This is octave.info, produced by makeinfo version 4.2 from octave.texi. START-INFO-DIR-ENTRY * Octave: (octave). Interactive language for numerical computations. END-INFO-DIR-ENTRY Copyright (C) 1996, 1997 John W. Eaton. Permission is granted to make and distribute verbatim copies of this manual provided the copyright notice and this permission notice are preserved on all copies. Permission is granted to copy and distribute modified versions of this manual under the conditions for verbatim copying, provided that the entire resulting derived work is distributed under the terms of a permission notice identical to this one. Permission is granted to copy and distribute translations of this manual into another language, under the above conditions for modified versions.  File: octave.info, Node: Coordinate Transformations, Next: Mathematical Constants, Prev: Special Functions, Up: Arithmetic Coordinate Transformations ========================== - Function File: [THETA, R] = cart2pol (X, Y) - Function File: [THETA, R, Z] = cart2pol (X, Y, Z) Transform cartesian to polar or cylindrical coordinates. X, Y (and Z) must be of same shape. THETA describes the angle relative to the x - axis. R is the distance to the z - axis (0, 0, z). - Function File: [X, Y] = pol2cart (THETA, R) - Function File: [X, Y, Z] = pol2cart (THETA, R, Z) Transform polar or cylindrical to cartesian coordinates. THETA, R (and Z) must be of same shape. THETA describes the angle relative to the x - axis. R is the distance to the z - axis (0, 0, z). - Function File: [THETA, PHI, R] = cart2sph (X, Y, Z) Transform cartesian to spherical coordinates. X, Y and Z must be of same shape. THETA describes the angle relative to the x - axis. PHI is the angle relative to the xy - plane. R is the distance to the origin (0, 0, 0). - Function File: [X, Y, Z] = sph2cart (THETA, PHI, R) Transform spherical to cartesian coordinates. X, Y and Z must be of same shape. THETA describes the angle relative to the x-axis. PHI is the angle relative to the xy-plane. R is the distance to the origin (0, 0, 0).  File: octave.info, Node: Mathematical Constants, Prev: Coordinate Transformations, Up: Arithmetic Mathematical Constants ====================== - Built-in Variable: I - Built-in Variable: J - Built-in Variable: i - Built-in Variable: j A pure imaginary number, defined as `sqrt (-1)'. The `I' and `J' forms are true constants, and cannot be modified. The `i' and `j' forms are like ordinary variables, and may be used for other purposes. However, unlike other variables, they once again assume their special predefined values if they are cleared *Note Status of Variables::. - Built-in Variable: Inf - Built-in Variable: inf Infinity. This is the result of an operation like 1/0, or an operation that results in a floating point overflow. - Built-in Variable: NaN - Built-in Variable: nan Not a number. This is the result of an operation like 0/0, or `Inf - Inf', or any operation with a NaN. Note that NaN always compares not equal to NaN. This behavior is specified by the IEEE standard for floating point arithmetic. To find NaN values, you must use the `isnan' function. - Built-in Variable: pi The ratio of the circumference of a circle to its diameter. Internally, `pi' is computed as `4.0 * atan (1.0)'. - Built-in Variable: e The base of natural logarithms. The constant E satisfies the equation `log' (E) = 1. - Built-in Variable: eps The machine precision. More precisely, `eps' is the largest relative spacing between any two adjacent numbers in the machine's floating point system. This number is obviously system-dependent. On machines that support 64 bit IEEE floating point arithmetic, `eps' is approximately 2.2204e-16. - Built-in Variable: realmax The largest floating point number that is representable. The actual value is system-dependent. On machines that support 64-bit IEEE floating point arithmetic, `realmax' is approximately 1.7977e+308 - Built-in Variable: realmin The smallest normalized floating point number that is representable. The actual value is system-dependent. On machines that support 64-bit IEEE floating point arithmetic, `realmin' is approximately 2.2251e-308  File: octave.info, Node: Linear Algebra, Next: Nonlinear Equations, Prev: Arithmetic, Up: Top Linear Algebra ************** This chapter documents the linear algebra functions of Octave. Reference material for many of these functions may be found in Golub and Van Loan, `Matrix Computations, 2nd Ed.', Johns Hopkins, 1989, and in `LAPACK Users' Guide', SIAM, 1992. * Menu: * Basic Matrix Functions:: * Matrix Factorizations:: * Functions of a Matrix::  File: octave.info, Node: Basic Matrix Functions, Next: Matrix Factorizations, Up: Linear Algebra Basic Matrix Functions ====================== - Loadable Function: AA = balance (A, OPT) - Loadable Function: [DD, AA] = balance (A, OPT) - Loadable Function: [CC, DD, AA, BB] = balance (A, B, OPT) `[dd, aa] = balance (a)' returns `aa = dd \ a * dd'. `aa' is a matrix whose row and column norms are roughly equal in magnitude, and `dd' = `p * d', where `p' is a permutation matrix and `d' is a diagonal matrix of powers of two. This allows the equilibration to be computed without roundoff. Results of eigenvalue calculation are typically improved by balancing first. `[cc, dd, aa, bb] = balance (a, b)' returns `aa = cc*a*dd' and `bb = cc*b*dd)', where `aa' and `bb' have non-zero elements of approximately the same magnitude and `cc' and `dd' are permuted diagonal matrices as in `dd' for the algebraic eigenvalue problem. The eigenvalue balancing option `opt' is selected as follows: `"N"', `"n"' No balancing; arguments copied, transformation(s) set to identity. `"P"', `"p"' Permute argument(s) to isolate eigenvalues where possible. `"S"', `"s"' Scale to improve accuracy of computed eigenvalues. `"B"', `"b"' Permute and scale, in that order. Rows/columns of a (and b) that are isolated by permutation are not scaled. This is the default behavior. Algebraic eigenvalue balancing uses standard LAPACK routines. Generalized eigenvalue problem balancing uses Ward's algorithm (SIAM Journal on Scientific and Statistical Computing, 1981). - Function File: cond (A) Compute the (two-norm) condition number of a matrix. `cond (a)' is defined as `norm (a) * norm (inv (a))', and is computed via a singular value decomposition. - Loadable Function: [D, RCOND] = det (A) Compute the determinant of A using LAPACK. Return an estimate of the reciprocal condition number if requested. - Function File: dmult (A, B) If A is a vector of length `rows (B)', return `diag (A) * B' (but computed much more efficiently). - Function File: dot (X, Y) Computes the dot product of two vectors. - Loadable Function: LAMBDA = eig (A) - Loadable Function: [V, LAMBDA] = eig (A) The eigenvalues (and eigenvectors) of a matrix are computed in a several step process which begins with a Hessenberg decomposition, followed by a Schur decomposition, from which the eigenvalues are apparent. The eigenvectors, when desired, are computed by further manipulations of the Schur decomposition. - Loadable Function: G = givens (X, Y) - Loadable Function: [C, S] = givens (X, Y) Return a 2 by 2 orthogonal matrix `G = [C S; -S' C]' such that `G [X; Y] = [*; 0]' with X and Y scalars. For example, givens (1, 1) => 0.70711 0.70711 -0.70711 0.70711 - Loadable Function: [X, RCOND] = inv (A) - Loadable Function: [X, RCOND] = inverse (A) Compute the inverse of the square matrix A. Return an estimate of the reciprocal condition number if requested, otherwise warn of an ill-conditioned matrix if the reciprocal condition number is small. - Function File: norm (A, P) Compute the p-norm of the matrix A. If the second argument is missing, `p = 2' is assumed. If A is a matrix: P = `1' 1-norm, the largest column sum of the absolute values of A. P = `2' Largest singular value of A. P = `Inf' Infinity norm, the largest row sum of the absolute values of A. P = `"fro"' Frobenius norm of A, `sqrt (sum (diag (A' * A)))'. If A is a vector or a scalar: P = `Inf' `max (abs (A))'. P = `-Inf' `min (abs (A))'. other p-norm of A, `(sum (abs (A) .^ P)) ^ (1/P)'. - Function File: null (A, TOL) Return an orthonormal basis of the null space of A. The dimension of the null space is taken as the number of singular values of A not greater than TOL. If the argument TOL is missing, it is computed as max (size (A)) * max (svd (A)) * eps - Function File: orth (A, TOL) Return an orthonormal basis of the range space of A. The dimension of the range space is taken as the number of singular values of A greater than TOL. If the argument TOL is missing, it is computed as max (size (A)) * max (svd (A)) * eps - Loadable Function: pinv (X, TOL) Return the pseudoinverse of X. Singular values less than TOL are ignored. If the second argument is omitted, it is assumed that tol = max (size (X)) * sigma_max (X) * eps, where `sigma_max (X)' is the maximal singular value of X. - Function File: rank (A, TOL) Compute the rank of A, using the singular value decomposition. The rank is taken to be the number of singular values of A that are greater than the specified tolerance TOL. If the second argument is omitted, it is taken to be tol = max (size (A)) * sigma(1) * eps; where `eps' is machine precision and `sigma(1)' is the largest singular value of A. - Function File: trace (A) Compute the trace of A, `sum (diag (A))'.  File: octave.info, Node: Matrix Factorizations, Next: Functions of a Matrix, Prev: Basic Matrix Functions, Up: Linear Algebra Matrix Factorizations ===================== - Loadable Function: chol (A) Compute the Cholesky factor, R, of the symmetric positive definite matrix A, where r' * r = a. - Loadable Function: H = hess (A) - Loadable Function: [P, H] = hess (A) Compute the Hessenberg decomposition of the matrix A. The Hessenberg decomposition is usually used as the first step in an eigenvalue computation, but has other applications as well (see Golub, Nash, and Van Loan, IEEE Transactions on Automatic Control, 1979. The Hessenberg decomposition is `p * h * p' = a' where `p' is a square unitary matrix (`p' * p = I', using complex-conjugate transposition) and `h' is upper Hessenberg (`i >= j+1 => h (i, j) = 0'). - Loadable Function: [L, U, P] = lu (A) Compute the LU decomposition of A, using subroutines from LAPACK. The result is returned in a permuted form, according to the optional return value P. For example, given the matrix `a = [1, 2; 3, 4]', [l, u, p] = lu (a) returns l = 1.00000 0.00000 0.33333 1.00000 u = 3.00000 4.00000 0.00000 0.66667 p = 0 1 1 0 The matrix is not required to be square.. - Loadable Function: [Q, R, P] = qr (A) Compute the QR factorization of A, using standard LAPACK subroutines. For example, given the matrix `a = [1, 2; 3, 4]', [q, r] = qr (a) returns q = -0.31623 -0.94868 -0.94868 0.31623 r = -3.16228 -4.42719 0.00000 -0.63246 The `qr' factorization has applications in the solution of least squares problems `min norm(A x - b)' for overdetermined systems of equations (i.e., `a' is a tall, thin matrix). The QR factorization is `q * r = a' where `q' is an orthogonal matrix and `r' is upper triangular. The permuted QR factorization `[Q, R, P] = qr (A)' forms the QR factorization such that the diagonal entries of `r' are decreasing in magnitude order. For example, given the matrix `a = [1, 2; 3, 4]', [q, r, p] = qr(a) returns q = -0.44721 -0.89443 -0.89443 0.44721 r = -4.47214 -3.13050 0.00000 0.44721 p = 0 1 1 0 The permuted `qr' factorization `[q, r, p] = qr (a)' factorization allows the construction of an orthogonal basis of `span (a)'. - Loadable Function: LAMBDA = qz (A, B) Generalized eigenvalue problem A x = s B x, QZ decomposition. Three ways to call: 1. `lambda = qz(A,B)' Computes the generalized eigenvalues LAMBDA of (A - sB). 2. `[AA, BB, Q, Z, V, W, lambda] = qz (A, B)' Computes qz decomposition, generalized eigenvectors, and generalized eigenvalues of (A - sB) A V = B V diag(lambda) W' A = diag(lambda) W' B AA = Q'*A*Z, BB = Q'*B*Z with Q, Z orthogonal (unitary)= I 3. `[AA,BB,Z{,lambda}] = qz(A,B,opt)' As in form [2], but allows ordering of generalized eigenpairs for (e.g.) solution of discrete time algebraic Riccati equations. Form 3 is not available for complex matrices and does not compute the generalized eigenvectors V, W, nor the orthogonal matrix Q. OPT for ordering eigenvalues of the GEP pencil. The leading block of the revised pencil contains all eigenvalues that satisfy: `"N"' = unordered (default) `"S"' = small: leading block has all |lambda| <=1 `"B"' = big: leading block has all |lambda >= 1 `"-"' = negative real part: leading block has all eigenvalues in the open left half-plant `"+"' = nonnegative real part: leading block has all eigenvalues in the closed right half-plane Note: qz performs permutation balancing, but not scaling (see balance). Order of output arguments was selected for compatibility with MATLAB See also: balance, dare, eig, schur - Function File: [AA, BB, Q, Z] = qzhess (A, B) Compute the Hessenberg-triangular decomposition of the matrix pencil `(A, B)', returning `AA = Q * A * Z', `BB = Q * B * Z', with Q and Z orthogonal. For example, [aa, bb, q, z] = qzhess ([1, 2; 3, 4], [5, 6; 7, 8]) => aa = [ -3.02244, -4.41741; 0.92998, 0.69749 ] => bb = [ -8.60233, -9.99730; 0.00000, -0.23250 ] => q = [ -0.58124, -0.81373; -0.81373, 0.58124 ] => z = [ 1, 0; 0, 1 ] The Hessenberg-triangular decomposition is the first step in Moler and Stewart's QZ decomposition algorithm. Algorithm taken from Golub and Van Loan, `Matrix Computations, 2nd edition'. - Loadable Function: S = schur (A) - Loadable Function: [U, S] = schur (A, OPT) The Schur decomposition is used to compute eigenvalues of a square matrix, and has applications in the solution of algebraic Riccati equations in control (see `are' and `dare'). `schur' always returns `s = u' * a * u' where `u' is a unitary matrix (`u'* u' is identity) and `s' is upper triangular. The eigenvalues of `a' (and `s') are the diagonal elements of `s' If the matrix `a' is real, then the real Schur decomposition is computed, in which the matrix `u' is orthogonal and `s' is block upper triangular with blocks of size at most `2 x 2' along the diagonal. The diagonal elements of `s' (or the eigenvalues of the `2 x 2' blocks, when appropriate) are the eigenvalues of `a' and `s'. The eigenvalues are optionally ordered along the diagonal according to the value of `opt'. `opt = "a"' indicates that all eigenvalues with negative real parts should be moved to the leading block of `s' (used in `are'), `opt = "d"' indicates that all eigenvalues with magnitude less than one should be moved to the leading block of `s' (used in `dare'), and `opt = "u"', the default, indicates that no ordering of eigenvalues should occur. The leading `k' columns of `u' always span the `a'-invariant subspace corresponding to the `k' leading eigenvalues of `s'. - Loadable Function: S = svd (A) - Loadable Function: [U, S, V] = svd (A) Compute the singular value decomposition of A a = u * sigma * v' The function `svd' normally returns the vector of singular values. If asked for three return values, it computes U, S, and V. For example, svd (hilb (3)) returns ans = 1.4083189 0.1223271 0.0026873 and [u, s, v] = svd (hilb (3)) returns u = -0.82704 0.54745 0.12766 -0.45986 -0.52829 -0.71375 -0.32330 -0.64901 0.68867 s = 1.40832 0.00000 0.00000 0.00000 0.12233 0.00000 0.00000 0.00000 0.00269 v = -0.82704 0.54745 0.12766 -0.45986 -0.52829 -0.71375 -0.32330 -0.64901 0.68867 If given a second argument, `svd' returns an economy-sized decomposition, eliminating the unnecessary rows or columns of U or V. - Function File: [HOUSV, BETA, ZER] = housh (X, J, Z) Computes householder reflection vector housv to reflect x to be jth column of identity, i.e., (I - beta*housv*housv')x =e(j) inputs x: vector j: index into vector z: threshold for zero (usually should be the number 0) outputs: (see Golub and Van Loan) beta: If beta = 0, then no reflection need be applied (zer set to 0) housv: householder vector - Function File: [U, H, NU] = krylov (A, V, K, EPS1, PFLG); construct orthogonal basis U of block Krylov subspace; [v a*v a^2*v ... a^(k+1)*v]; method used: householder reflections to guard against loss of orthogonality eps1: threshhold for 0 (default: 1e-12) pflg: flag to use row pivoting (improves numerical behavior) 0 [default]: no pivoting; prints a warning message if trivial null space is corrupted 1 : pivoting performed outputs: u: orthogonal basis of block krylov subspace h: Hessenberg matrix; if v is a vector then a u = u h otherwise h is meaningless nu: dimension of span of krylov subspace (based on eps1) if b is a vector and k > m-1, krylov returns h = the Hessenberg decompostion of a. Reference: Hodel and Misra, "Partial Pivoting in the Computation of Krylov Subspaces", to be submitted to Linear Algebra and its Applications  File: octave.info, Node: Functions of a Matrix, Prev: Matrix Factorizations, Up: Linear Algebra Functions of a Matrix ===================== - Loadable Function: expm (A) Return the exponential of a matrix, defined as the infinite Taylor series expm(a) = I + a + a^2/2! + a^3/3! + ... The Taylor series is _not_ the way to compute the matrix exponential; see Moler and Van Loan, `Nineteen Dubious Ways to Compute the Exponential of a Matrix', SIAM Review, 1978. This routine uses Ward's diagonal Pade' approximation method with three step preconditioning (SIAM Journal on Numerical Analysis, 1977). Diagonal Pade' approximations are rational polynomials of matrices -1 D (a) N (a) whose Taylor series matches the first `2q+1' terms of the Taylor series above; direct evaluation of the Taylor series (with the same preconditioning steps) may be desirable in lieu of the Pade' approximation when `Dq(a)' is ill-conditioned. - Function File: logm (A) Compute the matrix logarithm of the square matrix A. Note that this is currently implemented in terms of an eigenvalue expansion and needs to be improved to be more robust. - Loadable Function: [RESULT, ERROR_ESTIMATE] = sqrtm (A) Compute the matrix square root of the square matrix A. Ref: Nicholas J. Higham. A new sqrtm for MATLAB. Numerical Analysis Report No. 336, Manchester Centre for Computational Mathematics, Manchester, England, January 1999. - Function File: kron (A, B) Form the kronecker product of two matrices, defined block by block as x = [a(i, j) b] For example, kron (1:4, ones (3, 1)) => 1 2 3 4 1 2 3 4 1 2 3 4 - Loadable Function: X = syl (A, B, C) Solve the Sylvester equation A X + X B + C = 0 using standard LAPACK subroutines. For example, syl ([1, 2; 3, 4], [5, 6; 7, 8], [9, 10; 11, 12]) => [ -0.50000, -0.66667; -0.66667, -0.50000 ]  File: octave.info, Node: Nonlinear Equations, Next: Quadrature, Prev: Linear Algebra, Up: Top Nonlinear Equations ******************* Octave can solve sets of nonlinear equations of the form F (x) = 0 using the function `fsolve', which is based on the MINPACK subroutine `hybrd'. - Loadable Function: [X, INFO, MSG] = fsolve (FCN, X0) Given FCN, the name of a function of the form `f (X)' and an initial starting point X0, `fsolve' solves the set of equations such that `f(X) == 0'. If FCN is a two-element string array, the first element names the function f described above, and the second element names a function of the form `j (X)' to compute the Jacobian matrix with elements df_i jac(i,j) = ---- dx_j You can use the function `fsolve_options' to set optional parameters for `fsolve'. - Loadable Function: fsolve_options (OPT, VAL) When called with two arguments, this function allows you set options parameters for the function `fsolve'. Given one argument, `fsolve_options' returns the value of the corresponding option. If no arguments are supplied, the names of all the available options and their current values are displayed. Options include `"tolerance"' Nonnegative relative tolerance. Here is a complete example. To solve the set of equations -2x^2 + 3xy + 4 sin(y) = 6 3x^2 - 2xy^2 + 3 cos(x) = -4 you first need to write a function to compute the value of the given function. For example: function y = f (x) y(1) = -2*x(1)^2 + 3*x(1)*x(2) + 4*sin(x(2)) - 6; y(2) = 3*x(1)^2 - 2*x(1)*x(2)^2 + 3*cos(x(1)) + 4; endfunction Then, call `fsolve' with a specified initial condition to find the roots of the system of equations. For example, given the function `f' defined above, [x, info] = fsolve ("f", [1; 2]) results in the solution x = 0.57983 2.54621 info = 1 A value of `info = 1' indicates that the solution has converged. The function `perror' may be used to print English messages corresponding to the numeric error codes. For example, perror ("fsolve", 1) -| solution converged to requested tolerance  File: octave.info, Node: Quadrature, Next: Differential Equations, Prev: Nonlinear Equations, Up: Top Quadrature ********** * Menu: * Functions of One Variable:: * Orthogonal Collocation::  File: octave.info, Node: Functions of One Variable, Next: Orthogonal Collocation, Up: Quadrature Functions of One Variable ========================= - Loadable Function: [V, IER, NFUN, ERR] = quad (F, A, B, TOL, SING) Integrate a nonlinear function of one variable using Quadpack. The first argument is the name of the function to call to compute the value of the integrand. It must have the form y = f (x) where Y and X are scalars. The second and third arguments are limits of integration. Either or both may be infinite. The optional argument TOL is a vector that specifies the desired accuracy of the result. The first element of the vector is the desired absolute tolerance, and the second element is the desired relative tolerance. To choose a relative test only, set the absolute tolerance to zero. To choose an absolute test only, set the relative tolerance to zero. The optional argument SING is a vector of values at which the integrand is known to be singular. The result of the integration is returned in V and IER contains an integer error code (0 indicates a successful integration). The value of NFUN indicates how many function evaluations were required, and ERR contains an estimate of the error in the solution. You can use the function `quad_options' to set optional parameters for `quad'. - Loadable Function: quad_options (OPT, VAL) When called with two arguments, this function allows you set options parameters for the function `quad'. Given one argument, `quad_options' returns the value of the corresponding option. If no arguments are supplied, the names of all the available options and their current values are displayed. Options include `"absolute tolerance"' Absolute tolerance; may be zero for pure relative error test. `"relative tolerance"' Nonnegative relative tolerance. If the absolute tolerance is zero, the relative tolerance must be greater than or equal to `max (50*eps, 0.5e-28)'. Here is an example of using `quad' to integrate the function F(X) = X * sin (1/X) * sqrt (abs (1 - X)) from X = 0 to X = 3. This is a fairly difficult integration (plot the function over the range of integration to see why). The first step is to define the function: function y = f (x) y = x .* sin (1 ./ x) .* sqrt (abs (1 - x)); endfunction Note the use of the `dot' forms of the operators. This is not necessary for the call to `quad', but it makes it much easier to generate a set of points for plotting (because it makes it possible to call the function with a vector argument to produce a vector result). Then we simply call quad: [v, ier, nfun, err] = quad ("f", 0, 3) => 1.9819 => 1 => 5061 => 1.1522e-07 Although `quad' returns a nonzero value for IER, the result is reasonably accurate (to see why, examine what happens to the result if you move the lower bound to 0.1, then 0.01, then 0.001, etc.).  File: octave.info, Node: Orthogonal Collocation, Prev: Functions of One Variable, Up: Quadrature Orthogonal Collocation ====================== - Loadable Function: [R, AMAT, BMAT, Q] = colloc (N, "left", "right") Compute derivative and integral weight matrices for orthogonal collocation using the subroutines given in J. Villadsen and M. L. Michelsen, `Solution of Differential Equation Models by Polynomial Approximation'. Here is an example of using `colloc' to generate weight matrices for solving the second order differential equation U' - ALPHA * U" = 0 with the boundary conditions U(0) = 0 and U(1) = 1. First, we can generate the weight matrices for N points (including the endpoints of the interval), and incorporate the boundary conditions in the right hand side (for a specific value of ALPHA). n = 7; alpha = 0.1; [r, a, b] = colloc (n-2, "left", "right"); at = a(2:n-1,2:n-1); bt = b(2:n-1,2:n-1); rhs = alpha * b(2:n-1,n) - a(2:n-1,n); Then the solution at the roots R is u = [ 0; (at - alpha * bt) \ rhs; 1] => [ 0.00; 0.004; 0.01 0.00; 0.12; 0.62; 1.00 ]  File: octave.info, Node: Differential Equations, Next: Optimization, Prev: Quadrature, Up: Top Differential Equations ********************** Octave has two built-in functions for solving differential equations. Both are based on reliable ODE solvers written in Fortran. * Menu: * Ordinary Differential Equations:: * Differential-Algebraic Equations::  File: octave.info, Node: Ordinary Differential Equations, Next: Differential-Algebraic Equations, Up: Differential Equations Ordinary Differential Equations =============================== The function `lsode' can be used to solve ODEs of the form dx -- = f (x, t) dt using Hindmarsh's ODE solver LSODE. - Loadable Function: [X, ISTATE, MSG] lsode (FCN, X_0, T, T_CRIT) Solve the set of differential equations dx -- = f(x, t) dt with x(t_0) = x_0 The solution is returned in the matrix X, with each row corresponding to an element of the vector T. The first element of T should be t_0 and should correspond to the initial state of the system X_0, so that the first row of the output is X_0. The first argument, FCN, is a string that names the function to call to compute the vector of right hand sides for the set of equations. The function must have the form XDOT = f (X, T) in which XDOT and X are vectors and T is a scalar. If FCN is a two-element string array, the first element names the function f described above, and the second element names a function to compute the Jacobian of f. The Jacobian function must have the form JAC = j (X, T) in which JAC is the matrix of partial derivatives | df_1 df_1 df_1 | | ---- ---- ... ---- | | dx_1 dx_2 dx_N | | | | df_2 df_2 df_2 | | ---- ---- ... ---- | df_i | dx_1 dx_2 dx_N | jac = ---- = | | dx_j | . . . . | | . . . . | | . . . . | | | | df_N df_N df_N | | ---- ---- ... ---- | | dx_1 dx_2 dx_N | The second and third arguments specify the intial state of the system, x_0, and the initial value of the independent variable t_0. The fourth argument is optional, and may be used to specify a set of times that the ODE solver should not integrate past. It is useful for avoiding difficulties with singularities and points where there is a discontinuity in the derivative. After a successful computation, the value of ISTATE will be 2 (consistent with the Fortran version of LSODE). If the computation is not successful, ISTATE will be something other than 2 and MSG will contain additional information. You can use the function `lsode_options' to set optional parameters for `lsode'. - Loadable Function: lsode_options (OPT, VAL) When called with two arguments, this function allows you set options parameters for the function `lsode'. Given one argument, `lsode_options' returns the value of the corresponding option. If no arguments are supplied, the names of all the available options and their current values are displayed. Options include `"absolute tolerance"' Absolute tolerance. May be either vector or scalar. If a vector, it must match the dimension of the state vector. `"relative tolerance"' Relative tolerance parameter. Unlike the absolute tolerance, this parameter may only be a scalar. The local error test applied at each integration step is abs (local error in x(i)) <= rtol * abs (y(i)) + atol(i) `"integration method"' A string specifing the method of integration to use to solve the ODE system. Valid values are "adams" "non-stiff" No Jacobian used (even if it is available). "bdf" "stiff" Use stiff backward differentiation formula (BDF) method. If a function to compute the Jacobian is not supplied, `lsode' will compute a finite difference approximation of the Jacobian matrix. `"initial step size"' The step size to be attempted on the first step (default is determined automatically). `"maximum order"' Restrict the maximum order of the solution method. If using the Adams method, this option must be between 1 and 12. Otherwise, it must be between 1 and 5, inclusive. `"maximum step size"' Setting the maximum stepsize will avoid passing over very large regions (default is not specified). `"minimum step size"' The minimum absolute step size allowed (default is 0). `"step limit"' Maximum number of steps allowed (default is 100000). Here is an example of solving a set of three differential equations using `lsode'. Given the function function xdot = f (x, t) xdot = zeros (3,1); xdot(1) = 77.27 * (x(2) - x(1)*x(2) + x(1) \ - 8.375e-06*x(1)^2); xdot(2) = (x(3) - x(1)*x(2) - x(2)) / 77.27; xdot(3) = 0.161*(x(1) - x(3)); endfunction and the initial condition `x0 = [ 4; 1.1; 4 ]', the set of equations can be integrated using the command t = linspace (0, 500, 1000); y = lsode ("f", x0, t); If you try this, you will see that the value of the result changes dramatically between T = 0 and 5, and again around T = 305. A more efficient set of output points might be t = [0, logspace (-1, log10(303), 150), \ logspace (log10(304), log10(500), 150)]; Octave also includes ODESSA, a modification of LSODE that allows for the computation of parametric sensitivity information simultaneously with the solution of the set of ODEs. - Loadable Function: [X, SX, ISTATE, MSG] odessa (FCN, X_0, P, SX_0, T, T_CRIT) Solve the set of differential equations dx -- = f(x, t; p) dt with x(t_0) = x_0 and simultaneously compute the first-order sensitivity coefficients given by s'(t) = j(t)*s(t) + df/dp in which s(t) = dx(t)/dp (sensitivity functions) s'(t) = d(dx(t)/dp)/dt j(t) = df(x,t;p)/dx(t) (Jacobian matrix) df/dp = df(x,t;p)/dp (inhomogeneity matrix) The solution is returned in the matrix X, with each row corresponding to an element of the vector T. The first element of T should be t_0 and should correspond to the initial state of the system X_0, so that the first row of the output is X_0. The sensitivities are returned in a list of matrices, SX, with each element of the list corresponding to an element of the vector T. The first argument, FCN, is a string that names the function to call to compute the vector of right hand sides for the set of equations. The function must have the form XDOT = f (X, T, P) in which XDOT and X are vectors and T is a scalar. The variable P is a vector of parameters. The FCN argument may also be an array of strings ["f"; "j"; "b"] in which the first element names the function f described above, the second element names a function to compute the Jacobian of f, and the third element names a function to compute the inhomogeneity matrix. The Jacobian function must have the form JAC = j (X, T, P) in which X, T, and P have the same meanings as above for the function F, and JAC is the matrix of partial derivatives df_i jac = ---- dx_j The function B must have the form DFDP = b (X, T, P, C) in which X, T, and P have the same meanings as above for the function F, C indicates which partial derivatives to return in DFDP. For example, if C is 2, you should return the vector df_i dfdp = ----, i = 1:length(x) dp_2 The second argument, X_0, specifies the intial state of the system. The third argument, P, specifies the set of parameters. The fourth argument, SX_0 specifies the initial values of the sensitivities. The sixth argument is optional, and may be used to specify a set of times that the ODE solver should not integrate past. It is useful for avoiding difficulties with singularities and points where there is a discontinuity in the derivative. After a successful computation, the value of ISTATE will be 2 (consistent with the Fortran version of ODESSA). If the computation is not successful, ISTATE will be something other than 2 and MSG will contain additional information. You can use the function `lsode_options' to set optional parameters for `lsode'. - Loadable Function: odessa_options (OPT, VAL) When called with two arguments, this function allows you set options parameters for the function `odessa'. Given one argument, `odessa_options' returns the value of the corresponding option. If no arguments are supplied, the names of all the available options and their current values are displayed. Options include `"absolute tolerance"' Absolute tolerance. May be either vector or scalar. If a vector, it must match the dimension of the state vector. `"relative tolerance"' Relative tolerance parameter. Unlike the absolute tolerance, this parameter may only be a scalar. The local error test applied at each integration step is abs (local error in x(i)) <= rtol * abs (y(i)) + atol(i) `"integration method"' A string specifing the method of integration to use to solve the ODE system. Valid values are "adams" "non-stiff" No Jacobian used (even if it is available). "bdf" "stiff" Use stiff backward differentiation formula (BDF) method. If a function to compute the Jacobian is not supplied, `lsode' will compute a finite difference approximation of the Jacobian matrix. `"initial step size"' The step size to be attempted on the first step (default is determined automatically). `"maximum order"' Restrict the maximum order of the solution method. If using the Adams method, this option must be between 1 and 12. Otherwise, it must be between 1 and 5, inclusive. `"maximum step size"' Setting the maximum stepsize will avoid passing over very large regions (default is not specified). `"minimum step size"' The minimum absolute step size allowed (default is 0). `"step limit"' Maximum number of steps allowed (default is 100000). See Alan C. Hindmarsh, `ODEPACK, A Systematized Collection of ODE Solvers', in Scientific Computing, R. S. Stepleman, editor, (1983) for more information about the inner workings of `lsode'.  File: octave.info, Node: Differential-Algebraic Equations, Prev: Ordinary Differential Equations, Up: Differential Equations Differential-Algebraic Equations ================================ The function `daspk' can be used to solve DAEs of the form 0 = f (x-dot, x, t), x(t=0) = x_0, x-dot(t=0) = x-dot_0 using Petzold's DAE solver DASPK. - Loadable Function: [X, XDOT, ISTATE, MSG] = daspk (FCN, X_0, XDOT_0, T, T_CRIT) Solve the set of differential-algebraic equations 0 = f (xdot, x, t) with x(t_0) = x_0, xdot(t_0) = xdot_0 The solution is returned in the matrices X and XDOT, with each row in the result matrices corresponding to one of the elements in the vector T. The first element of T should be t_0 and correspond to the initial state of the system X_0 and its derivative XDOT_0, so that the first row of the output X is X_0 and the first row of the output XDOT is XDOT_0. The first argument, FCN, is a string that names the function to call to compute the vector of residuals for the set of equations. It must have the form RES = f (X, XDOT, T) in which X, XDOT, and RES are vectors, and T is a scalar. If FCN is a two-element string array, the first element names the function f described above, and the second element names a function to compute the modified Jacobian df df jac = -- + c ------ dx d xdot The modified Jacobian function must have the form JAC = j (X, XDOT, T, C) The second and third arguments to `daspk' specify the initial condition of the states and their derivatives, and the fourth argument specifies a vector of output times at which the solution is desired, including the time corresponding to the initial condition. The set of initial states and derivatives are not strictly required to be consistent. If they are not consistent, you must use the `daspk_options' function to provide additional information so that `daspk' can compute a consistent starting point. The fifth argument is optional, and may be used to specify a set of times that the DAE solver should not integrate past. It is useful for avoiding difficulties with singularities and points where there is a discontinuity in the derivative. After a successful computation, the value of ISTATE will be greater than zero (consistent with the Fortran version of DASPK). If the computation is not successful, the value of ISTATE will be less than zero and MSG will contain additional information. You can use the function `daspk_options' to set optional parameters for `daspk'. - Loadable Function: daspk_options (OPT, VAL) When called with two arguments, this function allows you set options parameters for the function `daspk'. Given one argument, `daspk_options' returns the value of the corresponding option. If no arguments are supplied, the names of all the available options and their current values are displayed. Options include `"absolute tolerance"' Absolute tolerance. May be either vector or scalar. If a vector, it must match the dimension of the state vector, and the relative tolerance must also be a vector of the same length. `"relative tolerance"' Relative tolerance. May be either vector or scalar. If a vector, it must match the dimension of the state vector, and the absolute tolerance must also be a vector of the same length. The local error test applied at each integration step is abs (local error in x(i)) <= rtol(i) * abs (Y(i)) + atol(i) `"compute consistent initial condition"' Denoting the differential variables in the state vector by `Y_d' and the algebraic variables by `Y_a', `ddaspk' can solve one of two initialization problems: 1. Given Y_d, calculate Y_a and Y'_d 2. Given Y', calculate Y. In either case, initial values for the given components are input, and initial guesses for the unknown components must also be provided as input. Set this option to 1 to solve the first problem, or 2 to solve the second (the default default is 0, so you must provide a set of initial conditions that are consistent). If this option is set to a nonzero value, you must also set the `"algebraic variables"' option to declare which variables in the problem are algebraic. `"use initial condition heuristics"' Set to a nonzero value to use the initial condition heuristics options described below. `"initial condition heuristics"' A vector of the following parameters that can be used to control the initial condition calculation. `MXNIT' Maximum number of Newton iterations (default is 5). `MXNJ' Maximum number of Jacobian evaluations (default is 6). `MXNH' Maximum number of values of the artificial stepsize parameter to be tried if the `"compute consistent initial condition"' option has been set to 1 (default is 5). Note that the maximum number of Newton iterations allowed in all is `MXNIT*MXNJ*MXNH' if the `"compute consistent initial condition"' option has been set to 1 and `MXNIT*MXNJ' if it is set to 2. `LSOFF' Set to a nonzero value to disable the linesearch algorithm (default is 0). `STPTOL' Minimum scaled step in linesearch algorithm (default is eps^(2/3)). `EPINIT' Swing factor in the Newton iteration convergence test. The test is applied to the residual vector, premultiplied by the approximate Jacobian. For convergence, the weighted RMS norm of this vector (scaled by the error weights) must be less than `EPINIT*EPCON', where `EPCON' = 0.33 is the analogous test constant used in the time steps. The default is `EPINIT' = 0.01. `"print initial condition info"' Set this option to a nonzero value to display detailed information about the initial condition calculation (default is 0). `"exclude algebraic variables from error test"' Set to a nonzero value to exclude algebraic variables from the error test. You must also set the `"algebraic variables"' option to declare which variables in the problem are algebraic (default is 0). `"algebraic variables"' A vector of the same length as the state vector. A nonzero element indicates that the corresponding element of the state vector is an algebraic variable (i.e., its derivative does not appear explicitly in the equation set. This option is required by the `compute consistent initial condition"' and `"exclude algebraic variables from error test"' options. `"enforce inequality constraints"' Set to one of the following values to enforce the inequality constraints specified by the `"inequality constraint types"' option (default is 0). 1. To have constraint checking only in the initial condition calculation. 2. To enforce constraint checking during the integration. 3. To enforce both options 1 and 2. `"inequality constraint types"' A vector of the same length as the state specifying the type of inequality constraint. Each element of the vector corresponds to an element of the state and should be assigned one of the following codes -2 Less than zero. -1 Less than or equal to zero. 0 Not constrained. 1 Greater than or equal to zero. 2 Greater than zero. This option only has an effect if the `"enforce inequality constraints"' option is nonzero. `"initial step size"' Differential-algebraic problems may occaisionally suffer from severe scaling difficulties on the first step. If you know a great deal about the scaling of your problem, you can help to alleviate this problem by specifying an initial stepsize (default is computed automatically). `"maximum order"' Restrict the maximum order of the solution method. This option must be between 1 and 5, inclusive (default is 5). `"maximum step size"' Setting the maximum stepsize will avoid passing over very large regions (default is not specified). Octave also includes DASSL, an earlier version of DASPK, and DASRT, which can be used to solve DAEs with constraints (stopping conditions). - Loadable Function: [X, XDOT, T_OUT, ISTAT, MSG] = dasrt (FCN [, G], X_0, XDOT_0, T [, T_CRIT]) Solve the set of differential-algebraic equations 0 = f (xdot, x, t) with x(t_0) = x_0, xdot(t_0) = xdot_0 with functional stopping criteria (root solving). The solution is returned in the matrices X and XDOT, with each row in the result matrices corresponding to one of the elements in the vector T_OUT. The first element of T should be t_0 and correspond to the initial state of the system X_0 and its derivative XDOT_0, so that the first row of the output X is X_0 and the first row of the output XDOT is XDOT_0. The vector T provides an upper limit on the length of the integration. If the stopping condition is met, the vector T_OUT will be shorter than T, and the final element of T_OUT will be the point at which the stopping condition was met, and may not correspond to any element of the vector T. The first argument, FCN, is a string that names the function to call to compute the vector of residuals for the set of equations. It must have the form RES = f (X, XDOT, T) in which X, XDOT, and RES are vectors, and T is a scalar. If FCN is a two-element string array, the first element names the function f described above, and the second element names a function to compute the modified Jacobian df df jac = -- + c ------ dx d xdot The modified Jacobian function must have the form JAC = j (X, XDOT, T, C) The optional second argument names a function that defines the constraint functions whose roots are desired during the integration. This function must have the form G_OUT = g (X, T) and return a vector of the constraint function values. If the value of any of the constraint functions changes sign, DASRT will attempt to stop the integration at the point of the sign change. If the name of the constraint function is omitted, `dasrt' solves the saem problem as `daspk' or `dassl'. Note that because of numerical errors in the constraint functions due to roundoff and integration error, DASRT may return false roots, or return the same root at two or more nearly equal values of T. If such false roots are suspected, the user should consider smaller error tolerances or higher precision in the evaluation of the constraint functions. If a root of some constraint function defines the end of the problem, the input to DASRT should nevertheless allow integration to a point slightly past that root, so that DASRT can locate the root by interpolation. The third and fourth arguments to `dasrt' specify the initial condition of the states and their derivatives, and the fourth argument specifies a vector of output times at which the solution is desired, including the time corresponding to the initial condition. The set of initial states and derivatives are not strictly required to be consistent. In practice, however, DASSL is not very good at determining a consistent set for you, so it is best if you ensure that the initial values result in the function evaluating to zero. The sixth argument is optional, and may be used to specify a set of times that the DAE solver should not integrate past. It is useful for avoiding difficulties with singularities and points where there is a discontinuity in the derivative. After a successful computation, the value of ISTATE will be greater than zero (consistent with the Fortran version of DASSL). If the computation is not successful, the value of ISTATE will be less than zero and MSG will contain additional information. You can use the function `dasrt_options' to set optional parameters for `dasrt'. - Loadable Function: dasrt_options (OPT, VAL) When called with two arguments, this function allows you set options parameters for the function `dasrt'. Given one argument, `dasrt_options' returns the value of the corresponding option. If no arguments are supplied, the names of all the available options and their current values are displayed. Options include `"absolute tolerance"' Absolute tolerance. May be either vector or scalar. If a vector, it must match the dimension of the state vector, and the relative tolerance must also be a vector of the same length. `"relative tolerance"' Relative tolerance. May be either vector or scalar. If a vector, it must match the dimension of the state vector, and the absolute tolerance must also be a vector of the same length. The local error test applied at each integration step is abs (local error in x(i)) <= rtol(i) * abs (Y(i)) + atol(i) `"initial step size"' Differential-algebraic problems may occaisionally suffer from severe scaling difficulties on the first step. If you know a great deal about the scaling of your problem, you can help to alleviate this problem by specifying an initial stepsize. `"maximum order"' Restrict the maximum order of the solution method. This option must be between 1 and 5, inclusive. `"maximum step size"' Setting the maximum stepsize will avoid passing over very large regions. `"step limit"' Maximum number of integration steps to attempt on a single call to the underlying Fortran code. See K. E. Brenan, et al., `Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations', North-Holland (1989) for more information about the implementation of DASSL.