path: root/gsl-1.9/doc/vectors.texi

@cindex blocks
@cindex vectors
@cindex matrices

The functions described in this chapter provide a simple vector and
matrix interface to ordinary C arrays. The memory management of these
arrays is implemented using a single underlying type, known as a
block. By writing your functions in terms of vectors and matrices you
can pass a single structure containing both data and dimensions as an
argument without needing additional function parameters.  The structures
are compatible with the vector and matrix formats used by @sc{blas}
routines.

@menu
* Data types::                  
* Blocks::                      
* Vectors::                     
* Matrices::                    
* Vector and Matrix References and Further Reading::  
@end menu

@node Data types
@section Data types

All the functions are available for each of the standard data-types.
The versions for @code{double} have the prefix @code{gsl_block},
@code{gsl_vector} and @code{gsl_matrix}.  Similarly the versions for
single-precision @code{float} arrays have the prefix
@code{gsl_block_float}, @code{gsl_vector_float} and
@code{gsl_matrix_float}.  The full list of available types is given
below,

@example
gsl_block                       double         
gsl_block_float                 float         
gsl_block_long_double           long double   
gsl_block_int                   int           
gsl_block_uint                  unsigned int  
gsl_block_long                  long          
gsl_block_ulong                 unsigned long 
gsl_block_short                 short         
gsl_block_ushort                unsigned short
gsl_block_char                  char          
gsl_block_uchar                 unsigned char 
gsl_block_complex               complex double        
gsl_block_complex_float         complex float         
gsl_block_complex_long_double   complex long double   
@end example

@noindent
Corresponding types exist for the @code{gsl_vector} and
@code{gsl_matrix} functions.



@node Blocks
@section Blocks

For consistency all memory is allocated through a @code{gsl_block}
structure.  The structure contains two components, the size of an area of
memory and a pointer to the memory.  The @code{gsl_block} structure looks
like this,

@example
typedef struct
@{
  size_t size;
  double * data;
@} gsl_block;
@end example
@comment

@noindent
Vectors and matrices are made by @dfn{slicing} an underlying block. A
slice is a set of elements formed from an initial offset and a
combination of indices and step-sizes. In the case of a matrix the
step-size for the column index represents the row-length.  The step-size
for a vector is known as the @dfn{stride}.

The functions for allocating and deallocating blocks are defined in
@file{gsl_block.h}

@menu
* Block allocation::            
* Reading and writing blocks::  
* Example programs for blocks::  
@end menu

@node Block allocation
@subsection Block allocation

The functions for allocating memory to a block follow the style of
@code{malloc} and @code{free}.  In addition they also perform their own
error checking.  If there is insufficient memory available to allocate a
block then the functions call the GSL error handler (with an error
number of @code{GSL_ENOMEM}) in addition to returning a null
pointer.  Thus if you use the library error handler to abort your program
then it isn't necessary to check every @code{alloc}.  

@deftypefun {gsl_block *} gsl_block_alloc (size_t @var{n})
This function allocates memory for a block of @var{n} double-precision
elements, returning a pointer to the block struct.  The block is not
initialized and so the values of its elements are undefined.  Use the
function @code{gsl_block_calloc} if you want to ensure that all the
elements are initialized to zero.

A null pointer is returned if insufficient memory is available to create
the block.
@end deftypefun

@deftypefun {gsl_block *} gsl_block_calloc (size_t @var{n})
This function allocates memory for a block and initializes all the
elements of the block to zero.
@end deftypefun

@deftypefun void gsl_block_free (gsl_block * @var{b})
This function frees the memory used by a block @var{b} previously
allocated with @code{gsl_block_alloc} or @code{gsl_block_calloc}.  The
block @var{b} must be a valid block object (a null pointer is not
allowed).
@end deftypefun

@node Reading and writing blocks
@subsection Reading and writing blocks

The library provides functions for reading and writing blocks to a file
as binary data or formatted text.

@deftypefun int gsl_block_fwrite (FILE * @var{stream}, const gsl_block * @var{b})
This function writes the elements of the block @var{b} to the stream
@var{stream} in binary format.  The return value is 0 for success and
@code{GSL_EFAILED} if there was a problem writing to the file.  Since the
data is written in the native binary format it may not be portable
between different architectures.
@end deftypefun

@deftypefun int gsl_block_fread (FILE * @var{stream}, gsl_block * @var{b})
This function reads into the block @var{b} from the open stream
@var{stream} in binary format.  The block @var{b} must be preallocated
with the correct length since the function uses the size of @var{b} to
determine how many bytes to read.  The return value is 0 for success and
@code{GSL_EFAILED} if there was a problem reading from the file.  The
data is assumed to have been written in the native binary format on the
same architecture.
@end deftypefun

@deftypefun int gsl_block_fprintf (FILE * @var{stream}, const gsl_block * @var{b}, const char * @var{format})
This function writes the elements of the block @var{b} line-by-line to
the stream @var{stream} using the format specifier @var{format}, which
should be one of the @code{%g}, @code{%e} or @code{%f} formats for
floating point numbers and @code{%d} for integers.  The function returns
0 for success and @code{GSL_EFAILED} if there was a problem writing to
the file.
@end deftypefun

@deftypefun int gsl_block_fscanf (FILE * @var{stream}, gsl_block * @var{b})
This function reads formatted data from the stream @var{stream} into the
block @var{b}.  The block @var{b} must be preallocated with the correct
length since the function uses the size of @var{b} to determine how many
numbers to read.  The function returns 0 for success and
@code{GSL_EFAILED} if there was a problem reading from the file.
@end deftypefun
@comment

@node Example programs for blocks
@subsection Example programs for blocks

The following program shows how to allocate a block,

@example
@verbatiminclude examples/block.c
@end example
@comment

@noindent
Here is the output from the program,

@example
@verbatiminclude examples/block.out
@end example
@comment

@node Vectors
@section Vectors
@cindex vectors
@cindex stride, of vector index

Vectors are defined by a @code{gsl_vector} structure which describes a
slice of a block.  Different vectors can be created which point to the
same block.  A vector slice is a set of equally-spaced elements of an
area of memory.

The @code{gsl_vector} structure contains five components, the
@dfn{size}, the @dfn{stride}, a pointer to the memory where the elements
are stored, @var{data}, a pointer to the block owned by the vector,
@var{block}, if any, and an ownership flag, @var{owner}.  The structure
is very simple and looks like this,

@example
typedef struct
@{
  size_t size;
  size_t stride;
  double * data;
  gsl_block * block;
  int owner;
@} gsl_vector;
@end example
@comment

@noindent
The @var{size} is simply the number of vector elements.  The range of
valid indices runs from 0 to @code{size-1}.  The @var{stride} is the
step-size from one element to the next in physical memory, measured in
units of the appropriate datatype.  The pointer @var{data} gives the
location of the first element of the vector in memory.  The pointer
@var{block} stores the location of the memory block in which the vector
elements are located (if any).  If the vector owns this block then the
@var{owner} field is set to one and the block will be deallocated when the
vector is freed.  If the vector points to a block owned by another
object then the @var{owner} field is zero and any underlying block will not be
deallocated with the vector.

The functions for allocating and accessing vectors are defined in
@file{gsl_vector.h}

@menu
* Vector allocation::           
* Accessing vector elements::   
* Initializing vector elements::  
* Reading and writing vectors::  
* Vector views::                
* Copying vectors::             
* Exchanging elements::         
* Vector operations::           
* Finding maximum and minimum elements of vectors::  
* Vector properties::           
* Example programs for vectors::  
@end menu

@node Vector allocation
@subsection Vector allocation

The functions for allocating memory to a vector follow the style of
@code{malloc} and @code{free}.  In addition they also perform their own
error checking.  If there is insufficient memory available to allocate a
vector then the functions call the GSL error handler (with an error
number of @code{GSL_ENOMEM}) in addition to returning a null
pointer.  Thus if you use the library error handler to abort your program
then it isn't necessary to check every @code{alloc}.

@deftypefun {gsl_vector *} gsl_vector_alloc (size_t @var{n})
This function creates a vector of length @var{n}, returning a pointer to
a newly initialized vector struct. A new block is allocated for the
elements of the vector, and stored in the @var{block} component of the
vector struct.  The block is ``owned'' by the vector, and will be
deallocated when the vector is deallocated.
@end deftypefun

@deftypefun {gsl_vector *} gsl_vector_calloc (size_t @var{n})
This function allocates memory for a vector of length @var{n} and
initializes all the elements of the vector to zero.
@end deftypefun

@deftypefun void gsl_vector_free (gsl_vector * @var{v})
This function frees a previously allocated vector @var{v}.  If the
vector was created using @code{gsl_vector_alloc} then the block
underlying the vector will also be deallocated.  If the vector has
been created from another object then the memory is still owned by
that object and will not be deallocated.  The vector @var{v} must be a
valid vector object (a null pointer is not allowed).
@end deftypefun

@node Accessing vector elements
@subsection Accessing vector elements
@cindex vectors, range-checking
@cindex range-checking for vectors
@cindex bounds checking, extension to GCC
@cindex gcc extensions, range-checking
@cindex Fortran range checking, equivalent in gcc

Unlike @sc{fortran} compilers, C compilers do not usually provide
support for range checking of vectors and matrices.  Range checking is
available in the GNU C Compiler bounds-checking extension, but it is not
part of the default installation of GCC.  The functions
@code{gsl_vector_get} and @code{gsl_vector_set} can perform portable
range checking for you and report an error if you attempt to access
elements outside the allowed range.

The functions for accessing the elements of a vector or matrix are
defined in @file{gsl_vector.h} and declared @code{extern inline} to
eliminate function-call overhead.  You must compile your program with
the macro @code{HAVE_INLINE} defined to use these functions.  

If necessary you can turn off range checking completely without
modifying any source files by recompiling your program with the
preprocessor definition @code{GSL_RANGE_CHECK_OFF}.  Provided your
compiler supports inline functions the effect of turning off range
checking is to replace calls to @code{gsl_vector_get(v,i)} by
@code{v->data[i*v->stride]} and calls to @code{gsl_vector_set(v,i,x)} by
@code{v->data[i*v->stride]=x}.  Thus there should be no performance
penalty for using the range checking functions when range checking is
turned off.

@deftypefun double gsl_vector_get (const gsl_vector * @var{v}, size_t @var{i})
This function returns the @var{i}-th element of a vector @var{v}.  If
@var{i} lies outside the allowed range of 0 to @math{@var{n}-1} then the error
handler is invoked and 0 is returned.
@end deftypefun

@deftypefun void gsl_vector_set (gsl_vector * @var{v}, size_t @var{i}, double @var{x})
This function sets the value of the @var{i}-th element of a vector
@var{v} to @var{x}.  If @var{i} lies outside the allowed range of 0 to
@math{@var{n}-1} then the error handler is invoked.
@end deftypefun

@deftypefun {double *} gsl_vector_ptr (gsl_vector * @var{v}, size_t @var{i})
@deftypefunx {const double *} gsl_vector_const_ptr (const gsl_vector * @var{v}, size_t @var{i})
These functions return a pointer to the @var{i}-th element of a vector
@var{v}.  If @var{i} lies outside the allowed range of 0 to @math{@var{n}-1}
then the error handler is invoked and a null pointer is returned.
@end deftypefun

@node Initializing vector elements
@subsection Initializing vector elements
@cindex vectors, initializing
@cindex initializing vectors

@deftypefun void gsl_vector_set_all (gsl_vector * @var{v}, double @var{x})
This function sets all the elements of the vector @var{v} to the value
@var{x}.
@end deftypefun

@deftypefun void gsl_vector_set_zero (gsl_vector * @var{v})
This function sets all the elements of the vector @var{v} to zero.
@end deftypefun

@deftypefun int gsl_vector_set_basis (gsl_vector * @var{v}, size_t @var{i})
This function makes a basis vector by setting all the elements of the
vector @var{v} to zero except for the @var{i}-th element which is set to
one.
@end deftypefun

@node Reading and writing vectors
@subsection Reading and writing vectors

The library provides functions for reading and writing vectors to a file
as binary data or formatted text.

@deftypefun int gsl_vector_fwrite (FILE * @var{stream}, const gsl_vector * @var{v})
This function writes the elements of the vector @var{v} to the stream
@var{stream} in binary format.  The return value is 0 for success and
@code{GSL_EFAILED} if there was a problem writing to the file.  Since the
data is written in the native binary format it may not be portable
between different architectures.
@end deftypefun

@deftypefun int gsl_vector_fread (FILE * @var{stream}, gsl_vector * @var{v})
This function reads into the vector @var{v} from the open stream
@var{stream} in binary format.  The vector @var{v} must be preallocated
with the correct length since the function uses the size of @var{v} to
determine how many bytes to read.  The return value is 0 for success and
@code{GSL_EFAILED} if there was a problem reading from the file.  The
data is assumed to have been written in the native binary format on the
same architecture.
@end deftypefun

@deftypefun int gsl_vector_fprintf (FILE * @var{stream}, const gsl_vector * @var{v}, const char * @var{format})
This function writes the elements of the vector @var{v} line-by-line to
the stream @var{stream} using the format specifier @var{format}, which
should be one of the @code{%g}, @code{%e} or @code{%f} formats for
floating point numbers and @code{%d} for integers.  The function returns
0 for success and @code{GSL_EFAILED} if there was a problem writing to
the file.
@end deftypefun

@deftypefun int gsl_vector_fscanf (FILE * @var{stream}, gsl_vector * @var{v})
This function reads formatted data from the stream @var{stream} into the
vector @var{v}.  The vector @var{v} must be preallocated with the correct
length since the function uses the size of @var{v} to determine how many
numbers to read.  The function returns 0 for success and
@code{GSL_EFAILED} if there was a problem reading from the file.
@end deftypefun

@node Vector views
@subsection Vector views

In addition to creating vectors from slices of blocks it is also
possible to slice vectors and create vector views.  For example, a
subvector of another vector can be described with a view, or two views
can be made which provide access to the even and odd elements of a
vector.

A vector view is a temporary object, stored on the stack, which can be
used to operate on a subset of vector elements.  Vector views can be
defined for both constant and non-constant vectors, using separate types
that preserve constness.  A vector view has the type
@code{gsl_vector_view} and a constant vector view has the type
@code{gsl_vector_const_view}.  In both cases the elements of the view
can be accessed as a @code{gsl_vector} using the @code{vector} component
of the view object.  A pointer to a vector of type @code{gsl_vector *}
or @code{const gsl_vector *} can be obtained by taking the address of
this component with the @code{&} operator.  

When using this pointer it is important to ensure that the view itself
remains in scope---the simplest way to do so is by always writing the
pointer as @code{&}@var{view}@code{.vector}, and never storing this value
in another variable.  

@deftypefun gsl_vector_view gsl_vector_subvector (gsl_vector * @var{v}, size_t @var{offset}, size_t @var{n})
@deftypefunx {gsl_vector_const_view} gsl_vector_const_subvector (const gsl_vector * @var{v}, size_t @var{offset}, size_t @var{n})
These functions return a vector view of a subvector of another vector
@var{v}.  The start of the new vector is offset by @var{offset} elements
from the start of the original vector.  The new vector has @var{n}
elements.  Mathematically, the @var{i}-th element of the new vector
@var{v'} is given by,

@example
v'(i) = v->data[(offset + i)*v->stride]
@end example

@noindent
where the index @var{i} runs from 0 to @code{n-1}.

The @code{data} pointer of the returned vector struct is set to null if
the combined parameters (@var{offset},@var{n}) overrun the end of the
original vector.

The new vector is only a view of the block underlying the original
vector, @var{v}.  The block containing the elements of @var{v} is not
owned by the new vector.  When the view goes out of scope the original
vector @var{v} and its block will continue to exist.  The original
memory can only be deallocated by freeing the original vector.  Of
course, the original vector should not be deallocated while the view is
still in use.

The function @code{gsl_vector_const_subvector} is equivalent to
@code{gsl_vector_subvector} but can be used for vectors which are
declared @code{const}.
@end deftypefun


@deftypefun gsl_vector_view gsl_vector_subvector_with_stride (gsl_vector * @var{v}, size_t @var{offset}, size_t @var{stride}, size_t @var{n})
@deftypefunx {gsl_vector_const_view} gsl_vector_const_subvector_with_stride (const gsl_vector * @var{v}, size_t @var{offset}, size_t @var{stride}, size_t @var{n})
These functions return a vector view of a subvector of another vector
@var{v} with an additional stride argument. The subvector is formed in
the same way as for @code{gsl_vector_subvector} but the new vector has
@var{n} elements with a step-size of @var{stride} from one element to
the next in the original vector.  Mathematically, the @var{i}-th element
of the new vector @var{v'} is given by,

@example
v'(i) = v->data[(offset + i*stride)*v->stride]
@end example

@noindent
where the index @var{i} runs from 0 to @code{n-1}.

Note that subvector views give direct access to the underlying elements
of the original vector. For example, the following code will zero the
even elements of the vector @code{v} of length @code{n}, while leaving the
odd elements untouched,

@example
gsl_vector_view v_even 
  = gsl_vector_subvector_with_stride (v, 0, 2, n/2);
gsl_vector_set_zero (&v_even.vector);
@end example

@noindent
A vector view can be passed to any subroutine which takes a vector
argument just as a directly allocated vector would be, using
@code{&}@var{view}@code{.vector}.  For example, the following code
computes the norm of the odd elements of @code{v} using the @sc{blas}
routine @sc{dnrm2},

@example
gsl_vector_view v_odd 
  = gsl_vector_subvector_with_stride (v, 1, 2, n/2);
double r = gsl_blas_dnrm2 (&v_odd.vector);
@end example

The function @code{gsl_vector_const_subvector_with_stride} is equivalent
to @code{gsl_vector_subvector_with_stride} but can be used for vectors
which are declared @code{const}.
@end deftypefun

@deftypefun gsl_vector_view gsl_vector_complex_real (gsl_vector_complex * @var{v})
@deftypefunx {gsl_vector_const_view} gsl_vector_complex_const_real (const gsl_vector_complex * @var{v})
These functions return a vector view of the real parts of the complex
vector @var{v}.

The function @code{gsl_vector_complex_const_real} is equivalent to
@code{gsl_vector_complex_real} but can be used for vectors which are
declared @code{const}.
@end deftypefun

@deftypefun gsl_vector_view gsl_vector_complex_imag (gsl_vector_complex * @var{v})
@deftypefunx {gsl_vector_const_view} gsl_vector_complex_const_imag (const gsl_vector_complex * @var{v})
These functions return a vector view of the imaginary parts of the
complex vector @var{v}.

The function @code{gsl_vector_complex_const_imag} is equivalent to
@code{gsl_vector_complex_imag} but can be used for vectors which are
declared @code{const}.
@end deftypefun

@deftypefun gsl_vector_view gsl_vector_view_array (double * @var{base}, size_t @var{n}) 
@deftypefunx {gsl_vector_const_view} gsl_vector_const_view_array (const double * @var{base}, size_t @var{n})
These functions return a vector view of an array.  The start of the new
vector is given by @var{base} and has @var{n} elements.  Mathematically,
the @var{i}-th element of the new vector @var{v'} is given by,

@example
v'(i) = base[i]
@end example

@noindent
where the index @var{i} runs from 0 to @code{n-1}.

The array containing the elements of @var{v} is not owned by the new
vector view.  When the view goes out of scope the original array will
continue to exist.  The original memory can only be deallocated by
freeing the original pointer @var{base}.  Of course, the original array
should not be deallocated while the view is still in use.

The function @code{gsl_vector_const_view_array} is equivalent to
@code{gsl_vector_view_array} but can be used for arrays which are
declared @code{const}.
@end deftypefun

@deftypefun gsl_vector_view gsl_vector_view_array_with_stride (double * @var{base}, size_t @var{stride}, size_t @var{n})
@deftypefunx gsl_vector_const_view gsl_vector_const_view_array_with_stride (const double * @var{base}, size_t @var{stride}, size_t @var{n})
These functions return a vector view of an array @var{base} with an
additional stride argument. The subvector is formed in the same way as
for @code{gsl_vector_view_array} but the new vector has @var{n} elements
with a step-size of @var{stride} from one element to the next in the
original array.  Mathematically, the @var{i}-th element of the new
vector @var{v'} is given by,

@example
v'(i) = base[i*stride]
@end example

@noindent
where the index @var{i} runs from 0 to @code{n-1}.

Note that the view gives direct access to the underlying elements of the
original array.  A vector view can be passed to any subroutine which
takes a vector argument just as a directly allocated vector would be,
using @code{&}@var{view}@code{.vector}.

The function @code{gsl_vector_const_view_array_with_stride} is
equivalent to @code{gsl_vector_view_array_with_stride} but can be used
for arrays which are declared @code{const}.
@end deftypefun


@comment @node Modifying subvector views
@comment @subsection Modifying subvector views
@comment 
@comment @deftypefun int gsl_vector_view_from_vector (gsl_vector * @var{v}, gsl_vector * @var{base}, size_t @var{offset}, size_t @var{n}, size_t @var{stride})
@comment This function modifies and existing vector view @var{v} to form a new
@comment view of a vector @var{base}, starting from element @var{offset}.  The
@comment vector has @var{n} elements separated by stride @var{stride}.  Any
@comment existing view in @var{v} will be lost as a result of this function.
@comment @end deftypefun
@comment 
@comment @deftypefun int gsl_vector_view_from_array (gsl_vector * @var{v}, double * @var{base}, size_t @var{offset}, size_t @var{n}, size_t @var{stride})
@comment This function modifies and existing vector view @var{v} to form a new
@comment view of an array @var{base}, starting from element @var{offset}.  The
@comment vector has @var{n} elements separated by stride @var{stride}.  Any
@comment existing view in @var{v} will be lost as a result of this function.
@comment @end deftypefun

@comment @deftypefun {gsl_vector *} gsl_vector_alloc_from_block (gsl_block * @var{b}, size_t @var{offset}, size_t @var{n}, size_t @var{stride})
@comment This function creates a vector as a slice of an existing block @var{b},
@comment returning a pointer to a newly initialized vector struct.  The start of
@comment the vector is offset by @var{offset} elements from the start of the
@comment block.  The vector has @var{n} elements, with a step-size of @var{stride}
@comment from one element to the next.  Mathematically, the @var{i}-th element of
@comment the vector is given by,
@comment 
@comment @example
@comment v(i) = b->data[offset + i*stride]
@comment @end example
@comment @noindent
@comment where the index @var{i} runs from 0 to @code{n-1}.
@comment 
@comment A null pointer is returned if the combined parameters
@comment (@var{offset},@var{n},@var{stride}) overrun the end of the block or if
@comment insufficient memory is available to store the vector.
@comment 
@comment The vector is only a view of the block @var{b}, and the block is not
@comment owned by the vector.  When the vector is deallocated the block @var{b}
@comment will continue to exist.  This memory can only be deallocated by freeing
@comment the block itself.  Of course, this block should not be deallocated while
@comment the vector is still in use.
@comment @end deftypefun
@comment 
@comment @deftypefun {gsl_vector *} gsl_vector_alloc_from_vector (gsl_vector * @var{v}, size_t @var{offset}, size_t @var{n}, size_t @var{stride})
@comment This function creates a vector as a slice of another vector @var{v},
@comment returning a pointer to a newly initialized vector struct.  The start of
@comment the new vector is offset by @var{offset} elements from the start of the
@comment original vector.  The new vector has @var{n} elements, with a step-size
@comment of @var{stride} from one element to the next in the original vector.
@comment Mathematically, the @var{i}-th element of the new vector @var{v'} is
@comment given by,
@comment 
@comment @example
@comment v'(i) = v->data[(offset + i*stride)*v->stride]
@comment @end example
@comment @noindent
@comment where the index @var{i} runs from 0 to @code{n-1}.
@comment 
@comment A null pointer is returned if the combined parameters
@comment (@var{offset},@var{n},@var{stride}) overrun the end of the original
@comment vector or if insufficient memory is available store the new vector.
@comment 
@comment The new vector is only a view of the block underlying the original
@comment vector, @var{v}.  The block is not owned by the new vector.  When the new
@comment vector is deallocated the original vector @var{v} and its block will
@comment continue to exist.  The original memory can only be deallocated by
@comment freeing the original vector.  Of course, the original vector should not
@comment be deallocated while the new vector is still in use.
@comment @end deftypefun

@node Copying vectors
@subsection Copying vectors

Common operations on vectors such as addition and multiplication are
available in the @sc{blas} part of the library (@pxref{BLAS
Support}).  However, it is useful to have a small number of utility
functions which do not require the full @sc{blas} code.  The following
functions fall into this category.

@deftypefun int gsl_vector_memcpy (gsl_vector * @var{dest}, const gsl_vector * @var{src})
This function copies the elements of the vector @var{src} into the
vector @var{dest}.  The two vectors must have the same length.
@end deftypefun

@deftypefun int gsl_vector_swap (gsl_vector * @var{v}, gsl_vector * @var{w})
This function exchanges the elements of the vectors @var{v} and @var{w}
by copying.  The two vectors must have the same length.
@end deftypefun


@node Exchanging elements
@subsection Exchanging elements

The following function can be used to exchange, or permute, the elements
of a vector.

@deftypefun int gsl_vector_swap_elements (gsl_vector * @var{v}, size_t @var{i}, size_t @var{j})
This function exchanges the @var{i}-th and @var{j}-th elements of the
vector @var{v} in-place.
@end deftypefun

@deftypefun int gsl_vector_reverse (gsl_vector * @var{v})
This function reverses the order of the elements of the vector @var{v}.
@end deftypefun


@node Vector operations
@subsection Vector operations

The following operations are only defined for real vectors.

@deftypefun int gsl_vector_add (gsl_vector * @var{a}, const gsl_vector * @var{b})
This function adds the elements of vector @var{b} to the elements of
vector @var{a}, @math{a'_i = a_i + b_i}. The two vectors must have the
same length.
@end deftypefun

@deftypefun int gsl_vector_sub (gsl_vector * @var{a}, const gsl_vector * @var{b})
This function subtracts the elements of vector @var{b} from the elements of
vector @var{a}, @math{a'_i = a_i - b_i}. The two vectors must have the
same length.
@end deftypefun

@deftypefun int gsl_vector_mul (gsl_vector * @var{a}, const gsl_vector * @var{b})
This function multiplies the elements of vector @var{a} by the elements of
vector @var{b}, @math{a'_i = a_i * b_i}. The two vectors must have the
same length.
@end deftypefun

@deftypefun int gsl_vector_div (gsl_vector * @var{a}, const gsl_vector * @var{b})
This function divides the elements of vector @var{a} by the elements of
vector @var{b}, @math{a'_i = a_i / b_i}. The two vectors must have the
same length.
@end deftypefun

@deftypefun int gsl_vector_scale (gsl_vector * @var{a}, const double @var{x})
This function multiplies the elements of vector @var{a} by the constant
factor @var{x}, @math{a'_i = x a_i}.
@end deftypefun

@deftypefun int gsl_vector_add_constant (gsl_vector * @var{a}, const double @var{x})
This function adds the constant value @var{x} to the elements of the
vector @var{a}, @math{a'_i = a_i + x}.
@end deftypefun

@node Finding maximum and minimum elements of vectors
@subsection Finding maximum and minimum elements of vectors

@deftypefun double gsl_vector_max (const gsl_vector * @var{v})
This function returns the maximum value in the vector @var{v}.
@end deftypefun

@deftypefun double gsl_vector_min (const gsl_vector * @var{v})
This function returns the minimum value in the vector @var{v}.
@end deftypefun

@deftypefun void gsl_vector_minmax (const gsl_vector * @var{v}, double * @var{min_out}, double * @var{max_out})
This function returns the minimum and maximum values in the vector
@var{v}, storing them in @var{min_out} and @var{max_out}.
@end deftypefun

@deftypefun size_t gsl_vector_max_index (const gsl_vector * @var{v})
This function returns the index of the maximum value in the vector @var{v}.
When there are several equal maximum elements then the lowest index is
returned.
@end deftypefun

@deftypefun size_t gsl_vector_min_index (const gsl_vector * @var{v})
This function returns the index of the minimum value in the vector @var{v}.
When there are several equal minimum elements then the lowest index is
returned.
@end deftypefun

@deftypefun void gsl_vector_minmax_index (const gsl_vector * @var{v}, size_t * @var{imin}, size_t * @var{imax})
This function returns the indices of the minimum and maximum values in
the vector @var{v}, storing them in @var{imin} and @var{imax}. When
there are several equal minimum or maximum elements then the lowest
indices are returned.
@end deftypefun

@node Vector properties
@subsection Vector properties

@deftypefun int gsl_vector_isnull (const gsl_vector * @var{v})
@deftypefunx int gsl_vector_ispos (const gsl_vector * @var{v})
@deftypefunx int gsl_vector_isneg (const gsl_vector * @var{v})
These functions return 1 if all the elements of the vector @var{v}
are zero, strictly positive, or strictly negative respectively, and 0
otherwise.  To test for a non-negative vector, use the expression
@code{!gsl_vector_isneg(v)}.
@end deftypefun

@node Example programs for vectors
@subsection Example programs for vectors

This program shows how to allocate, initialize and read from a vector
using the functions @code{gsl_vector_alloc}, @code{gsl_vector_set} and
@code{gsl_vector_get}.

@example
@verbatiminclude examples/vector.c
@end example
@comment

@noindent
Here is the output from the program.  The final loop attempts to read
outside the range of the vector @code{v}, and the error is trapped by
the range-checking code in @code{gsl_vector_get}.

@example
$ ./a.out
v_0 = 1.23
v_1 = 2.23
v_2 = 3.23
gsl: vector_source.c:12: ERROR: index out of range
Default GSL error handler invoked.
Aborted (core dumped)
@end example
@comment

@noindent
The next program shows how to write a vector to a file.

@example
@verbatiminclude examples/vectorw.c
@end example
@comment

@noindent
After running this program the file @file{test.dat} should contain the
elements of @code{v}, written using the format specifier
@code{%.5g}.  The vector could then be read back in using the function
@code{gsl_vector_fscanf (f, v)} as follows:

@example
@verbatiminclude examples/vectorr.c
@end example


@node Matrices
@section Matrices
@cindex matrices
@cindex physical dimension, matrices
@cindex trailing dimension, matrices
@cindex leading dimension, matrices

Matrices are defined by a @code{gsl_matrix} structure which describes a
generalized slice of a block.  Like a vector it represents a set of
elements in an area of memory, but uses two indices instead of one.

The @code{gsl_matrix} structure contains six components, the two
dimensions of the matrix, a physical dimension, a pointer to the memory
where the elements of the matrix are stored, @var{data}, a pointer to
the block owned by the matrix @var{block}, if any, and an ownership
flag, @var{owner}.  The physical dimension determines the memory layout
and can differ from the matrix dimension to allow the use of
submatrices.  The @code{gsl_matrix} structure is very simple and looks
like this,

@example
typedef struct
@{
  size_t size1;
  size_t size2;
  size_t tda;
  double * data;
  gsl_block * block;
  int owner;
@} gsl_matrix;
@end example
@comment

@noindent
Matrices are stored in row-major order, meaning that each row of
elements forms a contiguous block in memory.  This is the standard
``C-language ordering'' of two-dimensional arrays. Note that @sc{fortran}
stores arrays in column-major order. The number of rows is @var{size1}.
The range of valid row indices runs from 0 to @code{size1-1}.  Similarly
@var{size2} is the number of columns.  The range of valid column indices
runs from 0 to @code{size2-1}.  The physical row dimension @var{tda}, or
@dfn{trailing dimension}, specifies the size of a row of the matrix as
laid out in memory.

For example, in the following matrix @var{size1} is 3, @var{size2} is 4,
and @var{tda} is 8.  The physical memory layout of the matrix begins in
the top left hand-corner and proceeds from left to right along each row
in turn.

@example
@group
00 01 02 03 XX XX XX XX
10 11 12 13 XX XX XX XX
20 21 22 23 XX XX XX XX
@end group
@end example

@noindent
Each unused memory location is represented by ``@code{XX}''.  The
pointer @var{data} gives the location of the first element of the matrix
in memory.  The pointer @var{block} stores the location of the memory
block in which the elements of the matrix are located (if any).  If the
matrix owns this block then the @var{owner} field is set to one and the
block will be deallocated when the matrix is freed.  If the matrix is
only a slice of a block owned by another object then the @var{owner} field is
zero and any underlying block will not be freed.

The functions for allocating and accessing matrices are defined in
@file{gsl_matrix.h}

@menu
* Matrix allocation::           
* Accessing matrix elements::   
* Initializing matrix elements::  
* Reading and writing matrices::  
* Matrix views::                
* Creating row and column views::  
* Copying matrices::            
* Copying rows and columns::    
* Exchanging rows and columns::  
* Matrix operations::           
* Finding maximum and minimum elements of matrices::  
* Matrix properties::           
* Example programs for matrices::  
@end menu

@node Matrix allocation
@subsection Matrix allocation

The functions for allocating memory to a matrix follow the style of
@code{malloc} and @code{free}.  They also perform their own error
checking.  If there is insufficient memory available to allocate a vector
then the functions call the GSL error handler (with an error number of
@code{GSL_ENOMEM}) in addition to returning a null pointer.  Thus if you
use the library error handler to abort your program then it isn't
necessary to check every @code{alloc}.

@deftypefun {gsl_matrix *} gsl_matrix_alloc (size_t @var{n1}, size_t @var{n2})
This function creates a matrix of size @var{n1} rows by @var{n2}
columns, returning a pointer to a newly initialized matrix struct. A new
block is allocated for the elements of the matrix, and stored in the
@var{block} component of the matrix struct.  The block is ``owned'' by the
matrix, and will be deallocated when the matrix is deallocated.
@end deftypefun

@deftypefun {gsl_matrix *} gsl_matrix_calloc (size_t @var{n1}, size_t @var{n2})
This function allocates memory for a matrix of size @var{n1} rows by
@var{n2} columns and initializes all the elements of the matrix to zero.
@end deftypefun

@deftypefun void gsl_matrix_free (gsl_matrix * @var{m})
This function frees a previously allocated matrix @var{m}.  If the
matrix was created using @code{gsl_matrix_alloc} then the block
underlying the matrix will also be deallocated.  If the matrix has
been created from another object then the memory is still owned by
that object and will not be deallocated.  The matrix @var{m} must be a
valid matrix object (a null pointer is not allowed).
@end deftypefun

@node Accessing matrix elements
@subsection Accessing matrix elements
@cindex matrices, range-checking
@cindex range-checking for matrices

The functions for accessing the elements of a matrix use the same range
checking system as vectors.  You can turn off range checking by recompiling
your program with the preprocessor definition
@code{GSL_RANGE_CHECK_OFF}.

The elements of the matrix are stored in ``C-order'', where the second
index moves continuously through memory.  More precisely, the element
accessed by the function @code{gsl_matrix_get(m,i,j)} and
@code{gsl_matrix_set(m,i,j,x)} is 

@example
m->data[i * m->tda + j]
@end example
@comment 

@noindent
where @var{tda} is the physical row-length of the matrix.

@deftypefun double gsl_matrix_get (const gsl_matrix * @var{m}, size_t @var{i}, size_t @var{j})
This function returns the @math{(i,j)}-th element of a matrix
@var{m}.  If @var{i} or @var{j} lie outside the allowed range of 0 to
@math{@var{n1}-1} and 0 to @math{@var{n2}-1} then the error handler is invoked and 0
is returned.
@end deftypefun

@deftypefun void gsl_matrix_set (gsl_matrix * @var{m}, size_t @var{i}, size_t @var{j}, double @var{x})
This function sets the value of the @math{(i,j)}-th element of a
matrix @var{m} to @var{x}.  If @var{i} or @var{j} lies outside the
allowed range of 0 to @math{@var{n1}-1} and 0 to @math{@var{n2}-1} then the error
handler is invoked.
@end deftypefun

@deftypefun {double *} gsl_matrix_ptr (gsl_matrix * @var{m}, size_t @var{i}, size_t @var{j})
@deftypefunx {const double *} gsl_matrix_const_ptr (const gsl_matrix * @var{m}, size_t @var{i}, size_t @var{j})
These functions return a pointer to the @math{(i,j)}-th element of a
matrix @var{m}.  If @var{i} or @var{j} lie outside the allowed range of
0 to @math{@var{n1}-1} and 0 to @math{@var{n2}-1} then the error handler is invoked
and a null pointer is returned.
@end deftypefun

@node Initializing matrix elements
@subsection Initializing matrix elements
@cindex matrices, initializing
@cindex initializing matrices
@cindex identity matrix
@cindex matrix, identity
@cindex zero matrix
@cindex matrix, zero
@cindex constant matrix
@cindex matrix, constant

@deftypefun void gsl_matrix_set_all (gsl_matrix * @var{m}, double @var{x})
This function sets all the elements of the matrix @var{m} to the value
@var{x}.
@end deftypefun

@deftypefun void gsl_matrix_set_zero (gsl_matrix * @var{m})
This function sets all the elements of the matrix @var{m} to zero.
@end deftypefun

@deftypefun void gsl_matrix_set_identity (gsl_matrix * @var{m})
This function sets the elements of the matrix @var{m} to the
corresponding elements of the identity matrix, @math{m(i,j) =
\delta(i,j)}, i.e. a unit diagonal with all off-diagonal elements zero.
This applies to both square and rectangular matrices.
@end deftypefun

@node Reading and writing matrices
@subsection Reading and writing matrices

The library provides functions for reading and writing matrices to a file
as binary data or formatted text.

@deftypefun int gsl_matrix_fwrite (FILE * @var{stream}, const gsl_matrix * @var{m})
This function writes the elements of the matrix @var{m} to the stream
@var{stream} in binary format.  The return value is 0 for success and
@code{GSL_EFAILED} if there was a problem writing to the file.  Since the
data is written in the native binary format it may not be portable
between different architectures.
@end deftypefun

@deftypefun int gsl_matrix_fread (FILE * @var{stream}, gsl_matrix * @var{m})
This function reads into the matrix @var{m} from the open stream
@var{stream} in binary format.  The matrix @var{m} must be preallocated
with the correct dimensions since the function uses the size of @var{m} to
determine how many bytes to read.  The return value is 0 for success and
@code{GSL_EFAILED} if there was a problem reading from the file.  The
data is assumed to have been written in the native binary format on the
same architecture.
@end deftypefun

@deftypefun int gsl_matrix_fprintf (FILE * @var{stream}, const gsl_matrix * @var{m}, const char * @var{format})
This function writes the elements of the matrix @var{m} line-by-line to
the stream @var{stream} using the format specifier @var{format}, which
should be one of the @code{%g}, @code{%e} or @code{%f} formats for
floating point numbers and @code{%d} for integers.  The function returns
0 for success and @code{GSL_EFAILED} if there was a problem writing to
the file.
@end deftypefun

@deftypefun int gsl_matrix_fscanf (FILE * @var{stream}, gsl_matrix * @var{m})
This function reads formatted data from the stream @var{stream} into the
matrix @var{m}.  The matrix @var{m} must be preallocated with the correct
dimensions since the function uses the size of @var{m} to determine how many
numbers to read.  The function returns 0 for success and
@code{GSL_EFAILED} if there was a problem reading from the file.
@end deftypefun

@node Matrix views
@subsection Matrix views

A matrix view is a temporary object, stored on the stack, which can be
used to operate on a subset of matrix elements.  Matrix views can be
defined for both constant and non-constant matrices using separate types
that preserve constness.  A matrix view has the type
@code{gsl_matrix_view} and a constant matrix view has the type
@code{gsl_matrix_const_view}.  In both cases the elements of the view
can by accessed using the @code{matrix} component of the view object.  A
pointer @code{gsl_matrix *} or @code{const gsl_matrix *} can be obtained
by taking the address of the @code{matrix} component with the @code{&}
operator.  In addition to matrix views it is also possible to create
vector views of a matrix, such as row or column views.

@deftypefun gsl_matrix_view gsl_matrix_submatrix (gsl_matrix * @var{m}, size_t @var{k1}, size_t @var{k2}, size_t @var{n1}, size_t @var{n2})
@deftypefunx gsl_matrix_const_view gsl_matrix_const_submatrix (const gsl_matrix * @var{m}, size_t @var{k1}, size_t @var{k2}, size_t @var{n1}, size_t @var{n2})
These functions return a matrix view of a submatrix of the matrix
@var{m}.  The upper-left element of the submatrix is the element
(@var{k1},@var{k2}) of the original matrix.  The submatrix has @var{n1}
rows and @var{n2} columns.  The physical number of columns in memory
given by @var{tda} is unchanged.  Mathematically, the
@math{(i,j)}-th element of the new matrix is given by,

@example
m'(i,j) = m->data[(k1*m->tda + k2) + i*m->tda + j]
@end example

@noindent
where the index @var{i} runs from 0 to @code{n1-1} and the index @var{j}
runs from 0 to @code{n2-1}.

The @code{data} pointer of the returned matrix struct is set to null if
the combined parameters (@var{i},@var{j},@var{n1},@var{n2},@var{tda})
overrun the ends of the original matrix.

The new matrix view is only a view of the block underlying the existing
matrix, @var{m}.  The block containing the elements of @var{m} is not
owned by the new matrix view.  When the view goes out of scope the
original matrix @var{m} and its block will continue to exist.  The
original memory can only be deallocated by freeing the original matrix.
Of course, the original matrix should not be deallocated while the view
is still in use.

The function @code{gsl_matrix_const_submatrix} is equivalent to
@code{gsl_matrix_submatrix} but can be used for matrices which are
declared @code{const}.
@end deftypefun


@deftypefun gsl_matrix_view gsl_matrix_view_array (double * @var{base}, size_t @var{n1}, size_t @var{n2})
@deftypefunx gsl_matrix_const_view gsl_matrix_const_view_array (const double * @var{base}, size_t @var{n1}, size_t @var{n2})
These functions return a matrix view of the array @var{base}.  The
matrix has @var{n1} rows and @var{n2} columns.  The physical number of
columns in memory is also given by @var{n2}.  Mathematically, the
@math{(i,j)}-th element of the new matrix is given by,

@example
m'(i,j) = base[i*n2 + j]
@end example

@noindent
where the index @var{i} runs from 0 to @code{n1-1} and the index @var{j}
runs from 0 to @code{n2-1}.

The new matrix is only a view of the array @var{base}.  When the view
goes out of scope the original array @var{base} will continue to exist.
The original memory can only be deallocated by freeing the original
array.  Of course, the original array should not be deallocated while
the view is still in use.

The function @code{gsl_matrix_const_view_array} is equivalent to
@code{gsl_matrix_view_array} but can be used for matrices which are
declared @code{const}.
@end deftypefun


@deftypefun gsl_matrix_view gsl_matrix_view_array_with_tda (double * @var{base}, size_t @var{n1}, size_t @var{n2}, size_t @var{tda})
@deftypefunx gsl_matrix_const_view gsl_matrix_const_view_array_with_tda (const double * @var{base}, size_t @var{n1}, size_t @var{n2}, size_t @var{tda})
These functions return a matrix view of the array @var{base} with a
physical number of columns @var{tda} which may differ from the corresponding
dimension of the matrix.  The matrix has @var{n1} rows and @var{n2}
columns, and the physical number of columns in memory is given by
@var{tda}.  Mathematically, the @math{(i,j)}-th element of the new
matrix is given by,

@example
m'(i,j) = base[i*tda + j]
@end example

@noindent
where the index @var{i} runs from 0 to @code{n1-1} and the index @var{j}
runs from 0 to @code{n2-1}.

The new matrix is only a view of the array @var{base}.  When the view
goes out of scope the original array @var{base} will continue to exist.
The original memory can only be deallocated by freeing the original
array.  Of course, the original array should not be deallocated while
the view is still in use.

The function @code{gsl_matrix_const_view_array_with_tda} is equivalent
to @code{gsl_matrix_view_array_with_tda} but can be used for matrices
which are declared @code{const}.
@end deftypefun

@deftypefun gsl_matrix_view gsl_matrix_view_vector (gsl_vector * @var{v}, size_t @var{n1}, size_t @var{n2})
@deftypefunx gsl_matrix_const_view gsl_matrix_const_view_vector (const gsl_vector * @var{v}, size_t @var{n1}, size_t @var{n2})
These functions return a matrix view of the vector @var{v}.  The matrix
has @var{n1} rows and @var{n2} columns. The vector must have unit
stride. The physical number of columns in memory is also given by
@var{n2}.  Mathematically, the @math{(i,j)}-th element of the new
matrix is given by,

@example
m'(i,j) = v->data[i*n2 + j]
@end example

@noindent
where the index @var{i} runs from 0 to @code{n1-1} and the index @var{j}
runs from 0 to @code{n2-1}.

The new matrix is only a view of the vector @var{v}.  When the view
goes out of scope the original vector @var{v} will continue to exist.
The original memory can only be deallocated by freeing the original
vector.  Of course, the original vector should not be deallocated while
the view is still in use.

The function @code{gsl_matrix_const_view_vector} is equivalent to
@code{gsl_matrix_view_vector} but can be used for matrices which are
declared @code{const}.
@end deftypefun


@deftypefun gsl_matrix_view gsl_matrix_view_vector_with_tda (gsl_vector * @var{v}, size_t @var{n1}, size_t @var{n2}, size_t @var{tda})
@deftypefunx gsl_matrix_const_view gsl_matrix_const_view_vector_with_tda (const gsl_vector * @var{v}, size_t @var{n1}, size_t @var{n2}, size_t @var{tda})
These functions return a matrix view of the vector @var{v} with a
physical number of columns @var{tda} which may differ from the
corresponding matrix dimension.  The vector must have unit stride. The
matrix has @var{n1} rows and @var{n2} columns, and the physical number
of columns in memory is given by @var{tda}.  Mathematically, the
@math{(i,j)}-th element of the new matrix is given by,

@example
m'(i,j) = v->data[i*tda + j]
@end example

@noindent
where the index @var{i} runs from 0 to @code{n1-1} and the index @var{j}
runs from 0 to @code{n2-1}.

The new matrix is only a view of the vector @var{v}.  When the view
goes out of scope the original vector @var{v} will continue to exist.
The original memory can only be deallocated by freeing the original
vector.  Of course, the original vector should not be deallocated while
the view is still in use.

The function @code{gsl_matrix_const_view_vector_with_tda} is equivalent
to @code{gsl_matrix_view_vector_with_tda} but can be used for matrices
which are declared @code{const}.
@end deftypefun


@comment @node Modifying matrix views
@comment @subsection Modifying matrix views
@comment 
@comment @deftypefun int gsl_matrix_view_from_matrix (gsl_matrix * @var{m}, gsl_matrix * @var{mm}, const size_t @var{k1}, const size_t @var{k2}, const size_t @var{n1}, const size_t @var{n2})
@comment This function modifies and existing matrix view @var{m} to form a new
@comment view of a matrix @var{mm}, starting from element (@var{k1},@var{k2}).
@comment The matrix view has @var{n1} rows and @var{n2} columns.  Any existing
@comment view in @var{m} will be lost as a result of this function.
@comment @end deftypefun
@comment 
@comment @deftypefun int gsl_matrix_view_from_vector (gsl_matrix * @var{m}, gsl_vector * @var{v}, const size_t @var{offset}, const size_t @var{n1}, const size_t @var{n2})
@comment This function modifies and existing matrix view @var{m} to form a new
@comment view of a vector @var{v}, starting from element @var{offset}.  The
@comment vector has @var{n1} rows and @var{n2} columns.  Any
@comment existing view in @var{m} will be lost as a result of this function.
@comment @end deftypefun
@comment 
@comment @deftypefun int gsl_matrix_view_from_array (gsl_matrix * @var{m}, double * @var{base}, const size_t @var{offset}, const size_t @var{n1}, const size_t @var{n2})
@comment This function modifies and existing matrix view @var{m} to form a new
@comment view of an array @var{base}, starting from element @var{offset}.  The
@comment matrix has @var{n1} rows and @var{n2} columns.  Any
@comment existing view in @var{m} will be lost as a result of this function.
@comment @end deftypefun
@comment 
@comment @deftypefun {gsl_matrix *} gsl_matrix_alloc_from_block (gsl_block * @var{b}, size_t @var{offset}, size_t @var{n1}, size_t @var{n2}, size_t @var{tda})
@comment This function creates a matrix as a slice of the block @var{b},
@comment returning a pointer to a newly initialized matrix struct.  The start of
@comment the matrix is offset by @var{offset} elements from the start of the
@comment block.  The matrix has @var{n1} rows and @var{n2} columns, with the
@comment physical number of columns in memory given by @var{tda}.
@comment Mathematically, the (@var{i},@var{j})-th element of the matrix is given by,
@comment 
@comment @example
@comment m(i,j) = b->data[offset + i*tda + j]
@comment @end example
@comment @noindent
@comment where the index @var{i} runs from 0 to @code{n1-1} and the index @var{j}
@comment runs from 0 to @code{n2-1}.
@comment 
@comment A null pointer is returned if the combined parameters
@comment (@var{offset},@var{n1},@var{n2},@var{tda}) overrun the end of the block
@comment or if insufficient memory is available to store the matrix.
@comment 
@comment The matrix is only a view of the block @var{b}, and the block is not
@comment owned by the matrix.  When the matrix is deallocated the block @var{b}
@comment will continue to exist.  This memory can only be deallocated by freeing
@comment the block itself.  Of course, this block should not be deallocated while
@comment the matrix is still in use.
@comment @end deftypefun
@comment 
@comment @deftypefun {gsl_matrix *} gsl_matrix_alloc_from_matrix (gsl_matrix * @var{m}, size_t @var{k1}, size_t @var{k2}, size_t @var{n1}, size_t @var{n2})
@comment 
@comment This function creates a matrix as a submatrix of the matrix @var{m},
@comment returning a pointer to a newly initialized matrix struct.  The upper-left
@comment element of the submatrix is the element (@var{k1},@var{k2}) of the
@comment original matrix.  The submatrix has @var{n1} rows and @var{n2} columns.
@comment The physical number of columns in memory given by @var{tda} is
@comment unchanged.  Mathematically, the (@var{i},@var{j})-th element of the
@comment new matrix is given by,
@comment 
@comment @example
@comment m'(i,j) = m->data[(k1*m->tda + k2) + i*m->tda + j]
@comment @end example
@comment @noindent
@comment where the index @var{i} runs from 0 to @code{n1-1} and the index @var{j}
@comment runs from 0 to @code{n2-1}.
@comment 
@comment A null pointer is returned if the combined parameters
@comment (@var{k1},@var{k2},@var{n1},@var{n2},@var{tda}) overrun the end of the
@comment original matrix or if insufficient memory is available to store the matrix.
@comment 
@comment The new matrix is only a view of the block underlying the existing
@comment matrix, @var{m}.  The block is not owned by the new matrix.  When the new
@comment matrix is deallocated the original matrix @var{m} and its block will
@comment continue to exist.  The original memory can only be deallocated by
@comment freeing the original matrix.  Of course, the original matrix should not
@comment be deallocated while the new matrix is still in use.
@comment @end deftypefun

@node Creating row and column views
@subsection Creating row and column views

In general there are two ways to access an object, by reference or by
copying.  The functions described in this section create vector views
which allow access to a row or column of a matrix by reference.
Modifying elements of the view is equivalent to modifying the matrix,
since both the vector view and the matrix point to the same memory
block.

@deftypefun gsl_vector_view gsl_matrix_row (gsl_matrix * @var{m}, size_t @var{i})
@deftypefunx {gsl_vector_const_view} gsl_matrix_const_row (const gsl_matrix * @var{m}, size_t @var{i})
These functions return a vector view of the @var{i}-th row of the matrix
@var{m}.  The @code{data} pointer of the new vector is set to null if
@var{i} is out of range.

The function @code{gsl_vector_const_row} is equivalent to
@code{gsl_matrix_row} but can be used for matrices which are declared
@code{const}.
@end deftypefun

@deftypefun gsl_vector_view gsl_matrix_column (gsl_matrix * @var{m}, size_t @var{j})
@deftypefunx {gsl_vector_const_view} gsl_matrix_const_column (const gsl_matrix * @var{m}, size_t @var{j})
These functions return a vector view of the @var{j}-th column of the
matrix @var{m}.  The @code{data} pointer of the new vector is set to
null if @var{j} is out of range.

The function @code{gsl_vector_const_column} is equivalent to
@code{gsl_matrix_column} but can be used for matrices which are declared
@code{const}.
@end deftypefun

@cindex matrix diagonal
@cindex diagonal, of a matrix
@deftypefun gsl_vector_view gsl_matrix_diagonal (gsl_matrix * @var{m})
@deftypefunx {gsl_vector_const_view} gsl_matrix_const_diagonal (const gsl_matrix * @var{m})
These functions returns a vector view of the diagonal of the matrix
@var{m}. The matrix @var{m} is not required to be square. For a
rectangular matrix the length of the diagonal is the same as the smaller
dimension of the matrix.

The function @code{gsl_matrix_const_diagonal} is equivalent to
@code{gsl_matrix_diagonal} but can be used for matrices which are
declared @code{const}.
@end deftypefun

@cindex matrix subdiagonal
@cindex subdiagonal, of a matrix
@deftypefun gsl_vector_view gsl_matrix_subdiagonal (gsl_matrix * @var{m}, size_t @var{k}) 
@deftypefunx {gsl_vector_const_view} gsl_matrix_const_subdiagonal (const gsl_matrix * @var{m}, size_t @var{k})
These functions return a vector view of the @var{k}-th subdiagonal of
the matrix @var{m}. The matrix @var{m} is not required to be square.
The diagonal of the matrix corresponds to @math{k = 0}.

The function @code{gsl_matrix_const_subdiagonal} is equivalent to
@code{gsl_matrix_subdiagonal} but can be used for matrices which are
declared @code{const}.
@end deftypefun

@cindex matrix superdiagonal
@cindex superdiagonal, matrix
@deftypefun gsl_vector_view gsl_matrix_superdiagonal (gsl_matrix * @var{m}, size_t @var{k}) 
@deftypefunx {gsl_vector_const_view} gsl_matrix_const_superdiagonal (const gsl_matrix * @var{m}, size_t @var{k})
These functions return a vector view of the @var{k}-th superdiagonal of
the matrix @var{m}. The matrix @var{m} is not required to be square. The
diagonal of the matrix corresponds to @math{k = 0}.

The function @code{gsl_matrix_const_superdiagonal} is equivalent to
@code{gsl_matrix_superdiagonal} but can be used for matrices which are
declared @code{const}.
@end deftypefun

@comment @deftypefun {gsl_vector *} gsl_vector_alloc_row_from_matrix (gsl_matrix * @var{m}, size_t @var{i})
@comment This function allocates a new @code{gsl_vector} struct which points to
@comment the @var{i}-th row of the matrix @var{m}.
@comment @end deftypefun
@comment 
@comment @deftypefun {gsl_vector *} gsl_vector_alloc_col_from_matrix (gsl_matrix * @var{m}, size_t @var{j})
@comment This function allocates a new @code{gsl_vector} struct which points to
@comment the @var{j}-th column of the matrix @var{m}.
@comment @end deftypefun

@node Copying matrices
@subsection Copying matrices

@deftypefun int gsl_matrix_memcpy (gsl_matrix * @var{dest}, const gsl_matrix * @var{src})
This function copies the elements of the matrix @var{src} into the
matrix @var{dest}.  The two matrices must have the same size.
@end deftypefun

@deftypefun int gsl_matrix_swap (gsl_matrix * @var{m1}, gsl_matrix * @var{m2})
This function exchanges the elements of the matrices @var{m1} and
@var{m2} by copying.  The two matrices must have the same size.
@end deftypefun

@node Copying rows and columns
@subsection Copying rows and columns

The functions described in this section copy a row or column of a matrix
into a vector.  This allows the elements of the vector and the matrix to
be modified independently.  Note that if the matrix and the vector point
to overlapping regions of memory then the result will be undefined.  The
same effect can be achieved with more generality using
@code{gsl_vector_memcpy} with vector views of rows and columns.

@deftypefun int gsl_matrix_get_row (gsl_vector * @var{v}, const gsl_matrix * @var{m}, size_t @var{i})
This function copies the elements of the @var{i}-th row of the matrix
@var{m} into the vector @var{v}.  The length of the vector must be the
same as the length of the row.
@end deftypefun

@deftypefun int gsl_matrix_get_col (gsl_vector * @var{v}, const gsl_matrix * @var{m}, size_t @var{j})
This function copies the elements of the @var{j}-th column of the matrix
@var{m} into the vector @var{v}.  The length of the vector must be the
same as the length of the column.
@end deftypefun

@deftypefun int gsl_matrix_set_row (gsl_matrix * @var{m}, size_t @var{i}, const gsl_vector * @var{v})
This function copies the elements of the vector @var{v} into the
@var{i}-th row of the matrix @var{m}.  The length of the vector must be
the same as the length of the row.
@end deftypefun

@deftypefun int gsl_matrix_set_col (gsl_matrix * @var{m}, size_t @var{j}, const gsl_vector * @var{v})
This function copies the elements of the vector @var{v} into the
@var{j}-th column of the matrix @var{m}.  The length of the vector must be
the same as the length of the column.
@end deftypefun

@node Exchanging rows and columns
@subsection Exchanging rows and columns

The following functions can be used to exchange the rows and columns of
a matrix.

@deftypefun int gsl_matrix_swap_rows (gsl_matrix * @var{m}, size_t @var{i}, size_t @var{j})
This function exchanges the @var{i}-th and @var{j}-th rows of the matrix
@var{m} in-place.
@end deftypefun

@deftypefun int gsl_matrix_swap_columns (gsl_matrix * @var{m}, size_t @var{i}, size_t @var{j})
This function exchanges the @var{i}-th and @var{j}-th columns of the
matrix @var{m} in-place.
@end deftypefun

@deftypefun int gsl_matrix_swap_rowcol (gsl_matrix * @var{m}, size_t @var{i}, size_t @var{j})
This function exchanges the @var{i}-th row and @var{j}-th column of the
matrix @var{m} in-place.  The matrix must be square for this operation to
be possible.
@end deftypefun

@deftypefun int gsl_matrix_transpose_memcpy (gsl_matrix * @var{dest}, const gsl_matrix * @var{src})
This function makes the matrix @var{dest} the transpose of the matrix
@var{src} by copying the elements of @var{src} into @var{dest}.  This
function works for all matrices provided that the dimensions of the matrix
@var{dest} match the transposed dimensions of the matrix @var{src}.
@end deftypefun

@deftypefun int gsl_matrix_transpose (gsl_matrix * @var{m})
This function replaces the matrix @var{m} by its transpose by copying
the elements of the matrix in-place.  The matrix must be square for this
operation to be possible.
@end deftypefun

@node Matrix operations
@subsection Matrix operations

The following operations are defined for real and complex matrices.

@deftypefun int gsl_matrix_add (gsl_matrix * @var{a}, const gsl_matrix * @var{b})
This function adds the elements of matrix @var{b} to the elements of
matrix @var{a}, @math{a'(i,j) = a(i,j) + b(i,j)}. The two matrices must have the
same dimensions.
@end deftypefun

@deftypefun int gsl_matrix_sub (gsl_matrix * @var{a}, const gsl_matrix * @var{b})
This function subtracts the elements of matrix @var{b} from the elements of
matrix @var{a}, @math{a'(i,j) = a(i,j) - b(i,j)}. The two matrices must have the
same dimensions.
@end deftypefun

@deftypefun int gsl_matrix_mul_elements (gsl_matrix * @var{a}, const gsl_matrix * @var{b})
This function multiplies the elements of matrix @var{a} by the elements of
matrix @var{b}, @math{a'(i,j) = a(i,j) * b(i,j)}. The two matrices must have the
same dimensions.
@end deftypefun

@deftypefun int gsl_matrix_div_elements (gsl_matrix * @var{a}, const gsl_matrix * @var{b})
This function divides the elements of matrix @var{a} by the elements of
matrix @var{b}, @math{a'(i,j) = a(i,j) / b(i,j)}. The two matrices must have the
same dimensions.
@end deftypefun

@deftypefun int gsl_matrix_scale (gsl_matrix * @var{a}, const double @var{x})
This function multiplies the elements of matrix @var{a} by the constant
factor @var{x}, @math{a'(i,j) = x a(i,j)}.
@end deftypefun

@deftypefun int gsl_matrix_add_constant (gsl_matrix * @var{a}, const double @var{x})
This function adds the constant value @var{x} to the elements of the
matrix @var{a}, @math{a'(i,j) = a(i,j) + x}.
@end deftypefun

@node  Finding maximum and minimum elements of matrices
@subsection Finding maximum and minimum elements of matrices

The following operations are only defined for real matrices.

@deftypefun double gsl_matrix_max (const gsl_matrix * @var{m})
This function returns the maximum value in the matrix @var{m}.
@end deftypefun

@deftypefun double gsl_matrix_min (const gsl_matrix * @var{m})
This function returns the minimum value in the matrix @var{m}.
@end deftypefun

@deftypefun void gsl_matrix_minmax (const gsl_matrix * @var{m}, double * @var{min_out}, double * @var{max_out})
This function returns the minimum and maximum values in the matrix
@var{m}, storing them in @var{min_out} and @var{max_out}.
@end deftypefun

@deftypefun void gsl_matrix_max_index (const gsl_matrix * @var{m}, size_t * @var{imax}, size_t * @var{jmax})
This function returns the indices of the maximum value in the matrix
@var{m}, storing them in @var{imax} and @var{jmax}.  When there are
several equal maximum elements then the first element found is returned,
searching in row-major order.
@end deftypefun

@deftypefun void gsl_matrix_min_index (const gsl_matrix * @var{m}, size_t * @var{imin}, size_t * @var{jmin})
This function returns the indices of the minimum value in the matrix
@var{m}, storing them in @var{imin} and @var{jmin}.  When there are
several equal minimum elements then the first element found is returned,
searching in row-major order.
@end deftypefun

@deftypefun void gsl_matrix_minmax_index (const gsl_matrix * @var{m}, size_t * @var{imin}, size_t * @var{jmin}, size_t * @var{imax}, size_t * @var{jmax})
This function returns the indices of the minimum and maximum values in
the matrix @var{m}, storing them in (@var{imin},@var{jmin}) and
(@var{imax},@var{jmax}). When there are several equal minimum or maximum
elements then the first elements found are returned, searching in
row-major order.
@end deftypefun

@node Matrix properties
@subsection Matrix properties

@deftypefun int gsl_matrix_isnull (const gsl_matrix * @var{m})
@deftypefunx int gsl_matrix_ispos (const gsl_matrix * @var{m})
@deftypefunx int gsl_matrix_isneg (const gsl_matrix * @var{m})
These functions return 1 if all the elements of the matrix @var{m} are
zero, strictly positive, or strictly negative respectively, and 0
otherwise. To test for a non-negative matrix, use the expression
@code{!gsl_matrix_isneg(m)}.  To test whether a matrix is
positive-definite, use the Cholesky decomposition (@pxref{Cholesky Decomposition}).
@end deftypefun

@node Example programs for matrices
@subsection Example programs for matrices

The program below shows how to allocate, initialize and read from a matrix
using the functions @code{gsl_matrix_alloc}, @code{gsl_matrix_set} and
@code{gsl_matrix_get}.

@example
@verbatiminclude examples/matrix.c
@end example
@comment

@noindent
Here is the output from the program.  The final loop attempts to read
outside the range of the matrix @code{m}, and the error is trapped by
the range-checking code in @code{gsl_matrix_get}.

@example
$ ./a.out
m(0,0) = 0.23
m(0,1) = 1.23
m(0,2) = 2.23
m(1,0) = 100.23
m(1,1) = 101.23
m(1,2) = 102.23
...
m(9,2) = 902.23
gsl: matrix_source.c:13: ERROR: first index out of range
Default GSL error handler invoked.
Aborted (core dumped)
@end example
@comment

@noindent
The next program shows how to write a matrix to a file.

@example
@verbatiminclude examples/matrixw.c
@end example
@comment

@noindent
After running this program the file @file{test.dat} should contain the
elements of @code{m}, written in binary format.  The matrix which is read
back in using the function @code{gsl_matrix_fread} should be exactly
equal to the original matrix.

The following program demonstrates the use of vector views.  The program
computes the column norms of a matrix.

@example
@verbatiminclude examples/vectorview.c
@end example

@noindent
Here is the output of the program, 

@example
$ ./a.out
@verbatiminclude examples/vectorview.out
@end example

@noindent
The results can be confirmed using @sc{gnu octave},

@example
$ octave
GNU Octave, version 2.0.16.92
octave> m = sin(0:9)' * ones(1,10) 
               + ones(10,1) * cos(0:9); 
octave> sqrt(sum(m.^2))
ans =
  4.3146  3.1205  2.1932  3.2611  2.5342  2.5728
  4.2047  3.6520  2.0852  3.0731
@end example


@node Vector and Matrix References and Further Reading
@section References and Further Reading

The block, vector and matrix objects in GSL follow the @code{valarray}
model of C++.  A description of this model can be found in the following
reference,

@itemize @asis
@item
B. Stroustrup,
@cite{The C++ Programming Language} (3rd Ed), 
Section 22.4 Vector Arithmetic.
Addison-Wesley 1997, ISBN 0-201-88954-4.
@end itemize