'\"
'\" Generated from file 'quasirandom\&.man' by tcllib/doctools with format 'nroff'
'\"
.TH "math::quasirandom" 3tcl 1 tcllib "Tcl Math Library"
.\" The -*- nroff -*- definitions below are for supplemental macros used
.\" in Tcl/Tk manual entries.
.\"
.\" .AP type name in/out ?indent?
.\"	Start paragraph describing an argument to a library procedure.
.\"	type is type of argument (int, etc.), in/out is either "in", "out",
.\"	or "in/out" to describe whether procedure reads or modifies arg,
.\"	and indent is equivalent to second arg of .IP (shouldn't ever be
.\"	needed;  use .AS below instead)
.\"
.\" .AS ?type? ?name?
.\"	Give maximum sizes of arguments for setting tab stops.  Type and
.\"	name are examples of largest possible arguments that will be passed
.\"	to .AP later.  If args are omitted, default tab stops are used.
.\"
.\" .BS
.\"	Start box enclosure.  From here until next .BE, everything will be
.\"	enclosed in one large box.
.\"
.\" .BE
.\"	End of box enclosure.
.\"
.\" .CS
.\"	Begin code excerpt.
.\"
.\" .CE
.\"	End code excerpt.
.\"
.\" .VS ?version? ?br?
.\"	Begin vertical sidebar, for use in marking newly-changed parts
.\"	of man pages.  The first argument is ignored and used for recording
.\"	the version when the .VS was added, so that the sidebars can be
.\"	found and removed when they reach a certain age.  If another argument
.\"	is present, then a line break is forced before starting the sidebar.
.\"
.\" .VE
.\"	End of vertical sidebar.
.\"
.\" .DS
.\"	Begin an indented unfilled display.
.\"
.\" .DE
.\"	End of indented unfilled display.
.\"
.\" .SO ?manpage?
.\"	Start of list of standard options for a Tk widget. The manpage
.\"	argument defines where to look up the standard options; if
.\"	omitted, defaults to "options". The options follow on successive
.\"	lines, in three columns separated by tabs.
.\"
.\" .SE
.\"	End of list of standard options for a Tk widget.
.\"
.\" .OP cmdName dbName dbClass
.\"	Start of description of a specific option.  cmdName gives the
.\"	option's name as specified in the class command, dbName gives
.\"	the option's name in the option database, and dbClass gives
.\"	the option's class in the option database.
.\"
.\" .UL arg1 arg2
.\"	Print arg1 underlined, then print arg2 normally.
.\"
.\" .QW arg1 ?arg2?
.\"	Print arg1 in quotes, then arg2 normally (for trailing punctuation).
.\"
.\" .PQ arg1 ?arg2?
.\"	Print an open parenthesis, arg1 in quotes, then arg2 normally
.\"	(for trailing punctuation) and then a closing parenthesis.
.\"
.\"	# Set up traps and other miscellaneous stuff for Tcl/Tk man pages.
.if t .wh -1.3i ^B
.nr ^l \n(.l
.ad b
.\"	# Start an argument description
.de AP
.ie !"\\$4"" .TP \\$4
.el \{\
.   ie !"\\$2"" .TP \\n()Cu
.   el          .TP 15
.\}
.ta \\n()Au \\n()Bu
.ie !"\\$3"" \{\
\&\\$1 \\fI\\$2\\fP (\\$3)
.\".b
.\}
.el \{\
.br
.ie !"\\$2"" \{\
\&\\$1	\\fI\\$2\\fP
.\}
.el \{\
\&\\fI\\$1\\fP
.\}
.\}
..
.\"	# define tabbing values for .AP
.de AS
.nr )A 10n
.if !"\\$1"" .nr )A \\w'\\$1'u+3n
.nr )B \\n()Au+15n
.\"
.if !"\\$2"" .nr )B \\w'\\$2'u+\\n()Au+3n
.nr )C \\n()Bu+\\w'(in/out)'u+2n
..
.AS Tcl_Interp Tcl_CreateInterp in/out
.\"	# BS - start boxed text
.\"	# ^y = starting y location
.\"	# ^b = 1
.de BS
.br
.mk ^y
.nr ^b 1u
.if n .nf
.if n .ti 0
.if n \l'\\n(.lu\(ul'
.if n .fi
..
.\"	# BE - end boxed text (draw box now)
.de BE
.nf
.ti 0
.mk ^t
.ie n \l'\\n(^lu\(ul'
.el \{\
.\"	Draw four-sided box normally, but don't draw top of
.\"	box if the box started on an earlier page.
.ie !\\n(^b-1 \{\
\h'-1.5n'\L'|\\n(^yu-1v'\l'\\n(^lu+3n\(ul'\L'\\n(^tu+1v-\\n(^yu'\l'|0u-1.5n\(ul'
.\}
.el \}\
\h'-1.5n'\L'|\\n(^yu-1v'\h'\\n(^lu+3n'\L'\\n(^tu+1v-\\n(^yu'\l'|0u-1.5n\(ul'
.\}
.\}
.fi
.br
.nr ^b 0
..
.\"	# VS - start vertical sidebar
.\"	# ^Y = starting y location
.\"	# ^v = 1 (for troff;  for nroff this doesn't matter)
.de VS
.if !"\\$2"" .br
.mk ^Y
.ie n 'mc \s12\(br\s0
.el .nr ^v 1u
..
.\"	# VE - end of vertical sidebar
.de VE
.ie n 'mc
.el \{\
.ev 2
.nf
.ti 0
.mk ^t
\h'|\\n(^lu+3n'\L'|\\n(^Yu-1v\(bv'\v'\\n(^tu+1v-\\n(^Yu'\h'-|\\n(^lu+3n'
.sp
-1
.fi
.ev
.\}
.nr ^v 0
..
.\"	# Special macro to handle page bottom:  finish off current
.\"	# box/sidebar if in box/sidebar mode, then invoked standard
.\"	# page bottom macro.
.de ^B
.ev 2
'ti 0
'nf
.mk ^t
.if \\n(^b \{\
.\"	Draw three-sided box if this is the box's first page,
.\"	draw two sides but no top otherwise.
.ie !\\n(^b-1 \h'-1.5n'\L'|\\n(^yu-1v'\l'\\n(^lu+3n\(ul'\L'\\n(^tu+1v-\\n(^yu'\h'|0u'\c
.el \h'-1.5n'\L'|\\n(^yu-1v'\h'\\n(^lu+3n'\L'\\n(^tu+1v-\\n(^yu'\h'|0u'\c
.\}
.if \\n(^v \{\
.nr ^x \\n(^tu+1v-\\n(^Yu
\kx\h'-\\nxu'\h'|\\n(^lu+3n'\ky\L'-\\n(^xu'\v'\\n(^xu'\h'|0u'\c
.\}
.bp
'fi
.ev
.if \\n(^b \{\
.mk ^y
.nr ^b 2
.\}
.if \\n(^v \{\
.mk ^Y
.\}
..
.\"	# DS - begin display
.de DS
.RS
.nf
.sp
..
.\"	# DE - end display
.de DE
.fi
.RE
.sp
..
.\"	# SO - start of list of standard options
.de SO
'ie '\\$1'' .ds So \\fBoptions\\fR
'el .ds So \\fB\\$1\\fR
.SH "STANDARD OPTIONS"
.LP
.nf
.ta 5.5c 11c
.ft B
..
.\"	# SE - end of list of standard options
.de SE
.fi
.ft R
.LP
See the \\*(So manual entry for details on the standard options.
..
.\"	# OP - start of full description for a single option
.de OP
.LP
.nf
.ta 4c
Command-Line Name:	\\fB\\$1\\fR
Database Name:	\\fB\\$2\\fR
Database Class:	\\fB\\$3\\fR
.fi
.IP
..
.\"	# CS - begin code excerpt
.de CS
.RS
.nf
.ta .25i .5i .75i 1i
..
.\"	# CE - end code excerpt
.de CE
.fi
.RE
..
.\"	# UL - underline word
.de UL
\\$1\l'|0\(ul'\\$2
..
.\"	# QW - apply quotation marks to word
.de QW
.ie '\\*(lq'"' ``\\$1''\\$2
.\"" fix emacs highlighting
.el \\*(lq\\$1\\*(rq\\$2
..
.\"	# PQ - apply parens and quotation marks to word
.de PQ
.ie '\\*(lq'"' (``\\$1''\\$2)\\$3
.\"" fix emacs highlighting
.el (\\*(lq\\$1\\*(rq\\$2)\\$3
..
.\"	# QR - quoted range
.de QR
.ie '\\*(lq'"' ``\\$1''\\-``\\$2''\\$3
.\"" fix emacs highlighting
.el \\*(lq\\$1\\*(rq\\-\\*(lq\\$2\\*(rq\\$3
..
.\"	# MT - "empty" string
.de MT
.QW ""
..
.BS
.SH NAME
math::quasirandom \- Quasi-random points for integration and Monte Carlo type methods
.SH SYNOPSIS
package require \fBTcl  8\&.5\fR
.sp
package require \fBTclOO \fR
.sp
package require \fBmath::quasirandom  1\fR
.sp
\fB::math::quasirandom::qrpoint create\fR \fINAME\fR \fIDIM\fR ?ARGS?
.sp
\fBgen next\fR
.sp
\fBgen set-start\fR \fIindex\fR
.sp
\fBgen set-evaluations\fR \fInumber\fR
.sp
\fBgen integral\fR \fIfunc\fR \fIminmax\fR \fIargs\fR
.sp
.BE
.SH DESCRIPTION
.PP
In many applications pseudo-random numbers and pseudo-random points in a (limited)
sample space play an important role\&. For instance in any type of Monte Carlo simulation\&.
Pseudo-random numbers, however, may be too random and as a consequence a large
number of data points is required to reduce the error or fluctuation in the results
to the desired value\&.
.PP
Quasi-random numbers can be used as an alternative: instead of "completely" arbitrary
points, points are generated that are diverse enough to cover the entire sample space
in a more or less uniform way\&. As a consequence convergence to the limit can be
much faster, when such quasi-random numbers are well-chosen\&.
.PP
The package defines a \fIclass\fR "qrpoint" that creates a command to generate
quasi-random points in 1, 2 or more dimensions\&. The command can either generate
separate points, so that they can be used in a user-defined algorithm or use these
points to calculate integrals of functions defined over 1, 2 or more dimensions\&.
It also holds several other common algorithms\&. (NOTE: these are not implemented yet)
.PP
One particular characteristic of the generators is that there are no tuning parameters
involved, which makes the use particularly simple\&.
.SH COMMANDS
A quasi-random point generator is created using the \fIqrpoint\fR class:
.TP
\fB::math::quasirandom::qrpoint create\fR \fINAME\fR \fIDIM\fR ?ARGS?
This command takes the following arguments:
.RS
.TP
string \fINAME\fR
The name of the command to be created (alternatively: the \fInew\fR subcommand
will generate a unique name)
.TP
integer/string \fIDIM\fR
The number of dimensions or one of: "circle", "disk", "sphere" or "ball"
.TP
strings \fIARGS\fR
Zero or more key-value pairs\&. The supported options are:
.RS
.IP \(bu
\fI-start index\fR: The index for the next point to be generated (default: 1)
.IP \(bu
\fI-evaluations number\fR: The number of evaluations to be used by default (default: 100)
.RE
.RE
.PP
The points that are returned lie in the hyperblock [0,1[^n (n the number of dimensions)
or on the unit circle, within the unit disk, on the unit sphere or within the unit ball\&.
.PP
Each generator supports the following subcommands:
.TP
\fBgen next\fR
Return the coordinates of the next quasi-random point
.sp
.TP
\fBgen set-start\fR \fIindex\fR
Reset the index for the next quasi-random point\&. This is useful to control which list of points is returned\&.
Returns the new or the current value, if no value is given\&.
.sp
.TP
\fBgen set-evaluations\fR \fInumber\fR
Reset the default number of evaluations in compound algorithms\&. Note that the actual number is the
smallest 4-fold larger or equal to the given number\&. (The 4-fold plays a role in the detailed integration
routine\&.)
.sp
.TP
\fBgen integral\fR \fIfunc\fR \fIminmax\fR \fIargs\fR
Calculate the integral of the given function over the block (or the circle, sphere etc\&.)
.RS
.TP
string \fIfunc\fR
The name of the function to be integrated
.TP
list \fIminmax\fR
List of pairs of minimum and maximum coordinates\&. This can be used to
map the quasi-random coordinates to the desired hyper-block\&.
.sp
If the space is a circle, disk etc\&. then this argument should be a single value, the radius\&.
The circle, disk, etc\&. is centred at the origin\&. If this is not what is required, then a coordinate
transformation should be made within the function\&.
.TP
strings \fIargs\fR
Zero or more key-value pairs\&. The following options are supported:
.RS
.IP \(bu
\fI-evaluations number\fR: The number of evaluations to be used\&. If not specified use the
default of the generator object\&.
.RE
.RE
.PP
.SH TODO
Implement other algorithms and variants
.PP
Implement more unit tests\&.
.PP
Comparison to pseudo-random numbers for integration\&.
.SH REFERENCES
Various algorithms exist for generating quasi-random numbers\&. The generators created in this package are based on:
\fIhttp://extremelearning\&.com\&.au/unreasonable-effectiveness-of-quasirandom-sequences/\fR
.SH KEYWORDS
mathematics, quasi-random
.SH CATEGORY
Mathematics