Multivariable Taylor formula with a Lagrange remainder along a line segment

Depends on

• The multivariable Taylor polynomial in multi-index notation
• Repeated derivatives along a line expand by the multinomial formula
• The Lagrange and Cauchy forms of Taylor's remainder
• A convex subset of $\mathbb{R}^m$ contains every line segment between two of its points

Used by

• Constant-step BDF2 is second-order convergent with a consistent start and selected roots Corollary
• Multivariable Taylor formula with o(‖h‖ᵏ) remainder Corollary
• A curved double well has an exact circular minimum-energy path Example
• A damped Newton step precedes the local quadratic phase Example
• A regularized logistic finite sum with an exact minimizer Example
• Boundary-incompatible forcing reduces Strang order to five quarters in a Dirichlet modal model Example
• Fixed-time error slopes for Euler, midpoint, and classical RK4 Example
• Forward Euler on the scalar linear equation y'=y Example
• RATTLE gives an exact step and a cubic phase defect for a free rigid bond Example
• A Cᵏ⁺ʳ residual vanishing through order r-1 has a Cᵏ scaled extension Lemma
• A finite formal logarithm has Hamiltonian coefficients on a star-shaped domain Lemma
• A fixed finite modified flow matches its method map to the stated Taylor order Lemma
• A Gaussian collocation defect has a high-order weighted endpoint bound Lemma
• A Hessian lower bound gives strong convexity and gradient-gap bounds Lemma
• A Sobolev function vanishing on scattered sites has an h^ν bound Lemma
• Analytic derivative bounds give exponential polynomial approximation on a triangle Lemma
• Atoms have uniformly bounded Hᵖ quasi-norm and uniformly bounded test pairings Lemma
• Collocation polynomial derivatives scale with the step size Lemma
• Forward-difference Jacobian error bound Lemma
• Level decomposition of an Hᵖ distribution produces atoms Lemma
• Modified Patankar Euler has a uniform first-order defect and one-step bound on positive tubes Lemma
• MPRK22(1) has a uniform second-order defect and one-step bound on positive tubes Lemma
• Paired force evaluations give a second-order Hessian-vector estimate Lemma
• Positive Hessians make Bregman divergence nonnegative Lemma
• Quadratic interpolation gives the variable-step BDF2 formula and cubic residual Lemma
• Rough composition on a flat closed set Lemma
• Sard on the infinitely flat critical stratum Lemma
• Second-order centered and unequal-step tangent estimates Lemma
• Stationary phase with a compactly supported amplitude Lemma
• Taylor moment conditions characterize the order of a linear multistep method Lemma
• The mini-element Fortin map approximates in L2 Lemma
• One-node Gauss collocation is implicit midpoint of order two Proposition
• Sufficient explicit Runge–Kutta order conditions through order four Proposition
• The selected two-stage SDIRK method has order two and is scalar L-stable Proposition
• Two-stage Radau IIA has order three and is scalar L-stable Proposition
• For 0<p<1 the Hᵖ functional is a quasi-norm, and Hᵖ is a quasi-Banach space Remark
• A corrected regret bound for agile mirror descent Theorem
• Armijo gradient descent has an explicit global linear rate Theorem
• Augmented RBF-FD weights reproduce polynomials and obey a local Taylor bound Theorem
• Backward Euler, the selected SDIRK, and two-stage Radau IIA converge at their stated orders under controlled stage residuals Theorem

…and 23 more results.

Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• MAT237 notes: Taylor's theorem in several variables (standard reference, not scraped)