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
2 Radii Polynomial on Banach Spaces
2.1
Weighted \(\ell ^1_\nu \) Banach Algebra (Section 7.4)
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
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
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
Formal Power Series Connection
Analytic Evaluation
2.3.7
Branch Selection
2.3.8
Summary