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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 BLAS routines.

All the functions are available for each of the standard data-types.
The versions for `double`

have the prefix `gsl_block`

,
`gsl_vector`

and `gsl_matrix`

. Similarly the versions for
single-precision `float`

arrays have the prefix
`gsl_block_float`

, `gsl_vector_float`

and
`gsl_matrix_float`

. The full list of available types is given
below,

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

Corresponding types exist for the `gsl_vector`

and
`gsl_matrix`

functions.

For consistency all memory is allocated through a `gsl_block`

structure. The structure contains two components, the size of an area of
memory and a pointer to the memory. The `gsl_block`

structure looks
like this,

typedef struct { size_t size; double * data; } gsl_block;

Vectors and matrices are made by **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 **stride**.

The functions for allocating and deallocating blocks are defined in
``gsl_block.h'`

The functions for allocating memory to a block follow the style of
`malloc`

and `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 `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 `alloc`

.

__Function:__gsl_block ***gsl_block_alloc***(size_t*`n`)-
This function allocates memory for a block of
`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`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.

__Function:__gsl_block ***gsl_block_calloc***(size_t*`n`)- This function allocates memory for a block and initializes all the elements of the block to zero.

__Function:__void**gsl_block_free***(gsl_block **`b`)-
This function frees the memory used by a block
`b`previously allocated with`gsl_block_alloc`

or`gsl_block_calloc`

.

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

__Function:__int**gsl_block_fwrite***(FILE **`stream`, const gsl_block *`b`)-
This function writes the elements of the block
`b`to the stream`stream`in binary format. The return value is 0 for success and`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.

__Function:__int**gsl_block_fread***(FILE **`stream`, gsl_block *`b`)-
This function reads into the block
`b`from the open stream`stream`in binary format. The block`b`must be preallocated with the correct length since the function uses the size of`b`to determine how many bytes to read. The return value is 0 for success and`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.

__Function:__int**gsl_block_fprintf***(FILE **`stream`, const gsl_block *`b`, const char *`format`)-
This function writes the elements of the block
`b`line-by-line to the stream`stream`using the format specifier`format`, which should be one of the`%g`

,`%e`

or`%f`

formats for floating point numbers and`%d`

for integers. The function returns 0 for success and`GSL_EFAILED`

if there was a problem writing to the file.

__Function:__int**gsl_block_fscanf***(FILE **`stream`, gsl_block *`b`)-
This function reads formatted data from the stream
`stream`into the block`b`. The block`b`must be preallocated with the correct length since the function uses the size of`b`to determine how many numbers to read. The function returns 0 for success and`GSL_EFAILED`

if there was a problem reading from the file.

The following program shows how to allocate a block,

#include <stdio.h> #include <gsl/gsl_block.h> int main (void) { gsl_block * b = gsl_block_alloc (100); printf("length of block = %u\n", b->size); printf("block data address = %#x\n", b->data); gsl_block_free (b); return 0; }

Here is the output from the program,

length of block = 100 block data address = 0x804b0d8

Vectors are defined by a `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 `gsl_vector`

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

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

The `size` is simply the number of vector elements. The range of
valid indices runs from 0 to `size-1`

. The `stride` is the
step-size from one element to the next in physical memory, measured in
units of the appropriate datatype. The pointer `data` gives the
location of the first element of the vector in memory. The pointer
`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
`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 `owner` field is zero and any underlying block will not be
deallocated.

The functions for allocating and accessing vectors are defined in
``gsl_vector.h'`

The functions for allocating memory to a vector follow the style of
`malloc`

and `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 `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 `alloc`

.

__Function:__gsl_vector ***gsl_vector_alloc***(size_t*`n`)-
This function creates a vector of length
`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`block`component of the vector struct. The block is "owned" by the vector, and will be deallocated when the vector is deallocated.

__Function:__gsl_vector ***gsl_vector_calloc***(size_t*`n`)-
This function allocates memory for a vector of length
`n`and initializes all the elements of the vector to zero.

__Function:__void**gsl_vector_free***(gsl_vector **`v`)-
This function frees a previously allocated vector
`v`. If the vector was created using`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.

Unlike 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 extension `checkergcc`

but it is
not available on every platform. The functions `gsl_vector_get`

and `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 ``gsl_vector.h'` and declared `extern inline`

to
eliminate function-call overhead. If necessary you can turn off range
checking completely without modifying any source files by recompiling
your program with the preprocessor definition
`GSL_RANGE_CHECK_OFF`

. Provided your compiler supports inline
functions the effect of turning off range checking is to replace calls
to `gsl_vector_get(v,i)`

by `v->data[i*v->stride]`

and and
calls to `gsl_vector_set(v,i,x)`

by `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.

__Function:__double**gsl_vector_get***(const gsl_vector **`v`, size_t`i`)-
This function returns the
`i`-th element of a vector`v`. If`i`lies outside the allowed range of 0 to`n-1`then the error handler is invoked and 0 is returned.

__Function:__void**gsl_vector_set***(gsl_vector **`v`, size_t`i`, double`x`)-
This function sets the value of the
`i`-th element of a vector`v`to`x`. If`i`lies outside the allowed range of 0 to`n-1`then the error handler is invoked.

__Function:__double ***gsl_vector_ptr***(gsl_vector **`v`, size_t`i`)__Function:__const double ***gsl_vector_ptr***(const gsl_vector **`v`, size_t`i`)-
These functions return a pointer to the
`i`-th element of a vector`v`. If`i`lies outside the allowed range of 0 to`n-1`then the error handler is invoked and a null pointer is returned.

__Function:__void**gsl_vector_set_all***(gsl_vector **`v`, double`x`)-
This function sets all the elements of the vector
`v`to the value`x`.

__Function:__void**gsl_vector_set_zero***(gsl_vector **`v`)-
This function sets all the elements of the vector
`v`to zero.

__Function:__int**gsl_vector_set_basis***(gsl_vector **`v`, size_t`i`)-
This function makes a basis vector by setting all the elements of the
vector
`v`to zero except for the`i`-th element which is set to one.

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

__Function:__int**gsl_vector_fwrite***(FILE **`stream`, const gsl_vector *`v`)-
This function writes the elements of the vector
`v`to the stream`stream`in binary format. The return value is 0 for success and`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.

__Function:__int**gsl_vector_fread***(FILE **`stream`, gsl_vector *`v`)-
This function reads into the vector
`v`from the open stream`stream`in binary format. The vector`v`must be preallocated with the correct length since the function uses the size of`v`to determine how many bytes to read. The return value is 0 for success and`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.

__Function:__int**gsl_vector_fprintf***(FILE **`stream`, const gsl_vector *`v`, const char *`format`)-
This function writes the elements of the vector
`v`line-by-line to the stream`stream`using the format specifier`format`, which should be one of the`%g`

,`%e`

or`%f`

formats for floating point numbers and`%d`

for integers. The function returns 0 for success and`GSL_EFAILED`

if there was a problem writing to the file.

__Function:__int**gsl_vector_fscanf***(FILE **`stream`, gsl_vector *`v`)-
This function reads formatted data from the stream
`stream`into the vector`v`. The vector`v`must be preallocated with the correct length since the function uses the size of`v`to determine how many numbers to read. The function returns 0 for success and`GSL_EFAILED`

if there was a problem reading from the file.

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
`gsl_vector_view`

and a constant vector view has the type
`gsl_vector_const_view`

. In both cases the elements of the view
can be accessed as a `gsl_vector`

using the `vector`

component
of the view object. A pointer to a vector of type `gsl_vector *`

or `const gsl_vector *`

can be obtained by taking the address of
this component with the `&`

operator.

__Function:__gsl_vector_view**gsl_vector_subvector***(gsl_vector **`v`, size_t`offset`, size_t`n`)__Function:__gsl_vector_const_view**gsl_vector_const_subvector***(const gsl_vector **`v`, size_t`offset`, size_t`n`)-
These functions return a vector view of a subvector of another vector
`v`. The start of the new vector is offset by`offset`elements from the start of the original vector. The new vector has`n`elements. Mathematically, the`i`-th element of the new vector`v'`is given by,v'(i) = v->data[(offset + i)*v->stride]

where the index

`i`runs from 0 to`n-1`

.The

`data`

pointer of the returned vector struct is set to null if the combined parameters (`offset`,`n`) overrun the end of the original vector.The new vector is only a view of the block underlying the original vector,

`v`. The block containing the elements of`v`is not owned by the new vector. When the view goes out of scope the original vector`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

`gsl_vector_const_subvector`

is equivalent to`gsl_vector_subvector`

but can be used for vectors which are declared`const`

.

__Function:__gsl_vector**gsl_vector_subvector_with_stride***(gsl_vector **`v`, size_t`offset`, size_t`stride`, size_t`n`)__Function:__gsl_vector_const_view**gsl_vector_const_subvector_with_stride***(const gsl_vector **`v`, size_t`offset`, size_t`stride`, size_t`n`)-
These functions return a vector view of a subvector of another vector
`v`with an additional stride argument. The subvector is formed in the same way as for`gsl_vector_subvector`

but the new vector has`n`elements with a step-size of`stride`from one element to the next in the original vector. Mathematically, the`i`-th element of the new vector`v'`is given by,v'(i) = v->data[(offset + i*stride)*v->stride]

where the index

`i`runs from 0 to`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

`v`

of length`n`

, while leaving the odd elements untouched,gsl_vector_view v_even = gsl_vector_subvector_with_stride (v, 0, 2, n/2); gsl_vector_set_zero (&v_even.vector);

A vector view can be passed to any subroutine which takes a vector argument just as a directly allocated vector would be, using

`&`

`view``.vector`

. For example, the following code computes the norm of odd elements of`v`

using the BLAS routine DNRM2,gsl_vector_view v_odd = gsl_vector_subvector_with_stride (v, 1, 2, n/2); double r = gsl_blas_dnrm2 (&v_odd.vector);

The function

`gsl_vector_const_subvector_with_stride`

is equivalent to`gsl_vector_subvector_with_stride`

but can be used for vectors which are declared`const`

.

__Function:__gsl_vector_view**gsl_vector_complex_real***(gsl_vector_complex **`v`)__Function:__gsl_vector_const_view**gsl_vector_complex_const_real***(const gsl_vector_complex **`v`)-
These functions return a vector view of the real parts of the complex
vector
`v`.The function

`gsl_vector_complex_const_real`

is equivalent to`gsl_vector_complex_real`

but can be used for vectors which are declared`const`

.

__Function:__gsl_vector_view**gsl_vector_complex_imag***(gsl_vector_complex **`v`)__Function:__gsl_vector_const_view**gsl_vector_complex_const_imag***(const gsl_vector_complex **`v`)-
These functions return a vector view of the imaginary parts of the
complex vector
`v`.The function

`gsl_vector_complex_const_imag`

is equivalent to`gsl_vector_complex_imag`

but can be used for vectors which are declared`const`

.

__Function:__gsl_vector_view**gsl_vector_view_array***(double **`base`, size_t`n`)__Function:__gsl_vector_const_view**gsl_vector_const_view_array***(const double **`base`, size_t`n`)-
These functions return a vector view of an array. The start of the new
vector is given by
`base`and has`n`elements. Mathematically, the`i`-th element of the new vector`v'`is given by,v'(i) = base[i]

where the index

`i`runs from 0 to`n-1`

.The array containing the elements of

`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`base`. Of course, the original array should not be deallocated while the view is still in use.The function

`gsl_vector_const_view_array`

is equivalent to`gsl_vector_view_array`

but can be used for arrays which are declared`const`

.

__Function:__gsl_vector_view**gsl_vector_view_array_with_stride***(double **`base`, size_t`stride`, size_t`n`)__Function:__gsl_vector_const_view**gsl_vector_const_view_array_with_stride***(const double **`base`, size_t`stride`, size_t`n`)-
These functions return a vector view of an array
`base`with an additional stride argument. The subvector is formed in the same way as for`gsl_vector_view_array`

but the new vector has`n`elements with a step-size of`stride`from one element to the next in the original array. Mathematically, the`i`-th element of the new vector`v'`is given by,v'(i) = base[i*stride]

where the index

`i`runs from 0 to`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

`&`

`view``.vector`

.The function

`gsl_vector_const_view_array_with_stride`

is equivalent to`gsl_vector_view_array_with_stride`

but can be used for arrays which are declared`const`

.

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

__Function:__int**gsl_vector_memcpy***(gsl_vector **`dest`, const gsl_vector *`src`)-
This function copies the elements of the vector
`src`into the vector`dest`. The two vectors must have the same length.

__Function:__int**gsl_vector_swap***(gsl_vector **`v`, gsl_vector *`w`)-
This function exchanges the elements of the vectors
`v`and`w`by copying. The two vectors must have the same length.

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

__Function:__int**gsl_vector_swap_elements***(gsl_vector **`v`, size_t i, size_t j)-
This function exchanges the
`i`-th and`j`-th elements of the vector`v`in-place.

__Function:__int**gsl_vector_reverse***(gsl_vector **`v`)-
This function reverses the order of the elements of the vector
`v`.

The following operations are only defined for real vectors.

__Function:__int**gsl_vector_add***(gsl_vector **`a`, const gsl_vector *`b`)-
This function adds the elements of vector
`b`to the elements of vector`a`, a'_i = a_i + b_i. The two vectors must have the same length.

__Function:__int**gsl_vector_sub***(gsl_vector **`a`, const gsl_vector *`b`)-
This function subtracts the elements of vector
`b`from the elements of vector`a`, a'_i = a_i - b_i. The two vectors must have the same length.

__Function:__int**gsl_vector_mul***(gsl_vector **`a`, const gsl_vector *`b`)-
This function multiplies the elements of vector
`a`by the elements of vector`b`, a'_i = a_i * b_i. The two vectors must have the same length.

__Function:__int**gsl_vector_div***(gsl_vector **`a`, const gsl_vector *`b`)-
This function divides the elements of vector
`a`by the elements of vector`b`, a'_i = a_i / b_i. The two vectors must have the same length.

__Function:__int**gsl_vector_scale***(gsl_vector **`a`, const double`x`)-
This function multiplies the elements of vector
`a`by the constant factor`x`, a'_i = x a_i.

__Function:__int**gsl_vector_add_constant***(gsl_vector **`a`, const double`x`)-
This function adds the constant value
`x`to the elements of the vector`a`, a'_i = a_i + x.

__Function:__double**gsl_vector_max***(const gsl_vector **`v`)-
This function returns the maximum value in the vector
`v`.

__Function:__double**gsl_vector_min***(const gsl_vector **`v`)-
This function returns the minimum value in the vector
`v`.

__Function:__void**gsl_vector_minmax***(const gsl_vector **`v`, double *`min_out`, double *`max_out`)-
This function returns the minimum and maximum values in the vector
`v`, storing them in`min_out`and`max_out`.

__Function:__size_t**gsl_vector_max_index***(const gsl_vector **`v`)-
This function returns the index of the maximum value in the vector
`v`. When there are several equal maximum elements then the lowest index is returned.

__Function:__size_t**gsl_vector_min_index***(const gsl_vector **`v`)-
This function returns the index of the minimum value in the vector
`v`. When there are several equal minimum elements then the lowest index is returned.

__Function:__void**gsl_vector_minmax_index***(const gsl_vector **`v`, size_t *`imin`, size_t *`imax`)-
This function returns the indices of the minimum and maximum values in
the vector
`v`, storing them in`imin`and`imax`. When there are several equal minimum or maximum elements then the lowest indices are returned.

__Function:__int**gsl_vector_isnull***(const gsl_vector **`v`)-
This function returns 1 if all the elements of the vector
`v`are zero, and 0 otherwise.

This program shows how to allocate, initialize and read from a vector
using the functions `gsl_vector_alloc`

, `gsl_vector_set`

and
`gsl_vector_get`

.

#include <stdio.h> #include <gsl/gsl_vector.h> int main (void) { int i; gsl_vector * v = gsl_vector_alloc (3); for (i = 0; i < 3; i++) { gsl_vector_set (v, i, 1.23 + i); } for (i = 0; i < 100; i++) { printf("v_%d = %g\n", i, gsl_vector_get (v, i)); } return 0; }

Here is the output from the program. The final loop attempts to read
outside the range of the vector `v`

, and the error is trapped by
the range-checking code in `gsl_vector_get`

.

v_0 = 1.23 v_1 = 2.23 v_2 = 3.23 gsl: vector_source.c:12: ERROR: index out of range IOT trap/Abort (core dumped)

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

#include <stdio.h> #include <gsl/gsl_vector.h> int main (void) { int i; gsl_vector * v = gsl_vector_alloc (100); for (i = 0; i < 100; i++) { gsl_vector_set (v, i, 1.23 + i); } { FILE * f = fopen("test.dat", "w"); gsl_vector_fprintf (f, v, "%.5g"); fclose (f); } return 0; }

After running this program the file ``test.dat'` should contain the
elements of `v`

, written using the format specifier
`%.5g`

. The vector could then be read back in using the function
`gsl_vector_fscanf (f, v)`

.

Matrices are defined by a `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 `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, `data`, a pointer to
the block owned by the matrix `block`, if any, and an ownership
flag, `owner`. The physical dimension determines the memory layout
and can differ from the matrix dimension to allow the use of
submatrices. The `gsl_matrix`

structure is very simple and looks
like this,

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

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 FORTRAN
stores arrays in column-major order. The number of rows is `size1`.
The range of valid row indices runs from 0 to `size1-1`

. Similarly
`size2` is the number of columns. The range of valid column indices
runs from 0 to `size2-1`

. The physical row dimension `tda`, or
**trailing dimension**, specifies the size of a row of the matrix as
laid out in memory.

For example, in the following matrix `size1` is 3, `size2` is 4,
and `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.

00 01 02 03 XX XX XX XX 10 11 12 13 XX XX XX XX 20 21 22 23 XX XX XX XX

Each unused memory location is represented by "`XX`

". The
pointer `data` gives the location of the first element of the matrix
in memory. The pointer `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 `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 `owner` field is
zero and any underlying block will not be freed.

The functions for allocating and accessing matrices are defined in
``gsl_matrix.h'`

The functions for allocating memory to a matrix follow the style of
`malloc`

and `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
`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 `alloc`

.

__Function:__gsl_matrix ***gsl_matrix_alloc***(size_t*`n1`, size_t`n2`)-
This function creates a matrix of size
`n1`rows by`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`block`component of the matrix struct. The block is "owned" by the matrix, and will be deallocated when the matrix is deallocated.

__Function:__gsl_matrix ***gsl_matrix_calloc***(size_t*`n1`, size_t`n2`)-
This function allocates memory for a matrix of size
`n1`rows by`n2`columns and initializes all the elements of the matrix to zero.

__Function:__void**gsl_matrix_free***(gsl_matrix **`m`)-
This function frees a previously allocated matrix
`m`. If the matrix was created using`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 functions for accessing the elements of a matrix use the same range
checking system as vectors. You turn off range checking by recompiling
your program with the preprocessor definition
`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 `gsl_matrix_get(m,i,j)`

and
`gsl_matrix_set(m,i,j,x)`

is

m->data[i * m->tda + j]

where `tda` is the physical row-length of the matrix.

__Function:__double**gsl_matrix_get***(const gsl_matrix **`m`, size_t`i`, size_t`j`)-
This function returns the (
`i`,`j`)th element of a matrix`m`. If`i`or`j`lie outside the allowed range of 0 to`n1-1`and 0 to`n2-1`then the error handler is invoked and 0 is returned.

__Function:__void**gsl_matrix_set***(gsl_matrix **`m`, size_t`i`, size_t`j`, double`x`)-
This function sets the value of the (
`i`,`j`)th element of a matrix`m`to`x`. If`i`or`j`lies outside the allowed range of 0 to`n1-1`and 0 to`n2-1`then the error handler is invoked.

__Function:__double ***gsl_matrix_ptr***(gsl_matrix **`m`, size_t`i`, size_t`j`)__Function:__const double ***gsl_matrix_ptr***(const gsl_matrix **`m`, size_t`i`, size_t`j`)-
These functions return a pointer to the (
`i`,`j`)th element of a matrix`m`. If`i`or`j`lie outside the allowed range of 0 to`n1-1`and 0 to`n2-1`then the error handler is invoked and a null pointer is returned.

__Function:__void**gsl_matrix_set_all***(gsl_matrix **`m`, double`x`)-
This function sets all the elements of the matrix
`m`to the value`x`.

__Function:__void**gsl_matrix_set_zero***(gsl_matrix **`m`)-
This function sets all the elements of the matrix
`m`to zero.

__Function:__void**gsl_matrix_set_identity***(gsl_matrix **`m`)-
This function sets the elements of the matrix
`m`to the corresponding elements of the identity matrix, 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.

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

__Function:__int**gsl_matrix_fwrite***(FILE **`stream`, const gsl_matrix *`m`)-
This function writes the elements of the matrix
`m`to the stream`stream`in binary format. The return value is 0 for success and`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.

__Function:__int**gsl_matrix_fread***(FILE **`stream`, gsl_matrix *`m`)-
This function reads into the matrix
`m`from the open stream`stream`in binary format. The matrix`m`must be preallocated with the correct dimensions since the function uses the size of`m`to determine how many bytes to read. The return value is 0 for success and`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.

__Function:__int**gsl_matrix_fprintf***(FILE **`stream`, const gsl_matrix *`m`, const char *`format`)-
This function writes the elements of the matrix
`m`line-by-line to the stream`stream`using the format specifier`format`, which should be one of the`%g`

,`%e`

or`%f`

formats for floating point numbers and`%d`

for integers. The function returns 0 for success and`GSL_EFAILED`

if there was a problem writing to the file.

__Function:__int**gsl_matrix_fscanf***(FILE **`stream`, gsl_matrix *`m`)-
This function reads formatted data from the stream
`stream`into the matrix`m`. The matrix`m`must be preallocated with the correct dimensions since the function uses the size of`m`to determine how many numbers to read. The function returns 0 for success and`GSL_EFAILED`

if there was a problem reading from the file.

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
`gsl_matrix_view`

and a constant matrix view has the type
`gsl_matrix_const_view`

. In both cases the elements of the view
can by accessed using the `matrix`

component of the view object. A
pointer `gsl_matrix *`

or `const gsl_matrix *`

can be obtained
by taking the address of the `matrix`

component with the `&`

operator. In addition to matrix views it is also possible to create
vector views of a matrix, such as row or column views.

__Function:__gsl_matrix_view**gsl_matrix_submatrix***(gsl_matrix **`m`, size_t`i`, size_t`j`, size_t`n1`, size_t`n2`)__Function:__gsl_matrix_const_view**gsl_matrix_const_submatrix***(const gsl_matrix **`m`, size_t`i`, size_t`j`, size_t`n1`, size_t`n2`)-
These functions return a matrix view of a submatrix of the matrix
`m`. The upper-left element of the submatrix is the element (`k1`,`k2`) of the original matrix. The submatrix has`n1`rows and`n2`columns. The physical number of columns in memory given by`tda`is unchanged. Mathematically, the (`i`,`j`)-th element of the new matrix is given by,m'(i,j) = m->data[(k1*m->tda + k2) + i*m->tda + j]

where the index

`i`runs from 0 to`n1-1`

and the index`j`runs from 0 to`n2-1`

.The

`data`

pointer of the returned matrix struct is set to null if the combined parameters (`k1`,`k2`,`n1`,`n2`,`tda`) overrun the ends of the original matrix.The new matrix view is only a view of the block underlying the existing matrix,

`m`. The block containing the elements of`m`is not owned by the new matrix view. When the view goes out of scope the original matrix`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

`gsl_matrix_const_submatrix`

is equivalent to`gsl_matrix_submatrix`

but can be used for matrices which are declared`const`

.

__Function:__gsl_matrix_view**gsl_matrix_view_array***(double **`base`, size_t`n1`, size_t`n2`)__Function:__gsl_matrix_const_view**gsl_matrix_const_view_array***(const double **`base`, size_t`n1`, size_t`n2`)-
These functions return a matrix view of the array
`base`. The matrix has`n1`rows and`n2`columns. The physical number of columns in memory is also given by`n2`. Mathematically, the (`i`,`j`)-th element of the new matrix is given by,m'(i,j) = base[i*n2 + j]

where the index

`i`runs from 0 to`n1-1`

and the index`j`runs from 0 to`n2-1`

.The new matrix is only a view of the array

`base`. When the view goes out of scope the original array`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

`gsl_matrix_const_view_array`

is equivalent to`gsl_matrix_view_array`

but can be used for matrices which are declared`const`

.

__Function:__gsl_matrix_view**gsl_matrix_view_array_with_tda***(double **`base`, size_t`n1`, size_t`n2`, size_t`tda`)__Function:__gsl_matrix_const_view**gsl_matrix_const_view_array_with_tda***(const double **`base`, size_t`n1`, size_t`n2`, size_t`tda`)-
These functions return a matrix view of the array
`base`with a physical number of columns`tda`which may differ from corresponding the dimension of the matrix. The matrix has`n1`rows and`n2`columns, and the physical number of columns in memory is given by`tda`. Mathematically, the (`i`,`j`)-th element of the new matrix is given by,m'(i,j) = base[i*tda + j]

where the index

`i`runs from 0 to`n1-1`

and the index`j`runs from 0 to`n2-1`

.The new matrix is only a view of the array

`base`. When the view goes out of scope the original array`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

`gsl_matrix_const_view_array_with_tda`

is equivalent to`gsl_matrix_view_array_with_tda`

but can be used for matrices which are declared`const`

.

__Function:__gsl_matrix_view**gsl_matrix_view_vector***(gsl_vector **`v`, size_t`n1`, size_t`n2`)__Function:__gsl_matrix_const_view**gsl_matrix_const_view_vector***(const gsl_vector **`v`, size_t`n1`, size_t`n2`)-
These functions return a matrix view of the vector
`v`. The matrix has`n1`rows and`n2`columns. The vector must have unit stride. The physical number of columns in memory is also given by`n2`. Mathematically, the (`i`,`j`)-th element of the new matrix is given by,m'(i,j) = v->data[i*n2 + j]

where the index

`i`runs from 0 to`n1-1`

and the index`j`runs from 0 to`n2-1`

.The new matrix is only a view of the vector

`v`. When the view goes out of scope the original vector`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

`gsl_matrix_const_view_vector`

is equivalent to`gsl_matrix_view_vector`

but can be used for matrices which are declared`const`

.

__Function:__gsl_matrix_view**gsl_matrix_view_vector_with_tda***(gsl_vector **`v`, size_t`n1`, size_t`n2`, size_t`tda`)__Function:__gsl_matrix_const_view**gsl_matrix_const_view_vector_with_tda***(const gsl_vector **`v`, size_t`n1`, size_t`n2`, size_t`tda`)-
These functions return a matrix view of the vector
`v`with a physical number of columns`tda`which may differ from the corresponding matrix dimension. The vector must have unit stride. The matrix has`n1`rows and`n2`columns, and the physical number of columns in memory is given by`tda`. Mathematically, the (`i`,`j`)-th element of the new matrix is given by,m'(i,j) = v->data[i*tda + j]

where the index

`i`runs from 0 to`n1-1`

and the index`j`runs from 0 to`n2-1`

.The new matrix is only a view of the vector

`v`. When the view goes out of scope the original vector`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

`gsl_matrix_const_view_vector_with_tda`

is equivalent to`gsl_matrix_view_vector_with_tda`

but can be used for matrices which are declared`const`

.

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.

__Function:__gsl_vector_view**gsl_matrix_row***(gsl_matrix **`m`, size_t`i`)__Function:__gsl_vector_const_view**gsl_matrix_const_row***(const gsl_matrix **`m`, size_t`i`)-
These functions return a vector view of the
`i`-th row of the matrix`m`. The`data`

pointer of the new vector is set to null if`i`is out of range.The function

`gsl_vector_const_row`

is equivalent to`gsl_matrix_row`

but can be used for matrices which are declared`const`

.

__Function:__gsl_vector_view**gsl_matrix_column***(gsl_matrix **`m`, size_t`j`)__Function:__gsl_vector_const_view**gsl_matrix_const_column***(const gsl_matrix **`m`, size_t`j`)-
These functions return a vector view of the
`j`-th column of the matrix`m`. The`data`

pointer of the new vector is set to null if`j`is out of range.The function

`gsl_vector_const_column`

equivalent to`gsl_matrix_column`

but can be used for matrices which are declared`const`

.

__Function:__gsl_vector_view**gsl_matrix_diagonal***(gsl_matrix **`m`)__Function:__gsl_vector_const_view**gsl_matrix_const_diagonal***(const gsl_matrix **`m`)-
These functions returns a vector view of the diagonal of the matrix
`m`. The matrix`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

`gsl_matrix_const_diagonal`

is equivalent to`gsl_matrix_diagonal`

but can be used for matrices which are declared`const`

.

__Function:__gsl_vector_view**gsl_matrix_subdiagonal***(gsl_matrix **`m`, size_t`k`)__Function:__gsl_vector_const_view**gsl_matrix_const_subdiagonal***(const gsl_matrix **`m`, size_t`k`)-
These functions return a vector view of the
`k`-th subdiagonal of the matrix`m`. The matrix`m`is not required to be square. The diagonal of the matrix corresponds to k = 0.The function

`gsl_matrix_const_subdiagonal`

is equivalent to`gsl_matrix_subdiagonal`

but can be used for matrices which are declared`const`

.

__Function:__gsl_vector_view**gsl_matrix_superdiagonal***(gsl_matrix **`m`, size_t`k`)__Function:__gsl_vector_const_view**gsl_matrix_const_superdiagonal***(const gsl_matrix **`m`, size_t`k`)-
These functions return a vector view of the
`k`-th superdiagonal of the matrix`m`. The matrix`m`is not required to be square. The diagonal of the matrix corresponds to k = 0.The function

`gsl_matrix_const_superdiagonal`

is equivalent to`gsl_matrix_superdiagonal`

but can be used for matrices which are declared`const`

.

__Function:__int**gsl_matrix_memcpy***(gsl_matrix **`dest`, const gsl_matrix *`src`)-
This function copies the elements of the matrix
`src`into the matrix`dest`. The two matrices must have the same size.

__Function:__int**gsl_matrix_swap***(gsl_matrix **`m1`, const gsl_matrix *`m2`)-
This function exchanges the elements of the matrices
`m1`and`m2`by copying. The two matrices must have the same size.

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
`gsl_vector_memcpy`

with vector views of rows and columns.

__Function:__int**gsl_matrix_get_row***(gsl_vector **`v`, const gsl_matrix *`m`, size_t`i`)-
This function copies the elements of the
`i`-th row of the matrix`m`into the vector`v`. The length of the vector must be the same as the length of the row.

__Function:__int**gsl_matrix_get_col***(gsl_vector **`v`, const gsl_matrix *`m`, size_t`j`)-
This function copies the elements of the
`i`-th column of the matrix`m`into the vector`v`. The length of the vector must be the same as the length of the column.

__Function:__int**gsl_matrix_set_row***(gsl_matrix **`m`, size_t`i`, const gsl_vector *`v`)-
This function copies the elements of the vector
`v`into the`i`-th row of the matrix`m`. The length of the vector must be the same as the length of the row.

__Function:__int**gsl_matrix_set_col***(gsl_matrix **`m`, size_t`j`, const gsl_vector *`v`)-
This function copies the elements of the vector
`v`into the`i`-th column of the matrix`m`. The length of the vector must be the same as the length of the column.

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

__Function:__int**gsl_matrix_swap_rows***(gsl_matrix **`m`, size_t`i`, size_t`j`)-
This function exchanges the
`i`-th and`j`-th rows of the matrix`m`in-place.

__Function:__int**gsl_matrix_swap_columns***(gsl_matrix **`m`, size_t`i`, size_t`j`)-
This function exchanges the
`i`-th and`j`-th columns of the matrix`m`in-place.

__Function:__int**gsl_matrix_swap_rowcol***(gsl_matrix **`m`, size_t`i`, size_t`j`)-
This function exchanges the
`i`-th row and`j`-th column of the matrix`m`in-place. The matrix must be square for this operation to be possible.

__Function:__int**gsl_matrix_transpose_memcpy***(gsl_matrix **`dest`, gsl_matrix *`src`)-
This function makes the matrix
`dest`the transpose of the matrix`src`by copying the elements of`src`into`dest`. This function works for all matrices provided that the dimensions of the matrix`dest`match the transposed dimensions of the matrix`src`.

__Function:__int**gsl_matrix_transpose***(gsl_matrix **`m`)-
This function replaces the matrix
`m`by its transpose by copying the elements of the matrix in-place. The matrix must be square for this operation to be possible.

The following operations are only defined for real matrices.

__Function:__int**gsl_matrix_add***(gsl_matrix **`a`, const gsl_matrix *`b`)-
This function adds the elements of matrix
`b`to the elements of matrix`a`, a'(i,j) = a(i,j) + b(i,j). The two matrices must have the same dimensions.

__Function:__int**gsl_matrix_sub***(gsl_matrix **`a`, const gsl_matrix *`b`)-
This function subtracts the elements of matrix
`b`from the elements of matrix`a`, a'(i,j) = a(i,j) - b(i,j). The two matrices must have the same dimensions.

__Function:__int**gsl_matrix_mul_elements***(gsl_matrix **`a`, const gsl_matrix *`b`)-
This function multiplies the elements of matrix
`a`by the elements of matrix`b`, a'(i,j) = a(i,j) * b(i,j). The two matrices must have the same dimensions.

__Function:__int**gsl_matrix_div_elements***(gsl_matrix **`a`, const gsl_matrix *`b`)-
This function divides the elements of matrix
`a`by the elements of matrix`b`, a'(i,j) = a(i,j) / b(i,j). The two matrices must have the same dimensions.

__Function:__int**gsl_matrix_scale***(gsl_matrix **`a`, const double`x`)-
This function multiplies the elements of matrix
`a`by the constant factor`x`, a'(i,j) = x a(i,j).

__Function:__int**gsl_matrix_add_constant***(gsl_matrix **`a`, const double`x`)-
This function adds the constant value
`x`to the elements of the matrix`a`, a'(i,j) = a(i,j) + x.

__Function:__double**gsl_matrix_max***(const gsl_matrix **`m`)-
This function returns the maximum value in the matrix
`m`.

__Function:__double**gsl_matrix_min***(const gsl_matrix **`m`)-
This function returns the minimum value in the matrix
`m`.

__Function:__void**gsl_matrix_minmax***(const gsl_matrix **`m`, double *`min_out`, double *`max_out`)-
This function returns the minimum and maximum values in the matrix
`m`, storing them in`min_out`and`max_out`.

__Function:__void**gsl_matrix_max_index***(const gsl_matrix **`m`, size_t *`imax`, size_t *`jmax`)-
This function returns the indices of the maximum value in the matrix
`m`, storing them in`imax`and`jmax`. When there are several equal maximum elements then the first element found is returned.

__Function:__void**gsl_matrix_min_index***(const gsl_matrix **`m`, size_t *`imax`, size_t *`jmax`)-
This function returns the indices of the minimum value in the matrix
`m`, storing them in`imax`and`jmax`. When there are several equal minimum elements then the first element found is returned.

__Function:__void**gsl_matrix_minmax_index***(const gsl_matrix **`m`, size_t *`imin`, size_t *`imax`)-
This function returns the indices of the minimum and maximum values in
the matrix
`m`, storing them in (`imin`,`jmin`) and (`imax`,`jmax`). When there are several equal minimum or maximum elements then the first elements found are returned.

__Function:__int**gsl_matrix_isnull***(const gsl_matrix **`m`)-
This function returns 1 if all the elements of the matrix
`m`are zero, and 0 otherwise.

The program below shows how to allocate, initialize and read from a matrix
using the functions `gsl_matrix_alloc`

, `gsl_matrix_set`

and
`gsl_matrix_get`

.

#include <stdio.h> #include <gsl/gsl_matrix.h> int main (void) { int i, j; gsl_matrix * m = gsl_matrix_alloc (10, 3); for (i = 0; i < 10; i++) for (j = 0; j < 3; j++) gsl_matrix_set (m, i, j, 0.23 + 100*i + j); for (i = 0; i < 100; i++) for (j = 0; j < 3; j++) printf("m(%d,%d) = %g\n", i, j, gsl_matrix_get (m, i, j)); return 0; }

Here is the output from the program. The final loop attempts to read
outside the range of the matrix `m`

, and the error is trapped by
the range-checking code in `gsl_matrix_get`

.

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 IOT trap/Abort (core dumped)

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

#include <stdio.h> #include <gsl/gsl_matrix.h> int main (void) { int i, j, k = 0; gsl_matrix * m = gsl_matrix_alloc (100, 100); gsl_matrix * a = gsl_matrix_alloc (100, 100); for (i = 0; i < 100; i++) for (j = 0; j < 100; j++) gsl_matrix_set (m, i, j, 0.23 + i + j); { FILE * f = fopen("test.dat", "w"); gsl_matrix_fwrite (f, m); fclose (f); } { FILE * f = fopen("test.dat", "r"); gsl_matrix_fread (f, a); fclose (f); } for (i = 0; i < 100; i++) for (j = 0; j < 100; j++) { double mij = gsl_matrix_get(m, i, j); double aij = gsl_matrix_get(a, i, j); if (mij != aij) k++; } printf("differences = %d (should be zero)\n", k); return (k > 0); }

After running this program the file ``test.dat'` should contain the
elements of `m`

, written in binary format. The matrix which is read
back in using the function `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.

#include <math.h> #include <stdio.h> #include <gsl/gsl_matrix.h> #include <gsl/gsl_blas.h> int main (void) { size_t i,j; gsl_matrix *m = gsl_matrix_alloc (10, 10); for (i = 0; i < 10; i++) for (j = 0; j < 10; j++) gsl_matrix_set (m, i, j, sin (i) + cos (j)); for (j = 0; j < 10; j++) { gsl_vector_view column = gsl_matrix_column (m, j); double d; d = gsl_blas_dnrm2 (&column.vector); printf ("matrix column %d, norm = %g\n", j, d); } gsl_matrix_free (m); }

Here is the output of the program, which can be confirmed using GNU OCTAVE,

$ ./a.out matrix column 0, norm = 4.31461 matrix column 1, norm = 3.1205 matrix column 2, norm = 2.19316 matrix column 3, norm = 3.26114 matrix column 4, norm = 2.53416 matrix column 5, norm = 2.57281 matrix column 6, norm = 4.20469 matrix column 7, norm = 3.65202 matrix column 8, norm = 2.08524 matrix column 9, norm = 3.07313 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

The block, vector and matrix objects in GSL follow the `valarray`

model of C++. A description of this model can be found in the following
reference,

- B. Stroustrup, The C++ Programming Language (3rd Ed), Section 22.4 Vector Arithmetic. Addison-Wesley 1997, ISBN 0-201-88954-4.

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