RadiiPolynomial
Notations
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General Notations
Radii Polynomial Notations
Special Cases
Example 7.7: Weighted Sequence Space Notations
1
Foundations
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1.1
Contraction Mapping Theorem
1.2
Mean Value Theorem
1.3
Newton’s Method
1.4
Radii Polynomials in Finite Dimensions
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1.4.1
Example 2.4.5: Finding \(\sqrt{2}\)
2
Radii Polynomial on Banach Spaces
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2.1
Weighted \(\ell ^1_\nu \) Banach Algebra (Section 7.4)
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2.1.1
Positive Reals and Scaled Spaces
2.1.2
Weighted \(\ell ^p\) Spaces
2.1.3
Cauchy Product (Definition 7.4.2)
2.1.4
Scalar-Sequence Compatibility
2.1.5
Banach Algebra Structure (Theorem 7.4.4)
2.1.6
Typeclass Instances (Corollary 7.4.5)
2.1.7
Algebra Structure
2.1.8
Architecture Summary
2.2
Radii Polynomial Approach on Banach Spaces
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2.2.1
Banach Space Setup
2.2.2
Neumann Series and Operator Invertibility
2.2.3
Newton-Like Operators for E to F Maps
2.2.4
Radii Polynomial Definitions
2.2.5
Operator Bounds
2.2.6
Helper Lemmas
2.2.7
Main Theorems
2.3
Example 7.7: Square Root via Power Series
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2.3.1
The Fixed-Point Problem
2.3.2
Approximate Solution
2.3.3
Bound Definitions
2.3.4
Radii Polynomial
2.3.5
Verification
2.3.6
Power Series Interpretation
2.3.7
Branch Selection
2.3.8
Summary
Dependency graph
1 Foundations
1.1
Contraction Mapping Theorem
1.2
Mean Value Theorem
1.3
Newton’s Method
1.4
Radii Polynomials in Finite Dimensions
1.4.1
Example 2.4.5: Finding \(\sqrt{2}\)