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//! Chebyshev Polynomial Intelligence
//!
//! Mathematically accurate implementation of Chebyshev polynomials T_n(x), U_n(x)
//! for approximation theory with verified properties.
use crate::core::{Expression, Symbol};
use crate::functions::properties::*;
use std::collections::HashMap;
use std::sync::Arc;
/// Chebyshev Polynomial Intelligence
///
/// Complete mathematical intelligence for Chebyshev polynomials
pub struct ChebyshevIntelligence {
/// Function properties for Chebyshev polynomials
properties: HashMap<String, FunctionProperties>,
}
impl Default for ChebyshevIntelligence {
fn default() -> Self {
Self::new()
}
}
impl ChebyshevIntelligence {
/// Create new Chebyshev polynomial intelligence system
pub fn new() -> Self {
let mut intelligence = Self {
properties: HashMap::with_capacity(8),
};
intelligence.initialize_chebyshev_polynomials();
intelligence
}
/// Get all Chebyshev polynomial properties
pub fn get_properties(&self) -> HashMap<String, FunctionProperties> {
self.properties.clone()
}
/// Check if function is a Chebyshev polynomial
pub fn has_function(&self, name: &str) -> bool {
self.properties.contains_key(name)
}
/// Initialize Chebyshev polynomials with ABSOLUTE MATHEMATICAL ACCURACY
fn initialize_chebyshev_polynomials(&mut self) {
// Chebyshev Polynomials T_n(x) - MATHEMATICALLY VERIFIED
// Reference: Approximation Theory, Numerical Analysis textbooks
// Used for polynomial approximation and numerical integration
self.properties.insert(
"chebyshev_first".to_owned(),
FunctionProperties::Polynomial(Box::new(PolynomialProperties {
family: PolynomialFamily::Chebyshev,
// THREE-TERM RECURRENCE RELATION (MATHEMATICALLY VERIFIED)
// T_{n+1}(x) = 2x T_n(x) - T_{n-1}(x)
recurrence: ThreeTermRecurrence {
// Coefficient of x*T_n(x): 2x
alpha_coeff: Expression::mul(vec![
Expression::integer(2),
Expression::symbol("x"),
]),
// No constant term
beta_coeff: Expression::integer(0),
// Coefficient of T_{n-1}(x): -1
gamma_coeff: Expression::integer(-1),
// Initial conditions (mathematically verified)
// T_0(x) = 1, T_1(x) = x
initial_conditions: (Expression::integer(1), Expression::symbol("x")),
},
// Orthogonality properties (mathematically verified)
// ∫_{-1}^{1} T_m(x) T_n(x) / √(1-x²) dx = π/2 δ_{mn} (n > 0), π δ_{0n} (n = 0)
orthogonality: Some(OrthogonalityData {
// Weight function: w(x) = 1/√(1-x²)
weight_function: Expression::function(
"chebyshev_weight",
vec![Expression::symbol("x")],
),
// Orthogonality interval: [-1, 1]
interval: (Expression::integer(-1), Expression::integer(1)),
// Normalization depends on n
norm_squared: Expression::function(
"chebyshev_norm_squared",
vec![Expression::symbol("n")],
),
}),
// Rodrigues' formula (alternative representation)
rodrigues_formula: Some(RodriguesFormula {
formula: "T_n(x) = cos(n arccos(x))".to_owned(),
normalization: Expression::integer(1),
weight_function: Expression::function(
"chebyshev_weight",
vec![Expression::symbol("x")],
),
}),
// Generating function (mathematically verified)
// (1-tx)/(1-2tx+t²) = Σ_{n=0}^∞ T_n(x) t^n
generating_function: Some(GeneratingFunction {
function: Expression::function(
"chebyshev_generating",
vec![Expression::symbol("x"), Expression::symbol("t")],
),
gf_type: GeneratingFunctionType::Ordinary,
}),
// Special values (mathematically verified)
special_values: vec![
// T_n(1) = 1 for all n ≥ 0
SpecialValue {
input: "1".to_owned(),
output: Expression::integer(1),
latex_explanation: "T_n(1) = 1 \\text{ for all } n \\geq 0".to_owned(),
},
// T_n(-1) = (-1)^n for all n ≥ 0
SpecialValue {
input: "-1".to_owned(),
output: Expression::pow(Expression::integer(-1), Expression::symbol("n")),
latex_explanation: "T_n(-1) = (-1)^n \\text{ for all } n \\geq 0".to_owned(),
},
],
// Evaluation method: Recurrence is most stable
evaluation_method: EvaluationMethod::Recurrence,
// Numerical evaluator using recurrence relation),
// Symbolic expansion method for intelligence-driven computation
symbolic_expander: Some(super::super::properties::special::SymbolicExpander::Custom(
super::symbolic::expand_chebyshev_first_symbolic
)),
antiderivative_rule: AntiderivativeRule {
rule_type: AntiderivativeRuleType::Custom {
builder: Arc::new(|var: Symbol| {
Expression::integral(
Expression::function("chebyshev_first", vec![Expression::symbol(var.clone())]),
var
)
}),
},
result_template: "∫T_n(x) dx (symbolic - orthogonal polynomial integration requires specialized techniques)".to_owned(),
constant_handling: ConstantOfIntegration::AddConstant,
},
wolfram_name: None,
})),
);
// Chebyshev Polynomials of Second Kind U_n(x)
self.properties.insert(
"chebyshev_second".to_owned(),
FunctionProperties::Polynomial(Box::new(PolynomialProperties {
family: PolynomialFamily::Chebyshev,
// THREE-TERM RECURRENCE RELATION (MATHEMATICALLY VERIFIED)
// U_{n+1}(x) = 2x U_n(x) - U_{n-1}(x)
recurrence: ThreeTermRecurrence {
// Same recurrence as T_n but different initial conditions
alpha_coeff: Expression::mul(vec![
Expression::integer(2),
Expression::symbol("x"),
]),
beta_coeff: Expression::integer(0),
gamma_coeff: Expression::integer(-1),
// Initial conditions (mathematically verified)
// U_0(x) = 1, U_1(x) = 2x
initial_conditions: (
Expression::integer(1),
Expression::mul(vec![Expression::integer(2), Expression::symbol("x")]),
),
},
// Orthogonality properties (mathematically verified)
// ∫_{-1}^{1} U_m(x) U_n(x) √(1-x²) dx = π/2 δ_{mn}
orthogonality: Some(OrthogonalityData {
// Weight function: w(x) = √(1-x²)
weight_function: Expression::function(
"chebyshev_u_weight",
vec![Expression::symbol("x")],
),
// Orthogonality interval: [-1, 1]
interval: (Expression::integer(-1), Expression::integer(1)),
// Normalization: π/2
norm_squared: Expression::function("chebyshev_u_norm", vec![]),
}),
rodrigues_formula: None, // Different representation for U_n
generating_function: Some(GeneratingFunction {
function: Expression::function(
"chebyshev_u_generating",
vec![Expression::symbol("x"), Expression::symbol("t")],
),
gf_type: GeneratingFunctionType::Ordinary,
}),
special_values: vec![
// U_n(1) = n + 1
SpecialValue {
input: "1".to_owned(),
output: Expression::add(vec![
Expression::symbol("n"),
Expression::integer(1),
]),
latex_explanation: "U_n(1) = n + 1".to_owned(),
},
],
evaluation_method: EvaluationMethod::Recurrence,
// Numerical evaluator using recurrence relation),
// Symbolic expansion method for intelligence-driven computation
symbolic_expander: Some(super::super::properties::special::SymbolicExpander::Custom(
super::symbolic::expand_chebyshev_second_symbolic
)),
antiderivative_rule: AntiderivativeRule {
rule_type: AntiderivativeRuleType::Custom {
builder: Arc::new(|var: Symbol| {
Expression::integral(
Expression::function("chebyshev_second", vec![Expression::symbol(var.clone())]),
var
)
}),
},
result_template: "∫U_n(x) dx (symbolic - orthogonal polynomial integration requires specialized techniques)".to_owned(),
constant_handling: ConstantOfIntegration::AddConstant,
},
wolfram_name: None,
})),
);
}
}