Plotting¶

Plot2D(expr[, x0:x1][, option=value[, ...]])¶

Plot2D({expr1, expr2, ..., exprn}[, x0:x1][, option=value[, ...]])

adaptive two-dimensional plotting

Param f(x):	unevaluated expression containing one variables (function to be plotted)
Param list:	list of functions to plot
Param a}, {b:	numbers, plotting range in the \(x\) coordinate
Param option:	atom, option name
Param value:	atom, number or string (value of option)

The routine Plot2D() performs adaptive plotting of one or several functions of one variable in the specified range. The result is presented as a line given by the equation \(y=f(x)\). Several functions can be plotted at once. Various plotting options can be specified. Output can be directed to a plotting program (the default is to use {data}) to a list of values. The function parameter {f(x)} must evaluate to a yacas expression containing at most one variable. (The variable does not have to be called {x}.) Also, {N(f(x))} must evaluate to a real (not complex) numerical value when given a numerical value of the argument {x}. If the function {f(x)} does not satisfy these requirements, an error is raised. Several functions may be specified as a list and they do not have to depend on the same variable, for example, {{f(x), g(y)}}. The functions will be plotted on the same graph using the same coordinate ranges. If you have defined a function which accepts a number but does not accept an undefined variable, {Plot2D} will fail to plot it. Use {NFunction} to overcome this difficulty. Data files are created in a temporary directory {/tmp/plot.tmp/} unless otherwise requested. File names and other information is printed if {InVerboseMode()} returns True on using {V()}. The current algorithm uses Newton-Cotes quadratures and some heuristics for error estimation (see <yacasdoc://Algo/3/1/>). The initial grid of {points+1} points is refined between any grid points \(x0\), \(x1\) if the integral \(Integrate(x,a,b)f(x)\) is not approximated to the given precision by the existing grid. Default plotting range is {-5:5}. Range can also be specified as {x= -5:5} (note the mandatory space separating “{=}” and “{-}”); currently the variable name {x} is ignored in this case. Options are of the form {option=value}. Currently supported option names are: “points”, “precision”, “depth”, “output”, “filename”, “yrange”. Option values are either numbers or special unevaluated atoms such as {data}. If you need to use the names of these atoms in your script, strings can be used. Several option/value pairs may be specified (the function {Plot2D} has a variable number of arguments).

{yrange}: the range of ordinates to use for plotting, e.g. {yrange=0:20}. If no range is specified, the default is usually to leave the choice to the plotting backend.
{points}: initial number of points (default 23) – at least that many points will be plotted. The initial grid of this many points will be adaptively refined.
{precision}: graphing precision (default \(10^(-6)\)). This is interpreted as the relative precision of computing the integral of \(f(x)-Min(f(x))\) using the grid points. For a smooth, non-oscillating function this value should be roughly 1/(number of screen pixels in the plot).
{depth}: max. refinement depth, logarithmic (default 5) – means there will be at most \(2^depth\) extra points per initial grid point.
{output}: name of the plotting backend. Supported names: {data} (default). The {data} backend will return the data as a list of pairs such as {{{x1,y1}, {x2,y2}, …}}.
{filename}: specify name of the created data file. For example: {filename=”data1.txt”}. The default is the name {“output.data”}. Note that if several functions are plotted, the data files will have a number appended to the given name, for example {data.txt1}, {data.txt2}.

Other options may be supported in the future.

The current implementation can deal with a singularity within the plotting range only if the function {f(x)} returns {Infinity}, {-Infinity} or {Undefined} at the singularity. If the function {f(x)} generates a numerical error and fails at a singularity, {Plot2D} will fail if one of the grid points falls on thez singularity. (All grid points are generated by bisection so in principle the endpoints and the {points} parameter could be chosen to avoid numerical singularities.)

See also

V(), NFunction(), Plot3DS()

Plot3DS(expr[, option=value[, ...]])¶

Plot3DS({expr1, expr2, ..., exprn}[, option=value[, ...]])

three-dimensional (surface) plotting

The routine Plot3DS() performs adaptive plotting of a function of two variables in the specified ranges. The result is presented as a surface given by the equation \(z=f(x,y)\). Several functions can be plotted at once, by giving a list of functions. Various plotting options can be specified. Output can be directed to a plotting program (the default is to use {data}), to a list of values. The function parameter {f(x,y)} must evaluate to a Yacas expression containing at most two variables. (The variables do not have to be called {x} and {y}.) Also, {N(f(x,y))} must evaluate to a real (not complex) numerical value when given numerical values of the arguments {x}, {y}. If the function {f(x,y)} does not satisfy these requirements, an error is raised. Several functions may be specified as a list but they have to depend on the same symbolic variables, for example, {{f(x,y), g(y,x)}}, but not {{f(x,y), g(a,b)}}. The functions will be plotted on the same graph using the same coordinate ranges. If you have defined a function which accepts a number but does not accept an undefined variable, {Plot3DS} will fail to plot it. Use {NFunction} to overcome this difficulty. Data files are created in a temporary directory {/tmp/plot.tmp/} unless otherwise requested. File names and other information is printed if {InVerboseMode()} returns True on using {V()}. The current algorithm uses Newton-Cotes cubatures and some heuristics for error estimation (see <yacasdoc://Algo/3/1/>). The initial rectangular grid of {xpoints+1}*{ypoints+1} points is refined within any rectangle where the integral of \(f(x,y)\) is not approximated to the given precision by the existing grid. Default plotting range is {-5:5} in both coordinates. A range can also be specified with a variable name, e.g. {x= -5:5} (note the mandatory space separating “{=}” and “{-}”). The variable name {x} should be the same as that used in the function {f(x,y)}. If ranges are not given with variable names, the first variable encountered in the function {f(x,y)} is associated with the first of the two ranges. Options are of the form {option=value}. Currently supported option names are “points”, “xpoints”, “ypoints”, “precision”, “depth”, “output”, “filename”, “xrange”, “yrange”, “zrange”. Option values are either numbers or special unevaluated atoms such as {data}. If you need to use the names of these atoms in your script, strings can be used (e.g. {output=”data”}). Several option/value pairs may be specified (the function {Plot3DS} has a variable number of arguments).

{xrange}, {yrange}: optionally override coordinate ranges. Note that {xrange} is always the first variable and {yrange} the second variable, regardless of the actual variable names.
{zrange}: the range of the \(z\) axis to use for plotting, e.g. {zrange=0:20}. If no range is specified, the default is usually to leave the choice to the plotting backend. Automatic choice based on actual values may give visually inadequate plots if the function has a singularity.
{points}, {xpoints}, {ypoints}: initial number of points (default 10 each) – at least that many points will be plotted in each coordinate. The initial grid of this many points will be adaptively refined. If {points} is specified, it serves as a default for both {xpoints} and {ypoints}; this value may be overridden by {xpoints} and {ypoints} values.
{precision}: graphing precision (default \(0.01\)). This is interpreted as the relative precision of computing the integral of \(f(x,y)-Min(f(x,y))\) using the grid points. For a smooth, non-oscillating function this value should be roughly 1/(number of screen pixels in the plot).
{depth}: max. refinement depth, logarithmic (default 3) – means there will be at most \(2^depth\) extra points per initial grid point (in each coordinate).
{output}: name of the plotting backend. Supported names: {data} (default). The {data} backend will return the data as a list of triples such as {{{x1, y1, z1}, {x2, y2, z2}, …}}.

Other options may be supported in the future.

The current implementation can deal with a singularity within the plotting range only if the function {f(x,y)} returns {Infinity}, {-Infinity} or {Undefined} at the singularity. If the function {f(x,y)} generates a numerical error and fails at a singularity, {Plot3DS} will fail only if one of the grid points falls on the singularity. (All grid points are generated by bisection so in principle the endpoints and the {xpoints}, {ypoints} parameters could be chosen to avoid numerical singularities.)

The {filename} option is optional if using graphical backends, but can be used to specify the location of the created data file.

Example:

In> Plot3DS(a*b^2)
Out> True;
In> V(Plot3DS(Sin(x)*Cos(y),x=0:20, y=0:20,depth=3))
CachedConstant: Info: constant Pi is being
recalculated at precision 10
CachedConstant: Info: constant Pi is being
recalculated at precision 11
Plot3DS: using 1699  points for function Sin(x)*Cos(y)
Plot3DS: max. used 8 subdivisions for Sin(x)*Cos(y)
Plot3DS'datafile: created file '/tmp/plot.tmp/data1'
Out> True;