Faisal Saied
Keywords in Numerical Analysis
back to main page
back to Scientific Computing page

Numerical Linear Algebra
Linear Systems of Equations
     Gaussian elimination
     LU factors
     pivoting (partial, complete)
     forward solve, backsolve, triangular solve
     round-off errors
     condition number
     residual vector, error bound
     iterative improvement (refinement)
     vector norm, matrix norm, Frobenius norm
     growth factor, stability of GE
     Sherman-Morrison-Woodbury formula
     Cholesky facorization, L D L'
     GE for banded systems
     GE for tridiagonal systems
     Hilbert matrix
     Generalized inverse; Moore-Penrose
     Reverse Cuthill-McKee ordering
    
Numerical Linear Algebra
Least Squares Problems
     normal equations
     inner product, orthogonality
     orthogonal projectors
     orthonormal basis
     orthogonal matrix
     QR factorization
     QR factorization with column pivoting
     Householder transform
     Givens rotations
     Gram-Schmidt, modified Gram-Schmidt
     QR using modified Gram-Schmidt
     QR with column pivoting
     Singular Value Decomposition
    
Numerical Linear Algebra
Eigenvalue Problems
     eigenvalue, eigenvector
     characteristic polynomial
     diagonalization
     defective matrix
     Jordan canonical form
     eigenvalue localization
     Gerschgorin disks
     sensitivity of eigenvalues
     normal matrix
     similarity transformation
     orthogonal transformation
     symmetric matrix
     power method
     (shifted) inverse iteration
     Rayleigh quotient inverse iteration
     QR iteration
     Wilkinson shifts
     deflation
     reduction to tridiagonal form
     Jacobi method
     Sylvester's Law of Inertia
     congruence transformation
     Sturm sequences
     bisection; multisection
     divide and conquer
     Lanczos; Arnoldi
     block Lanczos
     Rayleigh-Ritz method
    
Interpolation
     polynomial interpolation
     monomial basis
     Horner's scheme (synthetic division)
     Vandermonde matrix
     ill-conditioned basis
     Runge phenomenon
     Hilbert matrix
     Lagrange interpolation
     Newton form
     divided differences
     interpolation error
     orthogonal polynomials
     Legendre, Chebyshev, Hermite
     piecewise polynomial interpolation
     splines
     Cubic Hermite, cubic spline
     B-splines
    
Numerical Quadrature
     Newton-Cotes
     midpoint rule
     trapezoidal rule
     Simpson's rule
     Gaussian quadrature
     Gauss-Legendre; Gauss-LObatto
     composite quadrature
     adaptive quadrature
     Romberg integration
     Richardson extrapolation
     multiple integrals
     Monte-Carlo method
    
Minimization Methods
     convexity, level sets
     gradient, Hessian
     necessary, sufficient conditions
     Lagrange multipliers
     Karush-Kuhn-Tucker (KKT) theorem
     critical point
     constrained optimization
    
     golden section search
     Newton's method
     steepest descent
     quasi-Newton
     (nonlinear) conjugate gradient
     BFGS; DFP
     inexact Newton
     nonlinear least squares
     Gauss-Newton
     Levenberg-Marquardt
     penatly methods
     linear programming
     simplex method
     Karamarkar algorithm
    
Nonlinear Equations
     contraction mapping
     fixed point theorem
     fixed point iteration
     ill-posed problems
     bisection (multisection)
     Newton's method
     secant method
     polynomial root finding algorithms
     Newton's method for systems of nonlinear eqns
     damped Newton, trust region
     Broyden
    
Ordinary Differential Equations
Initial Value Problems
     forward Euler
     backward Euler
     explicit, implicit methods
     single step methods
     multistep mthods
     Runge-Kutta
     stiff ODEs
     BDF formulas
     stability
     predictor-corrector methods
     Adams-Bashforth
     Adams-Moulton
     step size control
     predator-prey model
     PECE
     leapfrog method
    
Ordinary Differential Equations
Boundary Value Problems
    
    
Iterative solvers for linear systems
     Jacobi, Gauss-Seidel, SOR
     relaxation
     iteration matrix, spectral radius
     convergence criteria
     Krylov subspace methods
     symmetric positive definite matrices
     conjugate gradient method
     preconditioning
     A-norm
     non-symmetric solvers
     GMRES(k), BiCG/Stab, QMR, CGS, Orthomin
     (M)ILU, incomplete Cholesky
     Sparse Approximate Inverse (SPAI)
     Operator splitting, ADI
     block Jacobi
     multigrid
     Domain decomposition preconditioners
     alternating Schwarz
    
Sparse direct solvers
    
    
    
Hyperbolic PDEs
     Artificial viscosity
     Burgers' equation
     Characteristics
     Conservation Laws
     CFL condition (Courant, Friedrichs, Levy):
    (A necesary condition for stability)
     The domain of dependence of the PDE must lie within the domain
       of dependence of the numerical scheme
     Dispersion relation
     Dissipative difference scheme
     Explicit scheme
     Finite propagation speeds
     Flux function
     Godunov-Ryabinkii
     Group velocity
     Lax-Wendroff scheme
     Leap-frog scheme
     Phase error
     Reflection of waves
     Riemann invariants
     Rankine-Hugoniot conditions
     Shocks
     Spurious modes
     Symmetric hyperbolic system
     Three-level scheme
     Upwind scheme
     Wave equation (first order, second order)
    
Parabolic PDEs
     ADI (alternating direction implicit)
     Backward Euler
     Convection-diffusion equation
     mesh Reynolds number
     cell Peclet number
     upwind differencing
     Crank-Nicolson
     Diffusion equation
     D'yakonov scheme
     Heat equation
     Implicit scheme
     LOD (locally one-dimensional)
     Peaceman-Rachford
     Consistency, Stability, Convergence
     Consistent scheme
     Amplification factor (or amplification matrix)
     Energy method (stability analysis)
     Fourier stability analysis
     Hadamard
     Lax Equivalence Theorem:
      For a well-posed IVP for a PDE, a consistent numerical scheme is convergent if and only if it is stable.
     Order of Accuracy
     Stability
     Summation by parts
     Truncation error -- local, global
     Unconditional stability
     von Neumann stability condition
     Well-posed problem in the sense of Hadamard:
       existence + uniqueness + continuous dependence on inputs
    
Elliptic PDEs
     Advection equation
     Biharmonic equation
     Block tridiagonal matrix
     Convection-diffusion equation
     Corner discontinuity (rentrant corner)
     Essential b.c.
     Delta "function"
     Dirichlet boundary condition
     Discontinuous coefficient PDEs
     Elliptic Solvers
     Finite difference discretization
     Finite volume discretization
     Flux continuity
     5-point stencil
     Fundamental solution
     Green's function
     Helmholtz' equation
     Interfaces
     Laplace's equation
     Maximum principle
     Mixed b.c.
     Natural b.c.
     Natural ordering
     Nonlinear elliptic PDEs
     Neumann boundary condition
     Poisson's equation
     7-point stencil
     Self-adjoint problem
     Variable coefficients
    
The Finite Element Method
     Adaptive finite elements
     Angle condition
     Babuska-Brezzi condition
     Brick elements
     Cea's Lemma
     Reduces finite flement error estimation to a problem in approximation theory
     Delaunay, Voronoi
     Duality argument
     Element stiffness matrix
     Energy inner product, norm
     Essential boundary conditions
     Error estimates
     Finite element space
     Galerkin Approach
     Gauss' theorem (divergence theorem)
     Gaussian quadrature
     Global matrix assembly
     Global stiffness matrix
     Green's theorem
     Hierarchical basis
     Hilbert space, L_{2}
     h-p version of the FEM
     Integration by parts
     Interpolant (piecewise polynomial)
     Lax-Milgram lemma
     Mass matrix
     Master element (reference triangle)
     Multi-index
     Naturalal boundary conditions
     Nodal basis
     Numerical integration
     Petrov-Galerkin method
     Piecewise linear basis function
     Regularity
     Riesz representation theorem
     Ritz-Galerkin method
     Sobolev spaces, H_{0}^{1}
     Stiffness matrix
     Test functions
     Tetrahedral element
     Trial functions
     Triangulation
     Variational formulation
    
Multigrid
     Algebraic multigrid
     Coefficient averaging
     Coarse grid correction
     Coarse grid operators
     coarse grid solvers
     FMG (Full Multigrid)
     Full-weighting
     Galerkin coarsening
     Half-weighting
     Injection
     Interpolation (prolongation)
     high frequency errors
     MGGHAT
     Multigrid as a preconditioner
     MGLab
     Operator-based prolongation
     PLTMG, More info
     Restriction
     Recursion vs Iteration
     Semi-coarsening
     Smoothing (pre-, post-)
     Two-grid algorithm
     V-Cycle
     W-Cycle
     Variable V-Cycle
     nonlinear multigrid
     Full approxomation storage (FAS)
     truncation error accuracy
     optimal order O(N) solvers
     multilevel solvers
     multiscale solvers
    
Fast Poisson Solvers
     Block Cyclic Reduction
     FFT's and tridiagonal solves
     Multidimensional fast Fourier transforms
     O(N log(N))