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<Chapter Label="intro">
<Heading>Introduction</Heading>

<Section Label="philosophy">
<Heading>Philosophy</Heading>

<Package>Forms</Package> is a package for computing with
sesquilinear and quadratic forms on finite vector spaces.
It provides users with the basic algebraic tools to
work with classical groups and polar geometries,
and enables one to specify a form and its corresponding
geometry. The functionality of the package includes:
<List>
<Item>the construction of sesquilinear and quadratic forms;</Item>
<Item>operations which allow a user to change coordinates, that is,
to ``change form'' and work in an isometric (or similar)
formed vector space; and</Item>
<Item>a way to determine the form(s) left invariant by a matrix group (up to a scalar).</Item>
</List>

</Section>

<Section Label="overview">
<Heading>Overview over this manual</Heading>

The next chapter (<Ref Chap="examples"/>) gives some basic examples of
the use of this package. In "Background Theory of Forms" (Chapter
<Ref Chap="theory"/>) we revise the basic notions of the theory of
sesquilinear and quadratic forms, where we also set the notation
and conventions adopted by this package. In "Constructing forms and basic
functionality" (Chapter ), we describe all operations
to construct sesquilinear and quadratic forms and basic attributes and
properties that do not require morphisms. In "Morphims of forms" (Chapter
<Ref Chap="morphisms"/>) we revise the basic notions of morphisms of forms, and
the classification of sesquilinear and quadratic forms on vector spaces over
finite fields. Operations, attributes and properties that are related to the
computation of morphisms of forms, are also described in this chapter.
</Section>

<Section Label="howto">
<Heading>How to read this manual</Heading>

We have tried to make this manual pleasant to read for the general
reader. So it is inevitable that we will use Greek symbols and simple
mathematical formulas. To make these visible in the HTML version of this
documentation, you may have to change the default character set
of your browser to UTF-8.
</Section>

<Section Label="web">
<Heading>Web resources</Heading>
<List>
<Item>Find <Package>Forms</Package> on the Packages section of the GAP-website: <Homepage>https://www.gap-system.org/Packages/forms.html</Homepage>.</Item>
<Item>Find <Package>Forms</Package> on its homepage: <Homepage>https://gap-packages.github.io/forms</Homepage>.</Item>
<Item>Report bugs, questions and issues on the <Package>Forms</Package> issue tracker: <Homepage>https://github.com/gap-packages/forms/issues</Homepage></Item>
</List>
</Section>

<Section Label="release_notes">
<Heading>Release notes</Heading>

Version 1.2.1 of <Package>Forms</Package> contains some changed and extra
functionality with relation to trivial forms. The changed and new functionality
is described completely in Section <Ref Sect="trivialform"/>. We gratefully
acknowledge the useful feedback of Alice Niemeyer.
<P/>
In version 1.2.2 of <Package>Forms</Package> a minor bug, pointed out by John Bamberg,
in the code of <F>IsTotallyIsotropicSubspace</F> is repaired. On the occasion of the release
of the first beta versions of GAP4r5, we changed the names of some global functions such that
a name clash becomes unlikely. Version 1.2.2 of <Package>Forms</Package> is compatible with GAP4r4
and GAP4r5.
<P/>Version 1.2.3 contains a new operation <F>TypeOfForm</F>. Together with this addition, some parts of
the documentation, especially concerning degenerate and singular forms, have been edited. A bug found
in the methods for <F>\^</F> applicable on a pair of vectors and a hermitian form, and a pair of matrices
and a hermitian form has been fixed. A series of test files is now included in the tst directory.
Alexander Konovalov pointed out the the init.g and read.g files had windows line breaks, this is also
fixed. Finally, the documentation has been recompiled with the MathJax option.
<P/>Max Horn pointed out that we still used the deprecated GAP_ROOT_PATHS. This has been changed now
into GAPInfo.RootPaths in version 1.2.4. More tests have been added to reach a better code coverage.
Due to these tests, a bug in one of the methods for <C>EvaluateForm</C> was discovered and fixed.
Alexander Konovalov noted that we used the deprecated <F>ReadTest</F> in our test files. This has
been changed to <F>Test</F>. Finally some LaTeX issues were resolved in the documentation.
<P/>In version 1.2.5, some small changes were made to the recognition part.
Some new examples in the documentation explain better the functionality.
<P/>Version 1.2.6 is an intermediate update. It contains a number of corrections/additions suggested/implemented by Thomas Breuer and Max Horn, including an extension of some GAP library functions to create classical matrix groups. These additions are not yet documented.
<P/>In version 1.2.7, only an issue with the automatic release was fixed.
<P/>Version 1.2.8 optimizes the computation of base change matrices, making it
faster by several orders of magnitude for large inputs. Moreover, Max Horn was
added to the list of maintainers. Finally, various janitorial changes were made.
<P/>Version 1.2.9 fixes an unexpected error in <C>BaseChangeToCanonical</C> if the input
was over a finite field with more than 256 elements.
<P/>Version 1.2.10 optimizes the computation of base change matrices for symplectic and quadratic forms.
<P/>Version 1.2.11 optimizes the computation of base change matrices for hermitian and quadratic forms.
It also fixes a regression in the base change computation for bilinear forms introduced in 1.2.10;
and a long-standing bug in the base change computation for degenerate alternating forms.
<P/>Version 1.2.12 is to be considered as an intermediate release. Part of the recognition functionality has been
updated and corrected. This work in progress will be continued.
<P/>Version 1.2.13 fixes a bug <C>BaseChangeToCanonical</C> in dimension 1 where an error could be thrown.
This now works. Moreover the undocumented internal functions <C>IsSymplecticMatrix</C>, <C>IsOrthogonalMatrix</C>
and <C>IsHermitianMatrix</C> were renamed to prevent anyone from accidentally using them.
Various other internal changes were made that should not affect users.
</Section>


</Chapter>

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