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Nomenclature

The following will be used throughout this course

eps_step
If tex:$$x_k$$ and tex:$$x_{k + 1}$$ represent two successive approximations of a value tex:$$x$$, one condition for halting is that tex:$$\left | x_k - x_{k + 1} \right | < \epsilon_{\rm step}$$.
eps_abs
If tex:$$x_k$$ and tex:$$x_{k + 1}$$ represent two successive approximations of a value tex:$$x$$, one condition for halting is that tex:$$\left | f\left(x_k\right) - f\left(x_{k + 1}\right) \right | < \epsilon_{\rm abs}$$.
f
A function used in an algorithm, either for root finding (tex:$$f(x) = 0$$) or an initial value-problem (tex:$$y'(t) = f(t, y(t)))$$).
g
A forcing function of a 2nd-order ordinary differnetial equation.
i
An index over a single space variable, usually tex:$$x$$.
ix, iy, iz
Indices over two or three space variables, tex:$$x$$, tex:$$y$$, and tex:$$z$$, respectively.
k
An index iterating over a time variable.
M
A matrix.
m
Either the row-dimension of a matrix M (an m × n matrix) or an index of a vector.
n
Either the column-dimension of a matrix M (an m × n matrix) or a positive integer indicating the number of points into which an interval will be divided.
N_max
A maximum number of steps or iterations.
t_out
An output column vector indicating the partition of a time interval tex:$${\bf t}_{\rm rng} = [t_0, t_f]$$.
t_rng
A time interval where tex:$${\bf t}_{\rm rng} = [t_0, t_f]$$, that is, from an initial time tex:$$t_0$$ to a final time tex:$$t_f$$.
x_rng
A space interval over a variable tex:$$x$$ where tex:$${\bf x}_{\rm rng} = [a, b]$$.
u0
The initial values (a vector) of tex:$$u$$-values where usually U_out(1,:) = u0 for a solution over time and a single space variable tex:$$x$$.
U0
The initial values (a 2- or 3-dimensional array) of tex:$$u$$-values where usually U_out(1,:,:) = U0 or U_out(1,:,:,:) = U0 for a solution over time and two or three space variables tex:$$x$$, tex:$$y$$, and tex:$$z$$.
u_out
An output column vector of tex:$$y$$-values where usually y_out(1,:) = y0 for a solution over time.
U_out
An output column vector of tex:$$y$$-values where usually y_out(1,:) = y0 for a solution over time.
y0
An initial value (either a scalar or a column vector) of tex:$$y$$-values where usually y_out(1,:) = y0 for a solution over time.
y_out
An output column vector of tex:$$y$$-values where usually y_out(1,:) = y0 for a solution over time.