Computationally useful expression for the interpolating polynomial as a ratio of rational terms. (section-globalapprox-barycentric)
big-O
Relationship indicating that one function is bounded above by a multiple of another in some limit. (Efficiency of matrix computations)
boundary-value problem
A differential equation with which partial information about the solution is given at multiple points on the boundary of the domain. (section-bvp-tpbvp)
Subtraction of a shift followed by matrix inversion, used in power iteration to transform the eigenvalue closest to a target value into a dominant one. (section-krylov-inviter)
Jacobian matrix
Matrix of first partial derivatives that defines the linearization of a vector-valued function. (Newton for nonlinear systems)
Kronecker product
Alternative type of matrix multiplication useful for problems on a tensor-product domain. (section-twodim-laplace)
Krylov subspace
Vector space generated by powers of a square matrix that is often useful for reducing the dimension of large problems. (section-krylov-subspace)
Lagrange formula
Theoretically useful expression for an interpolating polynomial. (section-globalapprox-polynomial)
Lanczos iteration
Specialization of the Arnoldi iteration to the case of a hermitian (or real symmetric) matrix. (section-krylov-minrescg)
Laplace equation
Archetypical elliptic PDE describing a steady state. (section-twodim-laplace)
linear convergence
Sequence in which the difference between sequence value and limit asymptotically decreases by a constant factor at each term, making a straight line on a log-linear graph.\
(Fixed-point iteration)
linear least-squares problem
Minimization of the 2-norm of the residual for an overdetermined linear system. (Fitting functions to data)
Factorization of a square matrix into the product of a unit lower triangular matrix and an upper triangular matrix. (LU factorization)
machine epsilon
Distance from 1 to the next-largest floating-point number. Also called unit roundoff or machine precision, though the usages are not consistent across different references.\
(Floating-point numbers)
matrix condition number
Norm of the matrix times the norm of its inverse, equivalent to the condition number for solving a linear system. (Conditioning of linear systems)
method of lines
Solution technique for partial differential equations in which each independent variable is discretized separately. (section-diffusion-methodlines)
multistep
Formula using information over more than a single time step to advance the solution. (Multistep methods)
Neumann condition
Boundary condition specifying the derivative of the solution. (section-bvp-tpbvp)
Newton’s method
Rootfinding iteration that uses the linearization of the given function in order to define the next root approximation. (Newton’s method)
nodes
Values of the independent variable where an interpolant’s values are prescribed. (The interpolation problem)
nonlinear least-squares problem
Minimization of the 2-norm of the residual of a function that depends nonlinearly on a vector. (Nonlinear least squares)
Square ONC matrix, i.e., matrix whose transpose is its inverse. (The QR factorization)
orthogonal polynomials
Family of polynomials whose distinct members have an integral inner product equal to zero, as with Legendre and Chebyshev polynomials. (section-globalapprox-orthogonal)
orthonormal vectors
Vectors that are both mutually orthogonal and all of unit 2-norm. (The QR factorization)
outer product
Multiplication of two vectors resulting in a rank-1 matrix. (LU factorization)
overdetermined
Characterized by having more constraints than available degrees of freedom. (Fitting functions to data)
piecewise linear
Function that is linear between each consecutive pair of nodes, but whose slope may jump at the nodes. (Piecewise linear interpolation)
PLU factorization
LU factorization with row pivoting. (Row pivoting)
power iteration
Repeated application of a matrix to a vector, followed by normalization, resulting in convergence to an eigenvector for the dominant eigenvalue. (section-krylov-power)
preconditioning
Use of an approximate inverse to improve the convergence rate of Krylov iterations for a linear system. (section-krylov-precond)
pseudoinverse
Rectangular matrix that maps data to solution in the linear least-squares problem, generalizing the matrix inverse. (The normal equations)
QR factorization
Representation of a matrix as the product of an orthogonal and an upper triangular matrix. (The QR factorization)
quadratic convergence
Sequence in which the difference between sequence value and limit asymptotically decreases by a constant times the square of the preceding difference. (Newton’s method)
quasi-Newton methods
Rootfinding methods that overcome the issues of Jacobian computation and lack of global convergence in Newton’s method. (Quasi-Newton methods)
quasimatrix
Collection of functions (such as orthogonal polynomials) that have algebraic parallels to columns of a matrix. (section-globalapprox-orthogonal)
Rayleigh quotient
Function of vectors that equals an eigenvalue when given its eigenvector as input. (Symmetry and definiteness)
reduced QR factorization
See thin QR.
reduced SVD
See thin SVD.
residual
For a linear system, the difference between b and Ax~ for a computed solution approximation x~. More generally, the actual value of a quantity that is made zero by an exact solution. (Conditioning of linear systems, The rootfinding problem)
restarting
Technique used in GMRES to prevent the work per iteration and overall storage from growing uncontrollably. (section-krylov-gmres)
rootfinding problem
Finding the input value for a given function which makes that function zero. (The rootfinding problem)
row pivoting
Reordering rows during LU factorization to ensure that the factorization exists and can be computed stably. (Row pivoting)
Runge phenomenon
Manifestation of the instability of polynomial interpolation at equally spaced nodes as degree increases. (section-globalapprox-stability)
Runge--Kutta
One-step method for IVPs that evaluates the derivative of the solution more than once to advance a single step. (Runge–Kutta methods)
secant method
Scalar quasi-Newton method that uses a secant line rather than a tangent line to define a root estimate. (Interpolation-based methods)
shooting
Unstable technique for solving a boundary-value problem in which an initial value is sought for by a rootfinding algorithm. (section-bvp-shooting)
simple root
Root of a function at which the derivative of the function is nonzero. (The rootfinding problem)
singular value decomposition (SVD)
Expression of a matrix as a product of two orthogonal/unitary matrices and a nonnegative diagonal matrix. (Singular value decomposition)
sparse
Describing a matrix that has mostly zero elements for structural reasons. (Exploiting matrix structure, section-krylov-structure)
spectral convergence
Exponentially rapid decrease in error as the number of interpolation nodes increases, e.g., as observed in Chebyshev polynomial and trigonometric interpolation. (section-globalapprox-stability)
stability region
Region of the complex plane describing when numerical solution of a linear IVP is bounded as t→∞. (section-diffusion-absstab)
step size
Increment in time between successive solution values in a numerical IVP solver. (Euler’s method)
stiff
Describes an IVP in which stability is a greater restriction than accuracy for many solution methods, usually favoring the use of an implicit time stepping method. (Implementation of multistep methods, section-diffusion-stiffness)
subtractive cancellation
Growth in relative error that occurs when two numbers are added/subtracted to get a result that is much smaller in magnitude than the operands; also called loss of significance or cancellation error. (Problems and conditioning)
superlinear convergence
Sequence for which the convergence is asymptotically faster than any linear rate. (Interpolation-based methods)
Matrix that is symmetric and positive definite, thereby permitting a Cholesky factorization. Correspondingly called hermitian positive definite in the complex case. (Exploiting matrix structure)
tensor-product domain
A domain that can be parameterized using variables that lay in a logical rectangle or cuboid; i.e., each variable independently varies in an interval. (section-twodim-tensorprod)
thin QR factorization
Variant of the QR factorization that discards information not needed to fully represent the original matrix. (The QR factorization)
thin SVD
Variant of the singular value decomposition that discards information not needed to fully represent the original matrix. (Singular value decomposition)
Square matrix with complex-valued entries whose columns are orthonormal. (Eigenvalue decomposition)
unstable
Allowing perturbations of the data to have much larger effects on the results than can be explained by the problem’s condition number. (Stability)
upper Hessenberg
Describing a matrix that has nonzeros only in the upper triangle and first subdiagonal. (section-krylov-subspace)
Vandermonde matrix
Matrix whose columns are formed from elementwise powers of a given vector, important for polynomial interpolation and approximation of data. (Polynomial interpolation)
weights
Coefficients in a linear combination of function values in a finite-difference or integration method. (Finite differences, Numerical integration, section-globalapprox-integration)