abs absolute accuracy BVP ODE addition of matrices adjacency matrix and graphing Bucky ball defined distance between nodes node numbering nodes airflow modeling algorithms ODE solvers Adams-Bashworth-Moulton PECE Bogacki-Shampine (2,3) Dormand-Prince (4,5) modified Rosenbrock formula numerical differentiation formulas amp1dae analytical partial derivatives (BVP) angle arguments for ODE file arrays elements assignment statements A-stable differentiation formulas ballode ballode bandwidth of sparse matrix, reducing Basic Fitting interface batonode bicubic interpolation bicubic interpolation bilinear interpolation bilinear interpolation boundary condition, ODE Boundary Value Problems defined determining unknown parameters passing additional known parameters passing additional known parameters Boundary Value Problems, solver properties analytical partial derivatives analytical partial derivatives BCJacobian FJacobian bvpset function error tolerance error tolerance absolute accuracy AbsTol relative accuracy RelTol mesh mesh modifying property structure querying property structure solver output Stats Brusselator system (ODE example) brussode brussode Buckminster Fuller dome Bucky ball burgersode BVP. See Boundary Value Problems bvpget bvpset calling MATLAB functions ODE solvers cat characteristic polynomial characteristic roots of matrix chol chol Cholesky factorization for sparse matrices closest point searches colamd colmmd colmmd colon operator colperm column vector for polynomial roots representation of event locations (ODE) comparing interpolation methods sparse and full matrix storage complex conjugate transpose complex values in sparse matrix computational functions applying to sparse matrices condest condition, dimension compatibility conditions for ODEs boundary initial confidence intervals contents of sparse matrix continuous extension (ODE solvers) continuous extension (ODE solvers) contour contour plots, to compare interpolation methods conv convex hull multidimensional convhull convhulln convolution corrcoef correlation coefficients cov covariance creating sparse matrix cubic interpolation multidimensional one-dimensional cubic spline interpolation curve fitting confidence intervals using the Basic Fitting interface Cuthill-McKee, reverse ordering data filter monotonic multivariate preprocessing sorting data analysis finite differences triangulation data fitting confidence intervals error bounds exponential fit exponential fits polynomial fits using the Basic Fitting interface data gridding multidimensional data normalization decomposition eigenvalue Schur singular value deconv deconvolution Delaunay tesselations Delaunay triangulation closest point searches delaunayn density of sparse matrix derivative of polynomial descriptive statistics det determinant diag diagonal creating sparse matrix from of a matrix diff difference between successive vector elements difference equations differential equations boundary value problems for ODEs initial value problems for ODEs and DAEs partial differential equations dimension compatibility direct methods for systems of equations directories funfun discrete Fourier transform displaying sparse matrices distance between nodes division matrix of polynomials dot product dsearch eig eigenvalue decomposition eigenvalues of sparse matrix eigenvector error bound, for data fit tolerance boundary value problems BVP ODE ODE evaluating polynomials in matrix sense event location (ODE) event location (ODE) examples adjacency matrix (sparse) airflow modeling brussode Bucky ball Delaunay triangulation fem1ode interpolation ODE solvers orbitode rigidode second difference operator sparse matrix sparse matrix theoretical graph (sparse) van der Pol extra parameters stiff vdpode exponential fit to data exponentials, matrix eye eye factorization Cholesky for sparse matrices Cholesky LU triangular Hermitian positive definite incomplete LU positive definite QR fast Fourier transform. See Fourier transform, fast fem1ode fem1ode fem1ode example fem2ode fft fft FFT. See Fourier transform, fast fill-in of sparse matrix filtering find function and sparse matrices finding nonzero elements finite differences finite element discretization (ODE example) first-order differential equations, representation for ODE solvers fit. See data fitting fminbnd fminbnd fminsearch Fourier analysis Fourier transform fast FFT-based interpolation specifying length fplot full full function minimizing function functions function functions functions optimization funfun directory fzero Gaussian elimination Gaussian elimination geodesic dome gplot graph characteristics defined theoretical griddata hb1dae hb1dae hb1ode hccurve Hermitian positive definite factorization humps identity matrix importing sparse matrix improving solver performance incomplete factorization initial condition example initial condition vector initial condition (ODE) defined initial value problem defined initial-boundary value problem inner product integration double numerical See also ordinary differential equation solvers interp1 interp2 interp3 interpft interpn interpolation comparing methods cubic cubic spline defined FFT-based memory multidimensional multidimensional cubic linear nearest neighbor scattered data one-dimensional cubic cubic spline linear nearest neighbor polynomial smoothness speed three-dimensional nearest neighbor tricubic trilinear two-dimensional bicubic bicubic bilinear bilinear nearest neighbor nearest neighbor inv inverse isnan iterative methods for sparse matrices for systems of equations Jacobian matrix (ODE) Jacobian matrix (ODE) evaluated analytically sparsity pattern vectorized computation Jordan Canonical Form kron Kronecker tensor product least squares linear algebra and matrices linear interpolation multidimensional linear systems of equations direct methods iterative methods sparse linear transformation linear-in-the-parameters regression load Lobatto IIIa ODE solver log10 logarithm analysis with a second-order model lu lu LU factorization for sparse matrices and reordering magnitude Maple mass matrix (ODE) mass matrix (ODE) constant mass matrix returned by ODE file mat4bvp mathematical functions finding zeros MATLAB functions for minimizing minimizing numerical integration of one variable finding zeros of several variables plotting quadrature representing in MATLAB mathematical operations on sparse matrices MATLAB function functions representing functions matrices addition and linear algebra diagonal of dimension compatibility division full to sparse conversion full to sparse conversion identity multiplication orthagonal subtraction symmetric triangular matrix characteristic roots elements exponentials iterative methods multiplication powers max meshgrid meshgrid meshgrid M-files to represent mathematical functions min minimal norm minimizing functions of one variable of several variables setting minimization options minimum degree ordering missing values monotonic data for interpolation Moore-Penrose pseudoinverse multidimensional arrays interpolation interpolation multidimensional data gridding multidimensional interpolation multidimensional interpolation cubic linear nearest neighbor scattered data multiple regression multiplication matrix of polynomials multistep solver (ODE) multivariate data NaN NaNs removing from data ndgrid nearest neighbor interpolation nearest neighbor interpolation nearest neighbor interpolation nearest neighbor interpolation multidimensional nnz nnz nodes distance between numbering nonzero elements number of nonzero elements of sparse matrix maximum number in sparse matrix storage storage values visualizing with spy plot nonzeros norm Basic Fitting interface norm, minimal normalizing data null numerical integration nzmax nzmax ODE. See Ordinary Differential Equations ode113 description ode15s ode15s description ode23 description ode23s description ode23t description ode23tb description ode45 description odeget odephas2 odephas3 odeplot odeprint odeset one-dimensional interpolation one-dimensional interpolation cubic spline linear nearest neighbor ones one-step solver (ODE) operator second difference operators colon optimization calling sequence changes practicalities troubleshooting optimization code updating to MATLAB Version 5 syntax options minimization orbitode orbitode Ordinary Differential Equations coding in MATLAB definition passing additional parameters rewriting for ODE solvers solvers Ordinary Differential Equations, solver properties error tolerance error tolerance absolute accuracy AbsTol NormControl relative accuracy relative to norm of solution RelTol event location event location Events Jacobian matrix Jacobian matrix Jacobian JPattern Vectorized mass matrix mass matrix InitialSlope Mass MassSingular MStateDependence MvPattern modifying property structure ode15s BDF MaxOrder odeset function querying property structure solution components for output function solver output OutputFcn OutputSel Refine Stats Stats specifying (overview) specifying (overview) specifying (overview) step size step size InitialStep MaxStep Ordinary Differential Equations, solvers basic example nonstiff problem stiff problem boundary conditions calling different kinds of systems examples multistep solver nonstiff solvers obtaining solutions at specific times one-step solver overview representing problems rewriting problem as first-order system solution array stability stiff problems stiff solvers syntax, basic time interval time span vector van der Pol example extra parameters nonstiff stiff orthogonal matrix orthogonalization orthonormal columns outer product outliers output properties BVP solver ODE solvers overdetermined systems of simultaneous linear equations Partial Differential Equations definition representing solver partial fraction expansion partial pivoting pchip Basic Fitting interface PDE. See Partial Differential Equations pdex1 pdex2 pdex3 pdex4 pdex5 performance improving for solvers permutations phase pinv pivoting, partial plotting mathematical functions poly poly polyder polyfit polyfit polyfit polyfit polyfit Basic Fitting interface polynomial fits to data interpolation regression polynomials and curve fitting basic operations characteristics derivative of dividing evaluating in matrix sense multiplying representing roots polyval polyval polyval polyval positive definite factorization Hermitian powers matrix preconditioner for sparse matrix preprocessing data preprocessing data product dot inner outer property structure (BVP) creating modifying querying property structure (ODE) creating modifying querying pseudoinverses qr QR factorization QR factorization quad quad quad8 quad8 quadrature questions and answers, ODE solvers different kinds of systems rand rank deficiency deficiency rational format regression linear-in-the-parameters multiple polynomial relative accuracy BVP ODE Removing NaNs from data reorderings and LU factorization for sparser factorizations minimum degree ordering to reduce bandwidth representing polynomial roots polynomials problems for ODE solvers residuals for exponential data fit residue rigid body ODE example rigidode rigidode roots roots of polynomial row vector for polynomial representation save scalar scattered data multidimensional tesselation and interpolation triangulation and interpolation schur Schur decomposition second difference operator, example shockbvp singular value decomposition size solvers. See ODE solvers solving linear systems of equations sparse sort sorting data sparse sparse sparse matrix advantages and complex values Cholesky factorization computational considerations contents conversion from full conversion from full creating directly from diagonal elements defined density distance between nodes eigenvalues elementary example fill-in importing linear algebra linear equations linear systems of equations LU factorization and reordering mathematical operations nonzero elements maximum number specifying when creating matrix storage storage values nonzero elements of sparse matrix number of operations permutation preconditioner propagation through computations QR factorization reordering reordering storage for various permutations viewing theoretical graph triangular factorization viewing contents graphically viewing storage visualizing working with sparse ODE example spconvert spdiags speye speye speye spline Basic Fitting interface spones spparms sprand spy spy plot sqrtm stability (ODE solvers) statistics analyzing residuals correlation coefficients covariance descriptive preprocessing data step size (ODE) step size (ODE) first step upper bound stiff ODE example stiffness (ODE), defined storage for various permutations of sparse matrix of sparse matrix sparse and full, comparison viewing for sparse matrix subtraction of matrices sum sum surface plots to compare interpolation methods svd symamd Symbolic Math Toolbox symmetric matrix symmmd symmmd symrcm symrcm systems of equations. See linear systems of equations systems of ODEs tesselation Delaunay tessellation Voronoi diagrams theoretical graph example node three-dimensional interpolation nearest neighbor tricubic trilinear time interval (ODE) transformed data magnitude phase transforms discrete Fourier fast Fourier fft transpose complex conjugate unconjugated complex triangular factorization for sparse matrices triangular matrices triangulation closest point searches Delaunay Voronoi diagrams See also tesselation tricubic interpolation trilinear interpolation tsearch twobvp two-dimensional interpolation bicubic bilinear nearest neighbor unconjugated complex transpose unwrap updating optimization code to MATLAB Version 5 syntax van der Pol example extra parameters simple, nonstiff simple, stiff vdpode vdpode vector column initial condition (ODE) row time span vector (ODE) vector products vectorization for Jacobian matrix computation (ODE) visualizing ODE solver results sparse matrix spy plot voronoi Voronoi diagrams multidimensional voronoin whos zeros