + k + k2 + · · ·+ km−n−1)|p1 − p0| .
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 19 / 54
![Page 53: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/53.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
|pm − pn| ≤ kn(
1 + k + k2 + · · ·+ km−n−1)|p1 − p0| .
Proof of Corollary (3 of 3)
However, since limm→∞ pm = p, we obtain
|p − pn| = limm→∞
|pm − pn|
≤ kn |p1 − p0|∞∑
i=1
k i
=kn
1− k|p1 − p0| .
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 20 / 54
![Page 54: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/54.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
|pm − pn| ≤ kn(
1 + k + k2 + · · ·+ km−n−1)|p1 − p0| .
Proof of Corollary (3 of 3)However, since limm→∞ pm = p, we obtain
|p − pn| = limm→∞
|pm − pn|
≤ kn |p1 − p0|∞∑
i=1
k i
=kn
1− k|p1 − p0| .
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 20 / 54
![Page 55: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/55.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
|pm − pn| ≤ kn(
1 + k + k2 + · · ·+ km−n−1)|p1 − p0| .
Proof of Corollary (3 of 3)However, since limm→∞ pm = p, we obtain
|p − pn| = limm→∞
|pm − pn|
≤ kn |p1 − p0|∞∑
i=1
k i
=kn
1− k|p1 − p0| .
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 20 / 54
![Page 56: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/56.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
|pm − pn| ≤ kn(
1 + k + k2 + · · ·+ km−n−1)|p1 − p0| .
Proof of Corollary (3 of 3)However, since limm→∞ pm = p, we obtain
|p − pn| = limm→∞
|pm − pn|
≤ kn |p1 − p0|∞∑
i=1
k i
=kn
1− k|p1 − p0| .
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 20 / 54
![Page 57: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/57.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Example: g(x) = g(x) = 3−x
Consider the iteration function g(x) = 3−x over the interval [13 , 1]
starting with p0 = 13 . Determine a lower bound for the number of
iterations n required so that |pn − p| < 10−5?
Determine the Parameters of the Problem
Note that p1 = g(p0) = 3−13 = 0.6933612 and, since g′(x) = −3−x ln 3,
we obtain the bound
|g′(x)| ≤ 3−13 ln 3 ≤ .7617362 ≈ .762 = k .
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 21 / 54
![Page 58: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/58.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Example: g(x) = g(x) = 3−x
Consider the iteration function g(x) = 3−x over the interval [13 , 1]
starting with p0 = 13 . Determine a lower bound for the number of
iterations n required so that |pn − p| < 10−5?
Determine the Parameters of the Problem
Note that p1 = g(p0) = 3−13 = 0.6933612 and, since g′(x) = −3−x ln 3,
we obtain the bound
|g′(x)| ≤ 3−13 ln 3 ≤ .7617362 ≈ .762 = k .
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 21 / 54
![Page 59: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/59.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Example: g(x) = g(x) = 3−x
Consider the iteration function g(x) = 3−x over the interval [13 , 1]
starting with p0 = 13 . Determine a lower bound for the number of
iterations n required so that |pn − p| < 10−5?
Determine the Parameters of the Problem
Note that p1 = g(p0) = 3−13 = 0.6933612 and, since g′(x) = −3−x ln 3,
we obtain the bound
|g′(x)| ≤ 3−13 ln 3 ≤ .7617362 ≈ .762 = k .
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 21 / 54
![Page 60: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/60.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Use the Corollary
Therefore, we have
|pn − p| ≤ kn
1− k|p0 − p1|
≤ .762n
1− .762
∣∣∣∣13− .6933612
∣∣∣∣≤ 1.513× 0.762n
We require that
1.513× 0.762n < 10−5 or n > 43.88
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 22 / 54
![Page 61: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/61.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Use the CorollaryTherefore, we have
|pn − p| ≤ kn
1− k|p0 − p1|
≤ .762n
1− .762
∣∣∣∣13− .6933612
∣∣∣∣≤ 1.513× 0.762n
We require that
1.513× 0.762n < 10−5 or n > 43.88
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 22 / 54
![Page 62: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/62.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Use the CorollaryTherefore, we have
|pn − p| ≤ kn
1− k|p0 − p1|
≤ .762n
1− .762
∣∣∣∣13− .6933612
∣∣∣∣
≤ 1.513× 0.762n
We require that
1.513× 0.762n < 10−5 or n > 43.88
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 22 / 54
![Page 63: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/63.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Use the CorollaryTherefore, we have
|pn − p| ≤ kn
1− k|p0 − p1|
≤ .762n
1− .762
∣∣∣∣13− .6933612
∣∣∣∣≤ 1.513× 0.762n
We require that
1.513× 0.762n < 10−5 or n > 43.88
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 22 / 54
![Page 64: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/64.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Functional (Fixed-Point) Iteration
Use the CorollaryTherefore, we have
|pn − p| ≤ kn
1− k|p0 − p1|
≤ .762n
1− .762
∣∣∣∣13− .6933612
∣∣∣∣≤ 1.513× 0.762n
We require that
1.513× 0.762n < 10−5 or n > 43.88
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 22 / 54
![Page 65: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/65.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Footnote on the Estimate Obtained
It is important to realise that the estimate for the number ofiterations required given by the theorem is an upper bound.In the previous example, only 21 iterations are required inpractice, i.e. p21 = 0.54781 is accurate to 10−5.The reason, in this case, is that we used
g′(1) = 0.762
whereasg′(0.54781) = 0.602
If one had used k = 0.602 (were it available) to compute thebound, one would obtain N = 23 which is a more accurateestimate.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 23 / 54
![Page 66: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/66.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Footnote on the Estimate ObtainedIt is important to realise that the estimate for the number ofiterations required given by the theorem is an upper bound.
In the previous example, only 21 iterations are required inpractice, i.e. p21 = 0.54781 is accurate to 10−5.The reason, in this case, is that we used
g′(1) = 0.762
whereasg′(0.54781) = 0.602
If one had used k = 0.602 (were it available) to compute thebound, one would obtain N = 23 which is a more accurateestimate.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 23 / 54
![Page 67: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/67.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Footnote on the Estimate ObtainedIt is important to realise that the estimate for the number ofiterations required given by the theorem is an upper bound.In the previous example, only 21 iterations are required inpractice, i.e. p21 = 0.54781 is accurate to 10−5.
The reason, in this case, is that we used
g′(1) = 0.762
whereasg′(0.54781) = 0.602
If one had used k = 0.602 (were it available) to compute thebound, one would obtain N = 23 which is a more accurateestimate.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 23 / 54
![Page 68: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/68.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Footnote on the Estimate ObtainedIt is important to realise that the estimate for the number ofiterations required given by the theorem is an upper bound.In the previous example, only 21 iterations are required inpractice, i.e. p21 = 0.54781 is accurate to 10−5.The reason, in this case, is that we used
g′(1) = 0.762
whereasg′(0.54781) = 0.602
If one had used k = 0.602 (were it available) to compute thebound, one would obtain N = 23 which is a more accurateestimate.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 23 / 54
![Page 69: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/69.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Footnote on the Estimate ObtainedIt is important to realise that the estimate for the number ofiterations required given by the theorem is an upper bound.In the previous example, only 21 iterations are required inpractice, i.e. p21 = 0.54781 is accurate to 10−5.The reason, in this case, is that we used
g′(1) = 0.762
whereasg′(0.54781) = 0.602
If one had used k = 0.602 (were it available) to compute thebound, one would obtain N = 23 which is a more accurateestimate.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 23 / 54
![Page 70: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/70.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Outline
1 Functional (Fixed-Point) Iteration
2 Convergence Criteria for the Fixed-Point Method
3 Sample Problem: f (x) = x3 + 4x2 − 10 = 0
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 24 / 54
![Page 71: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/71.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
A Single Nonlinear Equation
Example 2We return to Example 1 and the equation
x3 + 4x2 − 10 = 0
which has a unique root in [1, 2]. Its value is approximately1.365230013.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 25 / 54
![Page 72: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/72.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
f (x) = x3 + 4x2 − 10 = 0 on [1, 2]
y
x1 2
14
−5
y = f(x) = x3 + 4x2 −10
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 26 / 54
![Page 73: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/73.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Solving f (x) = x3 + 4x2 − 10 = 0
Earlier, we listed 5 possible transpositions to x = g(x)
x = g1(x) = x − x3 − 4x2 + 10
x = g2(x) =
√10x− 4x
x = g3(x) =12
√10− x3
x = g4(x) =
√10
4 + x
x = g5(x) = x − x3 + 4x2 − 103x2 + 8x
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 27 / 54
![Page 74: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/74.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Solving f (x) = x3 + 4x2 − 10 = 0
Results Observed for x = g(x) with x0 = 1.5
x = g1(x) = x − x3 − 4x2 + 10 Does not Converge
x = g2(x) =
√10x− 4x Does not Converge
x = g3(x) =12
√10− x3 Converges after 31 Iterations
x = g4(x) =
√10
4 + xConverges after 12 Iterations
x = g5(x) = x − x3 + 4x2 − 103x2 + 8x
Converges after 5 Iterations
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 28 / 54
![Page 75: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/75.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Solving f (x) = x3 + 4x2 − 10 = 0
x = g(x) with x0 = 1.5
x = g1(x) = x − x3 − 4x2 + 10 Does not Converge
x = g2(x) =
√10x− 4x Does not Converge
x = g3(x) =12
√10− x3 Converges after 31 Iterations
x = g4(x) =
√10
4 + xConverges after 12 Iterations
x = g5(x) = x − x3 + 4x2 − 103x2 + 8x
Converges after 5 Iterations
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 29 / 54
![Page 76: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/76.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Solving f (x) = x3 + 4x2 − 10 = 0
x = g1(x) = x − x3 − 4x2 + 10
Iteration for x = g1(x) Does Not Converge
Since
g′1(x) = 1− 3x2 − 8x g′1(1) = −10 g′1(2) = −27
there is no interval [a, b] containing p for which∣∣g′1(x)
∣∣ < 1. Also, notethat
g1(1) = 6 and g2(2) = −12
so that g(x) /∈ [1, 2] for x ∈ [1, 2].
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 30 / 54
![Page 77: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/77.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Solving f (x) = x3 + 4x2 − 10 = 0
x = g1(x) = x − x3 − 4x2 + 10
Iteration for x = g1(x) Does Not ConvergeSince
g′1(x) = 1− 3x2 − 8x g′1(1) = −10 g′1(2) = −27
there is no interval [a, b] containing p for which∣∣g′1(x)
∣∣ < 1.
Also, notethat
g1(1) = 6 and g2(2) = −12
so that g(x) /∈ [1, 2] for x ∈ [1, 2].
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 30 / 54
![Page 78: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/78.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Solving f (x) = x3 + 4x2 − 10 = 0
x = g1(x) = x − x3 − 4x2 + 10
Iteration for x = g1(x) Does Not ConvergeSince
g′1(x) = 1− 3x2 − 8x g′1(1) = −10 g′1(2) = −27
there is no interval [a, b] containing p for which∣∣g′1(x)
∣∣ < 1. Also, notethat
g1(1) = 6 and g2(2) = −12
so that g(x) /∈ [1, 2] for x ∈ [1, 2].
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 30 / 54
![Page 79: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/79.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Iteration Function: x = g1(x) = x − x3 − 4x2 + 10
Iterations starting with p0 = 1.5
n pn−1 pn |pn − pn−1|1 1.5000000 -0.8750000 2.37500002 -0.8750000 6.7324219 7.60742193 6.7324219 -469.7200120 476.4524339
p4 ≈ 1.03× 108
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 31 / 54
![Page 80: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/80.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
g1 Does Not Map [1, 2] into [1, 2]
y
x
1 2
2
−10
6
g1(x) = x − x3− 4x
2 + 10
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 32 / 54
![Page 81: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/81.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
|g′1(x)| > 1 on [1, 2]
yx1 2
−10
−27
g′
1(x) = 1− 3x
2− 8x
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 33 / 54
![Page 82: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/82.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Solving f (x) = x3 + 4x2 − 10 = 0
x = g(x) with x0 = 1.5
x = g1(x) = x − x3 − 4x2 + 10 Does not Converge
x = g2(x) =
√10x− 4x Does not Converge
x = g3(x) =12
√10− x3 Converges after 31 Iterations
x = g4(x) =
√10
4 + xConverges after 12 Iterations
x = g5(x) = x − x3 + 4x2 − 103x2 + 8x
Converges after 5 Iterations
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 34 / 54
![Page 83: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/83.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Solving f (x) = x3 + 4x2 − 10 = 0
x = g2(x) =
√10x− 4x
Iteration for x = g2(x) is Not Defined
It is clear that g2(x) does not map [1, 2] onto [1, 2] and the sequence{pn}∞n=0 is not defined for p0 = 1.5. Also, there is no interval containingp such that ∣∣g′2(x)
∣∣ < 1
sinceg′(1) ≈ −2.86 g′(p) ≈ −3.43
and g′(x) is not defined for x > 1.58.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 35 / 54
![Page 84: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/84.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Solving f (x) = x3 + 4x2 − 10 = 0
x = g2(x) =
√10x− 4x
Iteration for x = g2(x) is Not DefinedIt is clear that g2(x) does not map [1, 2] onto [1, 2] and the sequence{pn}∞n=0 is not defined for p0 = 1.5.
Also, there is no interval containingp such that ∣∣g′2(x)
∣∣ < 1
sinceg′(1) ≈ −2.86 g′(p) ≈ −3.43
and g′(x) is not defined for x > 1.58.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 35 / 54
![Page 85: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/85.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Solving f (x) = x3 + 4x2 − 10 = 0
x = g2(x) =
√10x− 4x
Iteration for x = g2(x) is Not DefinedIt is clear that g2(x) does not map [1, 2] onto [1, 2] and the sequence{pn}∞n=0 is not defined for p0 = 1.5. Also, there is no interval containingp such that ∣∣g′2(x)
∣∣ < 1
sinceg′(1) ≈ −2.86 g′(p) ≈ −3.43
and g′(x) is not defined for x > 1.58.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 35 / 54
![Page 86: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/86.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Iteration Function: x = g2(x) =√
10x − 4x
Iterations starting with p0 = 1.5
n pn−1 pn |pn − pn−1|1 1.5000000 0.8164966 0.68350342 0.8164966 2.9969088 2.18041223 2.9969088
√−8.6509 —
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 36 / 54
![Page 87: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/87.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Solving f (x) = x3 + 4x2 − 10 = 0
x = g(x) with x0 = 1.5
x = g1(x) = x − x3 − 4x2 + 10 Does not Converge
x = g2(x) =
√10x− 4x Does not Converge
x = g3(x) =12
√10− x3 Converges after 31 Iterations
x = g4(x) =
√10
4 + xConverges after 12 Iterations
x = g5(x) = x − x3 + 4x2 − 103x2 + 8x
Converges after 5 Iterations
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 37 / 54
![Page 88: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/88.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Solving f (x) = x3 + 4x2 − 10 = 0
x = g3(x) =12
√10− x3
Iteration for x = g3(x) Converges (Slowly)
By differentiation,
g′3(x) = − 3x2
4√
10− x3< 0 for x ∈ [1, 2]
and so g=g3 is strictly decreasing on [1, 2]. However,∣∣g′3(x)
∣∣ > 1 forx > 1.71 and
∣∣g′3(2)∣∣ ≈ −2.12. A closer examination of {pn}∞n=0 will
show that it suffices to consider the interval [1, 1.7] where∣∣g′3(x)
∣∣ < 1and g(x) ∈ [1, 1.7] for x ∈ [1, 1.7].
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 38 / 54
![Page 89: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/89.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Solving f (x) = x3 + 4x2 − 10 = 0
x = g3(x) =12
√10− x3
Iteration for x = g3(x) Converges (Slowly)By differentiation,
g′3(x) = − 3x2
4√
10− x3< 0 for x ∈ [1, 2]
and so g=g3 is strictly decreasing on [1, 2].
However,∣∣g′3(x)
∣∣ > 1 forx > 1.71 and
∣∣g′3(2)∣∣ ≈ −2.12. A closer examination of {pn}∞n=0 will
show that it suffices to consider the interval [1, 1.7] where∣∣g′3(x)
∣∣ < 1and g(x) ∈ [1, 1.7] for x ∈ [1, 1.7].
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 38 / 54
![Page 90: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/90.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Solving f (x) = x3 + 4x2 − 10 = 0
x = g3(x) =12
√10− x3
Iteration for x = g3(x) Converges (Slowly)By differentiation,
g′3(x) = − 3x2
4√
10− x3< 0 for x ∈ [1, 2]
and so g=g3 is strictly decreasing on [1, 2]. However,∣∣g′3(x)
∣∣ > 1 forx > 1.71 and
∣∣g′3(2)∣∣ ≈ −2.12.
A closer examination of {pn}∞n=0 willshow that it suffices to consider the interval [1, 1.7] where
∣∣g′3(x)∣∣ < 1
and g(x) ∈ [1, 1.7] for x ∈ [1, 1.7].
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 38 / 54
![Page 91: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/91.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Solving f (x) = x3 + 4x2 − 10 = 0
x = g3(x) =12
√10− x3
Iteration for x = g3(x) Converges (Slowly)By differentiation,
g′3(x) = − 3x2
4√
10− x3< 0 for x ∈ [1, 2]
and so g=g3 is strictly decreasing on [1, 2]. However,∣∣g′3(x)
∣∣ > 1 forx > 1.71 and
∣∣g′3(2)∣∣ ≈ −2.12. A closer examination of {pn}∞n=0 will
show that it suffices to consider the interval [1, 1.7] where∣∣g′3(x)
∣∣ < 1and g(x) ∈ [1, 1.7] for x ∈ [1, 1.7].
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 38 / 54
![Page 92: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/92.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Iteration Function: x = g3(x) = 12
√10− x3
Iterations starting with p0 = 1.5
n pn−1 pn |pn − pn−1|1 1.500000000 1.286953768 0.2130462322 1.286953768 1.402540804 0.1155870363 1.402540804 1.345458374 0.0570824304 1.345458374 1.375170253 0.0297118795 1.375170253 1.360094193 0.0150760606 1.360094193 1.367846968 0.007752775
30 1.365230013 1.365230014 0.00000000131 1.365230014 1.365230013 0.000000000
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 39 / 54
![Page 93: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/93.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
g3 Maps [1, 1.7] into [1, 1.7]
y
x
1 2
2
1
g3(x) = 1
2
√
10− x3
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 40 / 54
![Page 94: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/94.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem∣∣g′3(x)∣∣ < 1 on [1, 1.7]
y
x1 2
1
−1
g′3(x) = −
3x2
4√
10−x3
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 41 / 54
![Page 95: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/95.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Solving f (x) = x3 + 4x2 − 10 = 0
x = g(x) with x0 = 1.5
x = g1(x) = x − x3 − 4x2 + 10 Does not Converge
x = g2(x) =
√10x− 4x Does not Converge
x = g3(x) =12
√10− x3 Converges after 31 Iterations
x = g4(x) =
√10
4 + xConverges after 12 Iterations
x = g5(x) = x − x3 + 4x2 − 103x2 + 8x
Converges after 5 Iterations
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 42 / 54
![Page 96: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/96.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Solving f (x) = x3 + 4x2 − 10 = 0
x = g4(x) =
√10
4 + x
Iteration for x = g4(x) Converges (Moderately)
By differentiation,
g′4(x) = −
√10
4(4 + x)3 < 0
and it is easy to show that
0.10 <∣∣g′4(x)
∣∣ < 0.15 ∀ x ∈ [1, 2]
The bound on the magnitude of∣∣g′4(x)
∣∣ is much smaller than that for∣∣g′3(x)∣∣ and this explains the reason for the much faster convergence.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 43 / 54
![Page 97: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/97.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Solving f (x) = x3 + 4x2 − 10 = 0
x = g4(x) =
√10
4 + x
Iteration for x = g4(x) Converges (Moderately)By differentiation,
g′4(x) = −
√10
4(4 + x)3 < 0
and it is easy to show that
0.10 <∣∣g′4(x)
∣∣ < 0.15 ∀ x ∈ [1, 2]
The bound on the magnitude of∣∣g′4(x)
∣∣ is much smaller than that for∣∣g′3(x)∣∣ and this explains the reason for the much faster convergence.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 43 / 54
![Page 98: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/98.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Solving f (x) = x3 + 4x2 − 10 = 0
x = g4(x) =
√10
4 + x
Iteration for x = g4(x) Converges (Moderately)By differentiation,
g′4(x) = −
√10
4(4 + x)3 < 0
and it is easy to show that
0.10 <∣∣g′4(x)
∣∣ < 0.15 ∀ x ∈ [1, 2]
The bound on the magnitude of∣∣g′4(x)
∣∣ is much smaller than that for∣∣g′3(x)∣∣ and this explains the reason for the much faster convergence.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 43 / 54
![Page 99: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/99.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Iteration Function: x = g4(x) =√
104+x
Iterations starting with p0 = 1.5
n pn−1 pn |pn − pn−1|1 1.500000000 1.348399725 0.1516002752 1.348399725 1.367376372 0.0189766473 1.367376372 1.364957015 0.0024193574 1.364957015 1.365264748 0.0003077335 1.365264748 1.365225594 0.0000391546 1.365225594 1.365230576 0.000004982
11 1.365230014 1.365230013 0.00000000012 1.365230013 1.365230013 0.000000000
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 44 / 54
![Page 100: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/100.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
g4 Maps [1, 2] into [1, 2]
y
x
1 2
2
1
g4(x) =√
10
4+x
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 45 / 54
![Page 101: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/101.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
|g′4(x)| < 1 on [1, 2]
y
x1 2
1
−1
g′
4(x) = −
√
104(4+x)3
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 46 / 54
![Page 102: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/102.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Solving f (x) = x3 + 4x2 − 10 = 0
x = g(x) with x0 = 1.5
x = g1(x) = x − x3 − 4x2 + 10 Does not Converge
x = g2(x) =
√10x− 4x Does not Converge
x = g3(x) =12
√10− x3 Converges after 31 Iterations
x = g4(x) =
√10
4 + xConverges after 12 Iterations
x = g5(x) = x − x3 + 4x2 − 103x2 + 8x
Converges after 5 Iterations
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 47 / 54
![Page 103: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/103.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Solving f (x) = x3 + 4x2 − 10 = 0
x = g5(x) = x − x3 + 4x2 − 103x2 + 8x
Iteration for x = g5(x) Converges (Rapidly)
For the iteration function g5(x), we obtain:
g5(x) = x − f (x)
f ′(x)⇒ g′5(x) =
f (x)f ′′(x)
[f ′(x)]2⇒ g′5(p) = 0
It is straightforward to show that 0 ≤∣∣g′5(x)
∣∣ < 0.28 ∀ x ∈ [1, 2] andthe order of convergence is quadratic since g′5(p) = 0.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 48 / 54
![Page 104: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/104.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Solving f (x) = x3 + 4x2 − 10 = 0
x = g5(x) = x − x3 + 4x2 − 103x2 + 8x
Iteration for x = g5(x) Converges (Rapidly)For the iteration function g5(x), we obtain:
g5(x) = x − f (x)
f ′(x)⇒ g′5(x) =
f (x)f ′′(x)
[f ′(x)]2⇒ g′5(p) = 0
It is straightforward to show that 0 ≤∣∣g′5(x)
∣∣ < 0.28 ∀ x ∈ [1, 2] andthe order of convergence is quadratic since g′5(p) = 0.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 48 / 54
![Page 105: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/105.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Solving f (x) = x3 + 4x2 − 10 = 0
x = g5(x) = x − x3 + 4x2 − 103x2 + 8x
Iteration for x = g5(x) Converges (Rapidly)For the iteration function g5(x), we obtain:
g5(x) = x − f (x)
f ′(x)⇒ g′5(x) =
f (x)f ′′(x)
[f ′(x)]2⇒ g′5(p) = 0
It is straightforward to show that 0 ≤∣∣g′5(x)
∣∣ < 0.28 ∀ x ∈ [1, 2] andthe order of convergence is quadratic since g′5(p) = 0.
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 48 / 54
![Page 106: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/106.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
Iteration Function: x = g5(x) = x − x3+4x2−103x2+8x
Iterations starting with p0 = 1.5
n pn−1 pn |pn − pn−1|1 1.500000000 1.373333333 0.1266666672 1.373333333 1.365262015 0.0080713183 1.365262015 1.365230014 0.0000320014 1.365230014 1.365230013 0.0000000015 1.365230013 1.365230013 0.000000000
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 49 / 54
![Page 107: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/107.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
g5 Maps [1, 2] into [1, 2]
y
x
1 2
2
1
g5(x) = x −x3+4x
2−10
3x2+8x
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 50 / 54
![Page 108: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/108.jpg)
Fixed-Point Iteration Convergence Criteria Sample Problem
|g′5(x)| < 1 on [1, 2]
y
x
1 2
1
−1
g′
5(x) = (x3+4x2−10)(6x+8)
(3x2+8x)2
Numerical Analysis (Chapter 2) Fixed-Point Iteration II R L Burden & J D Faires 51 / 54
![Page 109: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/109.jpg)
Questions?
![Page 110: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/110.jpg)
Reference Material
![Page 111: Solutions of Equations in One Variable …mamu/courses/231/Slides/ch02_2b.pdfSolutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden](https://reader033.fdocuments.in/reader033/viewer/2022042213/5eb761ec1b747d18234916c6/html5/thumbnails/111.jpg)
Mean Value Theorem
If f ∈ C[a, b] and f is differentiable on (a, b), then a number c existssuch that
f ′(c) =f (b)− f (a)
b − a
y
xa bc
Slope f 9(c)
Parallel lines
Slopeb 2 a
f (b) 2 f (a)
y 5 f (x)
Return to Fixed-Point Convergence Theorem