Foundations

Conditioning measures sensitivity of the exact answer to data perturbations.

Forward error compares a computed answer with the exact answer.

Newton iteration uses xₖ₊₁=xₖ−f(xₖ)/f′(xₖ).

An LU factorization writes A=LU with unit lower-triangular L and upper-triangular U.

Partial pivoting swaps rows to choose a largest-magnitude available entry in the pivot column before eliminating below it.

A real orthogonal square matrix Q satisfies QᵀQ=I and preserves Euclidean lengths.

For invertible A and an induced matrix norm, κ(A)=‖A‖‖A⁻¹‖.

For finite G and subgroup H, |G|=[G:H]|H|.

For a group homomorphism φ:G→H, ker φ is normal and G/ker φ is isomorphic to im φ by g ker φ↦φ(g).

Swapping rows, scaling a row by a nonzero scalar and adding a multiple of another row preserve a linear system's solution set.

With Aᵢⱼ formed by deleting row i and column j, cofactor expansion gives det A=Σⱼ(−1)ⁱ⁺ʲaᵢⱼdet Aᵢⱼ, starting from det[a]=a.

For x,y∈Rⁿ, ⟨x,y⟩=Σxᵢyᵢ and ‖x‖=√⟨x,x⟩.

The minimizers of ‖Ax−b‖₂² satisfy Aᵀ(Ax−b)=0, so their residual is orthogonal to the column space.

Interpolation chooses p with p(tᵢ)=yᵢ at specified distinct nodes.

On n equal subintervals with h=(b−a)/n, the trapezoidal estimate is T=h[f(a)/2+Σᵢ₌₁ⁿ⁻¹f(a+ih)+f(b)/2].

Euler stepping for y′=f(t,y) uses yₖ₊₁=yₖ+hf(tₖ,yₖ).

Set k₁=hf(t,y), k₂=hf(t+h/2,y+k₁/2), k₃=hf(t+h/2,y+k₂/2), k₄=hf(t+h,y+k₃); then y_next=y+(k₁+2k₂+2k₃+k₄)/6.

For y′=f(t)g(y) on a region where g(y)≠0, integrating dy/g(y)=f(t)dt gives an implicit solution.

For y′+p(t)y=q(t) with continuous p,q, μ(t)=exp(∫p(t)dt) transforms the equation into (μy)′=μq.

For ay″+by′+cy=0, a≠0, roots of ar²+br+c=0 give the solution modes: distinct real roots give eʳ¹ᵗ,eʳ²ᵗ; a repeated root gives eʳᵗ,teʳᵗ; conjugate α±iβ give eᵅᵗcos βt,eᵅᵗsin βt.

For a linear equation Ly=f, any one particular solution yₚ plus the general homogeneous solution y_c gives every solution y=y_c+yₚ.

In a C¹ planar autonomous system, equilibria are zeros of the vector field.

For a 2L-periodic piecewise smooth real f, aₙ=L⁻¹∫₋ᴸᴸf(t)cos(nπt/L)dt and bₙ=L⁻¹∫₋ᴸᴸf(t)sin(nπt/L)dt.

The one-sided Laplace transform is F(s)=∫₀∞e⁻ˢᵗf(t)dt.

For u_t=κu_xx on 0<x<L with zero endpoint temperatures, separation gives modes sin(nπx/L)e⁻κ⁽ⁿπ/L⁾²ᵗ.

For u_tt=c²u_xx with zero ends on [0,L], extend initial displacement f and velocity g oddly with period 2L to F,G.

If C¹ f:U⊆Rⁿ→Rⁿ has invertible derivative at a∈U, it restricts to a C¹ bijection between neighborhoods of a and f(a), with C¹ inverse and D(f⁻¹)(f(a))=[Df(a)]⁻¹.

For C¹ F:U⊆Rⁿ×Rᵐ→Rᵐ with F(a,b)=0 and invertible partial derivative D_yF(a,b), nearby zeros are uniquely y=g(x), where g is C¹ and g(a)=b.

A sigma-algebra contains the whole space and is closed under complements and countable unions.

A map between measurable spaces is measurable when every measurable target set has a measurable inverse image.