PPT4

What is the supremum norm denoted as in the given text?

• ∥f∥a,b,∞

• ∥f∥

• ∥f∥∞

• ∥f∥a,b

What is the characteristic Lagrange polynomial for the given function f?

• Lagrange Interpolation

• Piecewise Lagrange Interpolation

• Newton-Cotes Formulas

• Gaussian Quadrature

What is the error bound for the Lagrange interpolating polynomial?

• max(x∈[a,b])|f(x)|

• ∥f∥a,b,∞

• ∥f∥∞

• (b−a)n+1/(n+1)!

What is the degree of precision for the interpolatory quadrature QI_n?

• ∥f∥a,b,∞

• max(x∈[a,b])|f(x)|

• max{m∈N:E_n[p]=0 for all p∈Pm}

• ∥f∥∞

What is the approximate value of the integral Zxkxk−1f(x)dx using the given quadratic polynomial P2(x)?

• h(f(q)+f(s))

• h/4(f(q)+2f(r)+f(s))

• h/2(f(q)+f(r))

• h/3(f(q)+4f(r)+f(s))

What is the maximal degree of precision for numerical quadratures having n+1 nodes?

• n^2

• 2n+1

• 2n

• n

What is the largest possible degree of precision for a numerical quadrature Qn?

• n

• 2n

• 2n+1

• n^2

According to the Picard-Lindelöf Theorem, the differential equation u′(t)=f(t,u(t)) has a unique continuously differentiable solution u:[t0,tf]→R, provided that f:Ω→R is continuously differentiable and |f(t,v)−f(t,w)|≤L|v−w| for some r,L>0 and for every v, w∈[u0−r,u0+r]. What is the value of r?

• L

• 1

• 0

• r

Under the above framework, for every ε>0 small enough, there exists δ>0 such that for every ρ0∈R and ρ:[t0,tf]→R with |ρ0|<δ and |ρ(t)|<δt∈[t0,tf], the solutions of (8) and (9) satisfy |z(t)−u(t)|<ε for every t∈[t0,tf]. What is the value of ε?

• 0

• 1

• δ

• ε

If the single-step method (15) is consistent and zero-stable, then it is convergent in the sense that ∥eh∥∞→0 as h→0. What is the condition for the single-step method to be convergent?

• Consistency and zero-stability

• Consistency and stability

• Consistency and convergence

• Zero-stability and convergence