Foundations

Floating-point numbers use a sign, finite significand and exponent.

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‖.

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 continuous real f with f(a)f(b)<0, test the midpoint and retain a half interval with opposite endpoint signs, unless the midpoint is a root.

At distinct nodes t₀,…,tₙ, p(x)=Σyₖℓₖ(x), ℓₖ(x)=∏_(i≠k)(x−tᵢ)/(tₖ−tᵢ), is the unique interpolant of degree at most n.

A differentiable function has zero gradient at an interior local extremum.

For C¹ real f,g, a local constrained extremum of f on g(x)=c at a point where ∇g≠0 must satisfy ∇f=λ∇g together with g=c.

A differentiable function on an open convex domain is convex exactly when f(y)≥f(x)+∇f(x)·(y−x) for all domain points.

For differentiable convex f₀,fᵢ on Rⁿ and affine equalities Ax=b, feasible x is optimal if λ≥0, λᵢfᵢ(x)=0 and ∇f₀(x)+Σλᵢ∇fᵢ(x)+Aᵀν=0.

An n-element set has 2ⁿ subsets; its k-element subsets correspond to length-n binary strings with k ones.

For aₙ=saₙ₋₁+taₙ₋₂ with t≠0, distinct roots r₁,r₂ of r²−sr−t give aₙ=Ar₁ⁿ+Br₂ⁿ; a repeated root r gives aₙ=(A+Bn)rⁿ.

If columns of invertible P are an ordered basis B in standard coordinates, v=P[v]B.

From independent real vectors v₁,…,vₖ form wⱼ=vⱼ−Σ_(i<j)[(vⱼ·wᵢ)/(wᵢ·wᵢ)]wᵢ.

For uᵢ₊₁=uᵢ+hφ(tᵢ,uᵢ,h), if the exact-value one-step defect is bounded by Chᵖ⁺¹ and |∂ᵤφ|≤L, L>0, then exact initialization gives |u(tᵢ)−uᵢ|≤(Chᵖ/L)(eᴸ⁽ᵗⁱ⁻ᵃ⁾−1).

Disjoint finite alternatives have total count equal to the sum of their counts.

For finite A,B, |A∪B|=|A|+|B|−|A∩B|.

From n distinct objects choose k without repetition.