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* Octave: (octave).	Interactive language for numerical computations.
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   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: Ordinary Differential Equations,  Next: Differential-Algebraic Equations,  Prev: Differential Equations,  Up: Differential Equations

Ordinary Differential Equations
===============================

   The function `lsode' can be used Solve ODEs of the form

     dx
     -- = f (x, t)
     dt

using Hindmarsh's ODE solver LSODE.

 - Loadable Function:  lsode (FCN, X0, T, T_CRIT)
     Return a matrix of X as a function of T, given the initial state
     of the system X0.  Each row in the result matrix corresponds to
     one of the elements in the vector T.  The first element of T
     corresponds to the initial state X0, so that the first row of the
     output is X0.

     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.  It must have the form

          XDOT = f (X, T)

     where XDOT and X are vectors and T is a scalar.

     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.

   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)];

 - 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.

   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 `dassl' can be used 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 DASSL.

 - Loadable Function: [X, XDOT] = dassl (FCN, X0, XDOT0, T, T_CRIT)
     Return a matrix of states and their first derivatives with respect
     to T.  Each row in the result matrices correspond to one of the
     elements in the vector T.  The first element of T corresponds to
     the initial state X0 and derivative XDOT0, so that the first row
     of the output X is X0 and the first row of the output XDOT is
     XDOT0.

     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)

     where X, XDOT, and RES are vectors, and T is a scalar.

     The second and third arguments to `dassl' 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 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.

 - Loadable Function:  dassl_options (OPT, VAL)
     When called with two arguments, this function allows you set
     options parameters for the function `lsode'.  Given one argument,
     `dassl_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.

   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.


File: octave.info,  Node: Optimization,  Next: Statistics,  Prev: Differential Equations,  Up: Top

Optimization
************

* Menu:

* Quadratic Programming::
* Nonlinear Programming::
* Linear Least Squares::


File: octave.info,  Node: Quadratic Programming,  Next: Nonlinear Programming,  Prev: Optimization,  Up: Optimization

Quadratic Programming
=====================


File: octave.info,  Node: Nonlinear Programming,  Next: Linear Least Squares,  Prev: Quadratic Programming,  Up: Optimization

Nonlinear Programming
=====================


File: octave.info,  Node: Linear Least Squares,  Prev: Nonlinear Programming,  Up: Optimization

Linear Least Squares
====================

 - Function File: [BETA, V, R] = gls (Y, X, O)
     Generalized least squares estimation for the multivariate model `Y
     = X * B + E' with `mean (E) = 0' and `cov (vec (E)) = (S^2)*O',
     where Y is a T by P matrix, X is a T by K matrix, B is a K by P
     matrix, E is a T by P matrix, and O is a TP by TP matrix.

     Each row of Y and X is an observation and each column a variable.

     The return values BETA, V, and R are defined as follows.

    BETA
          The GLS estimator for B.

    V
          The GLS estimator for `S^2'.

    R
          The matrix of GLS residuals, `R = Y - X * BETA'.

 - Function File: [BETA, SIGMA, R] = ols (Y, X)
     Ordinary least squares estimation for the multivariate model `Y =
     X*B + E' with `mean (E) = 0' and `cov (vec (E)) = kron (S, I)'.
     where Y is a T by P matrix, X is a T by K matrix, B is a K by P
     matrix, and E is a T by P matrix.

     Each row of Y and X is an observation and each column a variable.

     The return values BETA, SIGMA, and R are defined as follows.

    BETA
          The OLS estimator for B, `BETA = pinv (X) * Y', where `pinv
          (X)' denotes the pseudoinverse of X.

    SIGMA
          The OLS estimator for the matrix S,

               SIGMA = (Y-X*BETA)' * (Y-X*BETA) / (T-rank(X))

    R
          The matrix of OLS residuals, `R = Y - X * BETA'.


File: octave.info,  Node: Statistics,  Next: Sets,  Prev: Optimization,  Up: Top

Statistics
**********

   I hope that someday Octave will include more statistics functions.
If you would like to help improve Octave in this area, please contact
<bug-octave@bevo.che.wisc.edu>.

 - Function File:  mean (X)
     If X is a vector, compute the mean of the elements of X

          mean (x) = SUM_i x(i) / N
     If X is a matrix, compute the mean for each column and return them
     in a row vector.

 - Function File:  median (X)
     If X is a vector, compute the median value of the elements of X.

                      x(ceil(N/2)),             N odd
          median(x) =
                      (x(N/2) + x((N/2)+1))/2,  N even
     If X is a matrix, compute the median value for each column
     and return them in a row vector.

 - Function File:  std (X)
     If X is a vector, compute the standard deviation of the elements
     of X.

          std (x) = sqrt (sumsq (x - mean (x)) / (n - 1))
     If X is a matrix, compute the standard deviation for each
     column and return them in a row vector.

 - Function File:  cov (X, Y)
     If each row of X and Y is an observation and each column is a
     variable, the (I,J)-th entry of `cov (X, Y)' is the covariance
     between the I-th variable in X and the J-th variable in Y.  If
     called with one argument, compute `cov (X, X)'.

 - Function File:  corrcoef (X, Y)
     If each row of X and Y is an observation and each column is a
     variable, the (I,J)-th entry of `corrcoef (X, Y)' is the
     correlation between the I-th variable in X and the J-th variable
     in Y.  If called with one argument, compute `corrcoef (X, X)'.

 - Function File:  kurtosis (X)
     If X is a vector of length N, return the kurtosis

          kurtosis (x) = N^(-1) std(x)^(-4) sum ((x - mean(x)).^4) - 3

     of X.  If X is a matrix, return the row vector containing the
     kurtosis of each column.

 - Function File:  mahalanobis (X, Y)
     Return the Mahalanobis' D-square distance between the multivariate
     samples X and Y, which must have the same number of components
     (columns), but may have a different number of observations (rows).

 - Function File:  skewness (X)
     If X is a vector of length N, return the skewness

          skewness (x) = N^(-1) std(x)^(-3) sum ((x - mean(x)).^3)

     of X.  If X is a matrix, return the row vector containing the
     skewness of each column.


File: octave.info,  Node: Sets,  Next: Polynomial Manipulations,  Prev: Statistics,  Up: Top

Sets
****

   Octave has a limited set of functions for managing sets of data,
where a set is defined as a collection unique elements.

 - Function File:  create_set (X)
     Return a row vector containing the unique values in X, sorted in
     ascending order.  For example,

          create_set ([ 1, 2; 3, 4; 4, 2 ])
               => [ 1, 2, 3, 4 ]

 - Function File:  union (X, Y)
     Return the set of elements that are in either of the sets X and Y.
     For example,

          union ([ 1, 2, 4 ], [ 2, 3, 5 ])
               => [ 1, 2, 3, 4, 5 ]

 - Function File:  intersection (X, Y)
     Return the set of elements that are in both sets X and Y.  For
     example,

          intersection ([ 1, 2, 3 ], [ 2, 3, 5 ])
               => [ 2, 3 ]

 - Function File:  complement (X, Y)
     Return the elements of set Y that are not in set X.  For example,

          complement ([ 1, 2, 3 ], [ 2, 3, 5 ])
               => 5


File: octave.info,  Node: Polynomial Manipulations,  Next: Control Theory,  Prev: Sets,  Up: Top

Polynomial Manipulations
************************

   In Octave, a polynomial is represented by its coefficients (arranged
in descending order).  For example, a vector  $c$ of length  N

     p(x) = C(1) x^N + ... + C(N) x + C(N+1).

 - Function File:  compan (C)
     Compute the companion matrix corresponding to polynomial
     coefficient vector C.

     The companion matrix is

               _                                                        _
              |  -c(2)/c(1)   -c(3)/c(1)  ...  -c(N)/c(1)  -c(N+1)/c(1)  |
              |       1            0      ...       0             0      |
              |       0            1      ...       0             0      |
          A = |       .            .   .            .             .      |
              |       .            .       .        .             .      |
              |       .            .           .    .             .      |
              |_      0            0      ...       1             0     _|

     The eigenvalues of the companion matrix are equal to the roots of
     the polynomial.

 - Function File:  conv (A, B)
     Convolve two vectors.

     `y = conv (a, b)' returns a vector of length equal to `length (a)
     + length (b) - 1'.  If A and B are polynomial coefficient vectors,
     `conv' returns the coefficients of the product polynomial.

 - Function File:  deconv (Y, A)
     Deconvolve two vectors.

     `[b, r] = deconv (y, a)' solves for B and R such that `y = conv
     (a, b) + r'.

     If Y and A are polynomial coefficient vectors, B will contain the
     coefficients of the polynomial quotient and R will be a remander
     polynomial of lowest order.

 - Function File:  poly (A)
     If A is a square N-by-N matrix, `poly (A)' is the row vector of
     the coefficients of `det (z * eye (N) - a)', the characteristic
     polynomial of A.  If X is a vector, `poly (X)' is a vector of
     coefficients of the polynomial whose roots are the elements of X.

 - Function File:  polyderiv (C)
     Return the coefficients of the derivative of the polynomial whose
     coefficients are given by vector C.

 - Function File:  polyfit (N, Y, N)
     Return the coefficients of a polynomial P(X) of degree N that
     minimizes `sumsq (p(x(i)) - y(i))',  to best fit the data in the
     least squares sense.

 - Function File:  polyinteg (C)
     Return the coefficients of the integral the polynomial whose
     coefficients are represented by the vector C.

     The constant of integration is set to zero.

 - Function File:  polyreduce (C)
     Reduces a polynomial coefficient vector to a minimum number of
     terms by stripping off any leading zeros.

 - Function File:  polyval (C, X)
     Evaluate a polynomial.

     `polyval (C, X)' will evaluate the polynomial at the specified
     value of X.

     If X is a vector or matrix, the polynomial is evaluated at each of
     the elements of X.

 - Function File:  polyvalm (C, X)
     Evaluate a polynomial in the matrix sense.

     `polyvalm (C, X)' will evaluate the polynomial in the matrix
     sense, i.e. matrix multiplication is used instead of element by
     element multiplication as is used in polyval.

     The argument X must be a square matrix.

 - Function File:  residue (B, A, TOL)
     If B and A are vectors of polynomial coefficients, then residue
     calculates the partial fraction expansion corresponding to the
     ratio of the two polynomials.

     The function `residue' returns R, P, K, and E, where the vector R
     contains the residue terms, P contains the pole values, K contains
     the coefficients of a direct polynomial term (if it exists) and E
     is a vector containing the powers of the denominators in the
     partial fraction terms.

     Assuming B and A represent polynomials  P (s) and Q(s)  we have:

           P(s)    M       r(m)         N
           ---- = SUM -------------  + SUM k(i)*s^(N-i)
           Q(s)   m=1 (s-p(m))^e(m)    i=1

     where M is the number of poles (the length of the R, P, and E
     vectors) and N is the length of the K vector.

     The argument TOL is optional, and if not specified, a default
     value of 0.001 is assumed.  The tolerance value is used to
     determine whether poles with small imaginary components are
     declared real.  It is also used to determine if two poles are
     distinct.  If the ratio of the imaginary part of a pole to the
     real part is less than TOL, the imaginary part is discarded.  If
     two poles are farther apart than TOL they are distinct.  For
     example,

           b = [1, 1, 1];
           a = [1, -5, 8, -4];
           [r, p, k, e] = residue (b, a);
               => r = [-2, 7, 3]
               => p = [2, 2, 1]
               => k = [](0x0)
               => e = [1, 2, 1]

     which implies the following partial fraction expansion

                  s^2 + s + 1       -2        7        3
             ------------------- = ----- + ------- + -----
             s^3 - 5s^2 + 8s - 4   (s-2)   (s-2)^2   (s-1)

 - Function File:  roots (V)
     For a vector V with N components, return the roots of the
     polynomial

          v(1) * z^(N-1) + ... + v(N-1) * z + v(N).


File: octave.info,  Node: Control Theory,  Next: Signal Processing,  Prev: Polynomial Manipulations,  Up: Top

Control Theory
**************

   Most of the functions described in this chapter were contributed by
A. Scottedward Hodel <A.S.Hodel@eng.auburn.edu> and R. Bruce Tenison
<Bruce.Tenison@eng.auburn.edu>.  They have also written a larger
collection of functions for solving linear control problems.  It is
currently being updated for Octave version 2, with snapshots of the
sources available from `ftp://ftp.eng.auburn.edu/pub/hodel'.

 - Function File: [N, M, P] = abcddim (A, B, C, D)
     Check for compatibility of the dimensions of the matrices defining
     the linear system [A, B, C, D] corresponding to

          dx/dt = a x + b u
          y = c x + d u

     or a similar discrete-time system.

     If the matrices are compatibly dimensioned, then `abcddim' returns

    N
          The number of system states.

    M
          The number of system inputs.

    P
          The number of system outputs.

     Otherwise `abcddim' returns N = M = P = -1.

 - Function File:  are (A, B, C, OPT)
     Return the solution, X, of the algebraic Riccati equation

          a' * x + x * a - x * b * x + c = 0

     for identically dimensioned square matrices A, B, and C.  If B is
     not square, `are' attempts to use `B*B'' instead.  If C is not
     square, `are' attempts to use `C'*C') instead.

     To form the solution, Laub's Schur method (IEEE Transactions on
     Automatic Control, 1979) is applied to the appropriate Hamiltonian
     matrix.

     The optional argument OPT is passed to the eigenvalue balancing
     routine.  If it is omitted, a value of `"B"' is assumed.

 - Function File:  c2d (A, B, T)
     Convert the continuous time system described by:

          dx/dt = a x + b u

     into a discrete time equivalent model

          x[k+1] = Ad x[k] + Bd u[k]

     via the matrix exponential assuming a zero-order hold on the input
     and sample time T.

 - Function File:  dare (A, B, C, R, OPT)
     Return the solution, X of the discrete-time algebraic Riccati
     equation

          a' x a - x + a' x b (r + b' x b)^(-1) b' x a + c = 0

     for matrices with dimensions:

    A
          N by N.

    B
          N by M.

    C
          N by N, symmetric positive semidefinite.

    R
          M by M, symmetric positive definite (invertible).

     If C is not square, then the function attempts to use `C'*C'
     instead.

     To form the solution, Laub's Schur method (IEEE Transactions on
     Automatic Control, 1979) is applied to the appropriate symplectic
     matrix.

     See also Ran and Rodman, `Stable Hermitian Solutions of Discrete
     Algebraic Riccati Equations', Mathematics of Control, Signals and
     Systems, Volume 5, Number 2 (1992).

     The optional argument OPT is passed to the eigenvalue balancing
     routine.  If it is omitted, a value of `"B"' is assumed.

 - Function File:  dgram (A, B)
     Return the discrete controllability or observability gramian for
     the discrete time system described by

          x[k+1] = A x[k] + B u[k]
            y[k] = C x[k] + D u[k]

     For example, `dgram (A, B)' returns the discrete controllability
     gramian and `dgram (A', C')' returns the observability gramian.

 - Function File: [L, M, P, E] = dlqe (A, G, C, SIGW, SIGV, Z)
     Construct the linear quadratic estimator (Kalman filter) for the
     discrete time system

          x[k+1] = A x[k] + B u[k] + G w[k]
            y[k] = C x[k] + D u[k] + w[k]

     where W, V are zero-mean gaussian noise processes with respective
     intensities `SIGW = cov (W, W)' and `SIGV = cov (V, V)'.

     If specified, Z is `cov (W, V)'.  Otherwise `cov (W, V) = 0'.

     The observer structure is

          z[k+1] = A z[k] + B u[k] + k(y[k] - C z[k] - D u[k])

     The following values are returned:

    L
          The observer gain, (A - ALC).  is stable.

    M
          The Riccati equation solution.

    P
          The estimate error covariance after the measurement update.

    E
          The closed loop poles of (A - ALC).

 - Function File: [K, P, E] = dlqr (A, B, Q, R, Z)
     Construct the linear quadratic regulator for the discrete time
     system

          x[k+1] = A x[k] + B u[k]

     to minimize the cost functional

          J = Sum (x' Q x + u' R u)

     Z omitted or

          J = Sum (x' Q x + u' R u + 2 x' Z u)

     Z included.

     The following values are returned:

    K
          The state feedback gain, (A - BK) is stable.

    P
          The solution of algebraic Riccati equation.

    E
          The closed loop poles of (A - BK).

 - Function File:  dlyap (A, B)
     Solve the discrete-time Lyapunov equation `a x a' - x + b = 0' for
     square matrices A, B.  If B is not square, then the function
     attempts to solve either `An Algorithm for Solving the Matrix
     Equation X = FXF' + S', International Journal of Control, Volume
     25, Number 5, pages 745-753 (1977); column-by-column solution
     method as suggested in Hammerling, `Numerical Solution of the
     Stable, Non-Negative Definite Lyapunov Equation', IMA Journal of
     Numerical Analysis, Volume 2, pages 303-323 (1982).

 - Function File:  is_controllable (A, B, TOL)
     Return 1 if the pair (A, B) is controllable.  Otherwise, return 0.

     The optional argument TOL is a roundoff parameter.  If it is
     omitted, a value of `2*eps' is used.

     Currently, `is_controllable' just constructs the controllability
     matrix and checks rank.

 - Function File:  is_observable (A, C, TOL)
     Return 1 if the pair (A, C) is observable.  Otherwise, return 0.

     The optional argument TOL is a roundoff parameter.  If it is
     omitted, a value of `2*eps' is used.

 - Function File: [K, P, E] = lqe (A, G, C, SIGW, SIGV, Z)
     Construct the linear quadratic estimator (Kalman filter) for the
     continuous time system

          dx
          -- = a x + b u
          dt
          
          y = c x + d u

     where W and V are zero-mean gaussian noise processes with
     respective intensities

          sigw = cov (w, w)
          sigv = cov (v, v)

     The optional argument Z is the cross-covariance `cov (W, V)'.  If
     it is omitted, `cov (W, V) = 0' is assumed.

     Observer structure is `dz/dt = A z + B u + k (y - C z - D u)'

     The following values are returned:

    K
          The observer gain, (A - KC) is stable.

    P
          The solution of algebraic Riccati equation.

    E
          The vector of closed loop poles of (A - KC).

 - Function File: [K, P, E] = lqr (A, B, Q, R, Z)
     construct the linear quadratic regulator for the continuous time
     system

          dx
          -- = A x + B u
          dt

     to minimize the cost functional

                infinity
                /
            J = |  x' Q x + u' R u
               /
              t=0

     Z omitted or

                infinity
                /
            J = |  x' Q x + u' R u + 2 x' Z u
               /
              t=0

     Z included.

     The following values are returned:

    K
          The state feedback gain, (A - BK) is stable.

    P
          The stabilizing solution of appropriate algebraic Riccati
          equation.

    E
          The vector of the closed loop poles of (A - BK).

 - Function File:  lyap (A, B, C)
     Solve the Lyapunov (or Sylvester) equation via the Bartels-Stewart
     algorithm (Communications of the ACM, 1972).

     If A, B, and C are specified, then `lyap' returns the solution of
     the  Sylvester equation

          a x + x b + c = 0

     If only `(a, b)' are specified, then `lyap' returns the solution
     of the Lyapunov equation

          a' x + x a + b = 0

     If B is not square, then `lyap' returns the solution of either

          a' x + x a + b' b = 0

     or

          a x + x a' + b b' = 0

     whichever is appropriate.

 - Function File:  tzero (A, B, C, D, OPT)
     Compute the transmission zeros of [A, B, C, D].

     The optional argument OPT is passed to the eigenvalue balancing
     routine.  If it is omitted, a value of `"B"' is assumed.


File: octave.info,  Node: Signal Processing,  Next: Image Processing,  Prev: Control Theory,  Up: Top

Signal Processing
*****************

   I hope that someday Octave will include more signal processing
functions.  If you would like to help improve Octave in this area,
please contact <bug-octave@bevo.che.wisc.edu>.

 - Function File:  detrend (X, P)
     If X is a vector, `detrend (X, P)' removes the best fit of a
     polynomial of order P from the data X.

     If X is a matrix, `detrend (X, P)' does the same for each column
     in X.

     The second argument is optional.  If it is not specified, a value
     of 1 is assumed.  This corresponds to removing a linear trend.

 - Function:  fft (A, N)
     Compute the FFT of A using subroutines from FFTPACK.  If A is a
     matrix, `fft' computes the FFT for each column of A.

     If called with two arguments, N is expected to be an integer
     specifying the number of elements of A to use.  If A is a matrix,
     N specifies the number of rows of A to use.  If N is larger than
     the size of A, A is resized and padded with zeros.

 - Loadable Function:  ifft (A, N)
     Compute the inverse FFT of A using subroutines from FFTPACK.  If A
     is a matrix, `fft' computes the inverse FFT for each column of A.

     If called with two arguments, N is expected to be an integer
     specifying the number of elements of A to use.  If A is a matrix,
     N specifies the number of rows of A to use.  If N is larger than
     the size of A, A is resized and padded with zeros.

 - Loadable Function:  fft2 (A, N, M)
     Compute the two dimensional FFT of A.

     The optional arguments N and M may be used specify the number of
     rows and columns of A to use.  If either of these is larger than
     the size of A, A is resized and padded with zeros.

 - Loadable Function:  ifft2 (A, N, M)
     Compute the two dimensional inverse FFT of A.

     The optional arguments N and M may be used specify the number of
     rows and columns of A to use.  If either of these is larger than
     the size of A, A is resized and padded with zeros.

 - Built-in Function:  fftconv (A, B, N)
     Return the convolution of the vectors A and B, as a vector with
     length equal to the `length (a) + length (b) - 1'.  If A and B are
     the coefficient vectors of two polynomials, the returned value is
     the coefficient vector of the product polynomial.

     The computation uses the FFT by calling the function `fftfilt'.  If
     the optional argument N is specified, an N-point FFT is used.

 - Function File:  fftfilt (B, X, N)
     With two arguments, `fftfilt' filters X with the FIR filter B
     using the FFT.

     Given the optional third argument, N, `fftfilt' uses the
     overlap-add method to filter X with B using an N-point FFT.

 - Loadable Function: y = filter (B, A, X)
     Return the solution to the following linear, time-invariant
     difference equation:

             N                   M
            SUM a(k+1) y(n-k) = SUM b(k+1) x(n-k)      for 1<=n<=length(x)
            k=0                 k=0

     where  N=length(a)-1 and M=length(b)-1.  An equivalent form of
     this equation is:

                      N                   M
            y(n) = - SUM c(k+1) y(n-k) + SUM d(k+1) x(n-k)  for 1<=n<=length(x)
                     k=1                 k=0

     where  c = a/a(1) and d = b/a(1).

     In terms of the z-transform, y is the result of passing the
     discrete- time signal x through a system characterized by the
     following rational system function:

                       M
                      SUM d(k+1) z^(-k)
                      k=0
            H(z) = ----------------------
                         N
                    1 + SUM c(k+1) z(-k)
                        k=1

 - Loadable Function: [Y, SF] = filter (B, A, X, SI)
     This is the same as the `filter' function described above, except
     that SI is taken as the initial state of the system and the final
     state is returned as SF.  The state vector is a column vector
     whose length is equal to the length of the longest coefficient
     vector minus one.  If SI is not set, the initial state vector is
     set to all zeros.

 - Function File: [H, W] = freqz (B, A, N, "whole")
     Return the complex frequency response H of the rational IIR filter
     whose numerator and denominator coefficients are B and A,
     respectively.  The response is evaluated at N angular frequencies
     between 0 and  2*pi.

     The output value W is a vector of the frequencies.

     If the fourth argument is omitted, the response is evaluated at
     frequencies between 0 and  pi.

     If N is omitted, a value of 512 is assumed.

     If A is omitted, the denominator is assumed to be 1 (this
     corresponds to a simple FIR filter).

     For fastest computation, N should factor into a small number of
     small primes.

 - Function File:  sinc (X)
     Return  sin(pi*x)/(pi*x).


File: octave.info,  Node: Image Processing,  Next: Audio Processing,  Prev: Signal Processing,  Up: Top

Image Processing
****************

   Octave can display images with the X Window System using the
`xloadimage' program.  You do not need to be running X in order to
manipulate images, however, so some of these functions may be useful
even if you are not able to view the results.

   Loading images only works with Octave's image format (a file with a
matrix containing the image data, and a matrix containing the
colormap).  Contributions of robust, well-written functions to read
other image formats are welcome.  If you can provide them, or would like
to improve Octave's image processing capabilities in other ways, please
contact <bug-octave@bevo.che.wisc.edu>.

 - Function File:  colormap (MAP)
 - Function File:  colormap ("default")
     Set the current colormap.

     `colormap (MAP)' sets the current colormap to MAP.  The color map
     should be an N row by 3 column matrix.  The columns contain red,
     green, and blue intensities respectively.  All entries should be
     between 0 and 1 inclusive.  The new colormap is returned.

     `colormap ("default")' restores the default colormap (a gray scale
     colormap with 64 entries).  The default colormap is returned.

     With no arguments, `colormap' returns the current color map.

 - Function File:  gray (N)
     Return a gray colormap with N entries corresponding to values from
     0 to N.  The argument N should be a scalar.  If it is omitted, 64
     is assumed.

 - Function File: [IMG, MAP] = gray2ind ()
     Convert a gray scale intensity image to an Octave indexed image.

 - Function File:  image (X, ZOOM)
     Display a matrix as a color image.  The elements of X are indices
     into the current colormap and should have values between 1 and the
     length of the colormap.  If ZOOM is omitted, a value of 4 is
     assumed.

 - Function File:  imagesc (X, ZOOM)
     Display a scaled version of the matrix X as a color image.  The
     matrix is scaled so that its entries are indices into the current
     colormap.  The scaled matrix is returned.  If ZOOM is omitted, a
     value of 4 is assumed.

 - Function File:  imshow (X, MAP)
 - Function File:  imshow (X, N)
 - Function File:  imshow (I, N)
 - Function File:  imshow (R, G, B)
     Display images.

     `imshow (X)' displays an indexed image using the current colormap.

     `imshow (X, MAP)' displays an indexed image using the specified
     colormap.

     `imshow (I, N)' displays a gray scale intensity image.

     `imshow (R, G, B)' displays an RGB image.

 - Function File:  ind2gray (X, MAP)
     Convert an Octave indexed image to a gray scale intensity image.
     If MAP is omitted, the current colormap is used to determine the
     intensities.

 - Function File: [R, G, B] = ind2rgb (X, MAP)
     Convert an indexed image to red, green, and blue color components.
     If MAP is omitted, the current colormap is used for the conversion.

 - Function File: [X, MAP] = loadimage (FILE)
     Load an image file and it's associated color map from the specified
     FILE.  The image must be stored in Octave's image format.

 - Function File:  rgb2ntsc (RGB)
     Image format conversion.

 - Function File:  ntsc2rgb (YIQ)
     Image format conversion.

 - Function File:  ocean (N)
     Create color colormap.  The argument N should be a scalar.  If it
     is omitted, 64 is assumed.

 - Function File: [X, MAP] = rgb2ind (R, G, B)
     Convert and RGB image to an Octave indexed image.

 - Function File:  saveimage (FILE, X, FMT, MAP)
     Save the matrix X to FILE in image format FMT.  Valid values for
     FMT are

    `"img"'
          Octave's image format.  The current colormap is also saved in
          the file.

    `"ppm"'
          Portable pixmap format.

    `"ps"'
          PostScript format.  Note that images saved in PostScript
          format can not be read back into Octave with loadimage.

     If the fourth argument is supplied, the specified colormap will
     also be saved along with the image.

     Note: if the colormap contains only two entries and these entries
     are black and white, the bitmap ppm and PostScript formats are
     used.  If the image is a gray scale image (the entries within each
     row of the colormap are equal) the gray scale ppm and PostScript
     image formats are used, otherwise the full color formats are used.

 - Built-in Variable: IMAGEPATH
     A colon separated list of directories in which to search for image
     files.


File: octave.info,  Node: Audio Processing,  Next: System Utilities,  Prev: Image Processing,  Up: Top

Audio Processing
****************

   Octave provides a few functions for dealing with audio data.  An
audio `sample' is a single output value from an A/D converter, i.e., a
small integer number (usually 8 or 16 bits), and audio data is just a
series of such samples.  It can be characterized by three parameters:
the sampling rate (measured in samples per second or Hz, e.g. 8000 or
44100), the number of bits per sample (e.g. 8 or 16), and the number of
channels (1 for mono, 2 for stereo, etc.).

   There are many different formats for representing such data.
Currently, only the two most popular, *linear encoding* and *mu-law
encoding*, are supported by Octave.  There is an excellent FAQ on audio
formats by Guido van Rossum <guido@cwi.nl> which can be found at any
FAQ ftp site, in particular in the directory
`/pub/usenet/news.answers/audio-fmts' of the archive site
`rtfm.mit.edu'.

   Octave simply treats audio data as vectors of samples (non-mono data
are not supported yet).  It is assumed that audio files using linear
encoding have one of the extensions `lin' or `raw', and that files
holding data in mu-law encoding end in `au', `mu', or `snd'.

 - Function File:  lin2mu (X)
     If the vector X represents mono audio data in 8- or 16-bit linear
     encoding, `lin2mu (X)' is the corresponding mu-law encoding.

 - Function File:  mu2lin (X, BPS)
     If the vector X represents mono audio data in mu-law encoding,
     `mu2lin' converts it to linear encoding.  The optional argument
     BPS specifies whether the input data uses 8 bit per sample
     (default) or 16 bit.

 - Function File:  loadaudio (NAME, EXT, BPS)
     Loads audio data from the file `NAME.EXT' into the vector X.

     The extension EXT determines how the data in the audio file is
     interpreted;  the extensions `lin' (default) and `raw' correspond
     to linear, the extensions `au', `mu', or `snd' to mu-law encoding.

     The argument BPS can be either 8 (default) or 16, and specifies
     the number of bits per sample used in the audio file.

 - Function File:  saveaudio (NAME, X, EXT, BPS)
     Saves a vector X of audio data to the file `NAME.EXT'.  The
     optional parameters EXT and BPS determine the encoding and the
     number of bits per sample used in the audio file (see
     `loadaudio');  defaults are `lin' and 8, respectively.

   The following functions for audio I/O require special A/D hardware
and operating system support.  It is assumed that audio data in linear
encoding can be played and recorded by reading from and writing to
`/dev/dsp', and that similarly `/dev/audio' is used for mu-law
encoding.  These file names are system-dependent.  Improvements so that
these functions will work without modification on a wide variety of
hardware are welcome.

 - Function File:  playaudio (NAME, EXT)
 - Function File:  playaudio (X)
     Plays the audio file `NAME.EXT' or the audio data stored in the
     vector X.

 - Function File:  record (SEC, SAMPLING_RATE)
     Records SEC seconds of audio input into the vector X.  The default
     value for SAMPLING_RATE is 8000 samples per second, or 8kHz.  The
     program waits until the user types <RET> and then immediately
     starts to record.

 - Function File:  setaudio (TYPE)
 - Function File:  setaudio (TYPE, VALUE)
     Set or display various properties of your mixer hardware.

     For example, if `vol' corresponds to the volume property, you can
     set it to 50 (percent) by `setaudio ("vol", 50)'.

     This is an simple experimental program to control the audio
     hardware settings.  It assumes that there is a `mixer' program
     which can be used as `mixer TYPE VALUE', and simply executes
     `system ("mixer TYPE VALUE")'.  Future releases might get rid of
     this assumption by using the `fcntl' interface.


File: octave.info,  Node: System Utilities,  Next: Tips,  Prev: Audio Processing,  Up: Top

System Utilities
****************

   This chapter describes the functions that are available to allow you
to get information about what is happening outside of Octave, while it
is still running, and use this information in your program.  For
example, you can get information about environment variables, the
current time, and even start other programs from the Octave prompt.

* Menu:

* Timing Utilities::
* Filesystem Utilities::
* Controlling Subprocesses::
* Process ID Information::
* Environment Variables::
* Current Working Directory::
* Password Database Functions::
* Group Database Functions::
* System Information::


File: octave.info,  Node: Timing Utilities,  Next: Filesystem Utilities,  Prev: System Utilities,  Up: System Utilities

Timing Utilities
================

   Octave's core set of functions for manipulating time values are
patterned after the corresponding functions from the standard C library.
Several of these functions use a data structure for time that includes
the following elements:

`usec'
     Microseconds after the second (0-999999).

`sec'
     Seconds after the minute (0-61).  This number can be 61 to account
     for leap seconds.

`min'
     Minutes after the hour (0-59).

`hour'
     Hours since midnight (0-23).

`mday'
     Day of the month (1-31).

`mon'
     Months since January (0-11).

`year'
     Years since 1900.

`wday'
     Days since Sunday (0-6).

`yday'
     Days since January 1 (0-365).

`isdst'
     Daylight Savings Time flag.

`zone'
     Time zone.

In the descriptions of the following functions, this structure is
referred to as a TM_STRUCT.

 - Loadable Function:  time ()
     Return the current time as the number of seconds since the epoch.
     The epoch is referenced to 00:00:00 CUT (Coordinated Universal
     Time) 1 Jan 1970.  For example, on Monday February 17, 1997 at
     07:15:06 CUT, the value returned by `time' was 856163706.

 - Function File:  ctime (T)
     Convert a value returned from `time' (or any other nonnegative
     integer), to the local time and return a string of the same form as
     `asctime'.  The function `ctime (time)' is equivalent to `asctime
     (localtime (time))'.  For example,

          ctime (time ())
               => "Mon Feb 17 01:15:06 1997"

 - Loadable Function:  gmtime (T)
     Given a value returned from time (or any nonnegative integer),
     return a time structure corresponding to CUT.  For example,

          gmtime (time ())
               => {
                     usec = 0
                     year = 97
                     mon = 1
                     mday = 17
                     sec = 6
                     zone = CST
                     min = 15
                     wday = 1
                     hour = 7
                     isdst = 0
                     yday = 47
                   }

 - Loadable Function:  localtime (T)
     Given a value returned from time (or any nonnegative integer),
     return a time structure corresponding to the local time zone.

          localtime (time ())
               => {
                     usec = 0
                     year = 97
                     mon = 1
                     mday = 17
                     sec = 6
                     zone = CST
                     min = 15
                     wday = 1
                     hour = 1
                     isdst = 0
                     yday = 47
                   }

 - Loadable Function:  mktime (TM_STRUCT)
     Convert a time structure corresponding to the local time to the
     number of seconds since the epoch.  For example,

          mktime (localtime (time ())
               => 856163706

 - Function File:  asctime (TM_STRUCT)
     Convert a time structure to a string using the following five-field
     format: Thu Mar 28 08:40:14 1996.  For example,

          asctime (localtime (time ())
               => "Mon Feb 17 01:15:06 1997\n"

     This is equivalent to `ctime (time ())'.

 - Loadable Function:  strftime (TM_STRUCT)
     Format a time structure in a flexible way using `%' substitutions
     similar to those in `printf'.  Except where noted, substituted
     fields have a fixed size; numeric fields are padded if necessary.
     Padding is with zeros by default; for fields that display a single
     number, padding can be changed or inhibited by following the `%'
     with one of the modifiers described below.  Unknown field
     specifiers are copied as normal characters.  All other characters
     are copied to the output without change.  For example,

          strftime ("%r (%Z) %A %e %B %Y", localtime (time ())
               => "01:15:06 AM (CST) Monday 17 February 1997"

     Octave's `strftime' function supports a superset of the ANSI C
     field specifiers.

     Literal character fields:

    `%'
          % character.

    `n'
          Newline character.

    `t'
          Tab character.

     Numeric modifiers (a nonstandard extension):

    `- (dash)'
          Do not pad the field.

    `_ (underscore)'
          Pad the field with spaces.

     Time fields:

    `%H'
          Hour (00-23).

    `%I'
          Hour (01-12).

    `%k'
          Hour (0-23).

    `%l'
          Hour (1-12).

    `%M'
          Minute (00-59).

    `%p'
          Locale's AM or PM.

    `%r'
          Time, 12-hour (hh:mm:ss [AP]M).

    `%R'
          Time, 24-hour (hh:mm).

    `%s'
          Time in seconds since 00:00:00, Jan 1, 1970 (a nonstandard
          extension).

    `%S'
          Second (00-61).

    `%T'
          Time, 24-hour (hh:mm:ss).

    `%X'
          Locale's time representation (%H:%M:%S).

    `%Z'
          Time zone (EDT), or nothing if no time zone is determinable.

     Date fields:

    `%a'
          Locale's abbreviated weekday name (Sun-Sat).

    `%A'
          Locale's full weekday name, variable length (Sunday-Saturday).

    `%b'
          Locale's abbreviated month name (Jan-Dec).

    `%B'
          Locale's full month name, variable length (January-December).

    `%c'
          Locale's date and time (Sat Nov 04 12:02:33 EST 1989).

    `%C'
          Century (00-99).

    `%d'
          Day of month (01-31).

    `%e'
          Day of month ( 1-31).

    `%D'
          Date (mm/dd/yy).

    `%h'
          Same as %b.

    `%j'
          Day of year (001-366).

    `%m'
          Month (01-12).

    `%U'
          Week number of year with Sunday as first day of week (00-53).

    `%w'
          Day of week (0-6).

    `%W'
          Week number of year with Monday as first day of week (00-53).

    `%x'
          Locale's date representation (mm/dd/yy).

    `%y'
          Last two digits of year (00-99).

    `%Y'
          Year (1970-).

   Most of the remaining functions described in this section are not
patterned after the standard C library.  Some are available for
compatiblity with MATLAB and others are provided because they are
useful.

 - Function File:  clock ()
     Return a vector containing the current year, month (1-12), day
     (1-31), hour (0-23), minute (0-59) and second (0-61).  For example,

          clock ()
               => [ 1993, 8, 20, 4, 56, 1 ]

     The function clock is more accurate on systems that have the
     `gettimeofday' function.

 - Function File:  date ()
     Return the date as a character string in the form DD-MMM-YY.  For
     example,

          date ()
               => "20-Aug-93"

 - Function File:  etime (T1, T2)
     Return the difference (in seconds) between two time values
     returned from `clock'.  For example:

          t0 = clock ();
          # many computations later...
          elapsed_time = etime (clock (), t0);

     will set the variable `elapsed_time' to the number of seconds since
     the variable `t0' was set.

 - Built-in Function: [TOTAL, USER, SYSTEM] = cputime ();
     Return the CPU time used by your Octave session.  The first output
     is the total time spent executing your process and is equal to the
     sum of second and third outputs, which are the number of CPU
     seconds spent executing in user mode and the number of CPU seconds
     spent executing in system mode, respectively.  If your system does
     not have a way to report CPU time usage, `cputime' returns 0 for
     each of its output values.  Note that because Octave used some CPU
     time to start, it is reasonable to check to see if `cputime' works
     by checking to see if the total CPU time used is nonzero.

 - Function File:  is_leap_year (YEAR)
     Return 1 if the given year is a leap year and 0 otherwise.  If no
     arguments are provided, `is_leap_year' will use the current year.
     For example,

          is_leap_year (2000)
               => 1

 - Function File:  tic ()
 - Function File:  toc ()
     These functions set and check a wall-clock timer.  For example,

          tic ();
          # many computations later...
          elapsed_time = toc ();

     will set the variable `elapsed_time' to the number of seconds since
     the most recent call to the function `tic'.

     If you are more interested in the CPU time that your process used,
     you should use the `cputime' function instead.  The `tic' and
     `toc' functions report the actual wall clock time that elapsed
     between the calls.  This may include time spent processing other
     jobs or doing nothing at all.  For example,

          tic (); sleep (5); toc ()
               => 5
          t = cputime (); sleep (5); cputime () - t
               => 0

     (This example also illustrates that the CPU timer may have a fairly
     coarse resolution.)

 - Built-in Function:  pause (SECONDS)
     Suspend the execution of the program.  If invoked without any
     arguments, Octave waits until you type a character.  With a
     numeric argument, it pauses for the given number of seconds.  For
     example, the following statement prints a message and then waits 5
     seconds before clearing the screen.

          fprintf (stderr, "wait please...\n");
          pause (5);
          clc;

 - Built-in Function:  sleep (SECONDS)
     Suspend the execution of the program for the given number of
     seconds.

 - Built-in Function:  usleep (MICROSECONDS)
     Suspend the execution of the program for the given number of
     microseconds.  On systems where it is not possible to sleep for
     periods of time less than one second, `usleep' will pause the
     execution for `round (MICROSECONDS / 1e6)' seconds.

