NAME    mat - keyword to create a matrix valueSYNOPSIS    mat [index-range-list] [ = {value_0. ...} ]    mat [] [= {value_0, ...}]    mat variable_1 ... [index-range-list] [ = {value_0, ...} ]    mat variable_1 ... [] [ = {value_0, ...} ]    mat [index-range-list_1[index-ranges-list_2] ... [ = { { ...} ...}  ]    decl id_1 id_2 ... [index-range-list] ...TYPES    index-range-list	range_1 [, range_2, ...]  up to 4 ranges    range_1, ...		integer, or integer_1 : integer_2    value, value_1, ...	any    variable_1 ...	lvalue    decl			declarator = global, static or local    id_1, ...		identifierDESCRIPTION    The expression  mat [index-range-list]  returns a matrix value.    This may be assigned to one or more lvalues A, B, ... by either	mat A B ... [index-range-list]    or	A = B = ... = mat[index-range-list]    If a variable is specified by an expression that is not a symbol with    possibly object element specifiers, the expression should be enclosed    in parentheses.  For example, parentheses are required in    mat (A[2]) [3]  and	mat (*p) [3]  but  mat P.x [3] is acceptable.    When an index-range is specified as integer_1 : integer_2, where    integer_1 and integer_2 are expressions which evaluate to integers,    the index-range consists of all integers from the minimum of the    two integers to the maximum of the two integers.  For example,    mat[2:5, 0:4] and mat[5:2, 4:0] return the same matrix value.    If an index-range is an expression which evaluates to an integer,    the range is as if specified by 0 : integer - 1.  For example,    mat[4] and mat[0:3] return the same 4-element matrix; mat[-2] and    mat[-3:0] return the same 4-element matrix.    If the variable A has a matrix value, then for integer indices    i_1, i_2, ..., equal in number to the number of ranges specified at    its creation, and such that each index is in the corresponding range,    the matrix element associated with those index list is given as an    lvalue by the expressions A[i_1, i_2, ...].    The elements of the matrix are stored internally as a linear array    in which locations are arranged in order of increasing indices.    For example, in order of location, the six element of A = mat [2,3]    are	A[0,0], A[0,1], A[0,2], A[1,0], A[1,,1], A[1,2].    These elements may also be specified using the double-bracket operator    with a single integer index as in A[[0]], A[[1]], ..., A[[5]].    If p is assigned the value &A[0.0], the address of A[[i]] for 0 <= i < 6    is p + i as long as A exists and a new value is not assigned to A.    When a matrix is created, each element is initially assigned the    value zero.	Other values may be assigned then or later using the    "= {...}" assignment operation.  Thus	A = {value_0, value_1, ...}    assigns the values value_0, value_1, ... to the elements A[[0]],    A[[1]], ...	Any blank "value" is passed over.  For example,	A = {1, , 2}    will assign the value 1 to A[[0]], 2 to A[[2]] and leave all other    elements unchanged.	Values may also be assigned to elements by    simple assignments, as in A[0,0] = 1, A[0,2] = 2;    If the index-range is left blank but an initializer list is specified    as in:	; mat A[] = {1, 2 }	; B = mat[] = {1, , 3, }    the matrix created is one-dimensional.  If the list contains a    positive number n of values or blanks, the result is as if the    range were specified by [n], i.e. the range of indices is from    0 to n - 1.	In the above examples, A is of size 2 with A[0] = 1    and A[1] = 2;  B is of size 4 with B[0] = 1, B[1] = B[3] = 0,    B[2] = 3.  The specification mat[] = { } creates the same as mat[1].    If the index-range is left blank and no initializer list is specified,    as in  mat C[]  or	C = mat[], the matrix assigned to C has zero    dimension; this has one element C[].    To assign a value using "= { ...}" at the same time as creating C,    parentheses are required as in (mat[]) = {value}  or  (mat C[]) =    {value}. Later a value may be assigned to C[] by  C[] = value  or    C = {value}.    The value assigned at any time to any element of a matrix can be of    any type - number, string, list, matrix, object of previously specified    type, etc.  For some matrix operations there are of course conditions    that elements may have to satisfy: for example, addition of matrices    requires that addition of corresponding elements be possible.    If an element of a matrix is a structure for which indices or an    object element specifier is required, an element of that structure is    referred to by appropriate uses of [ ] or ., and so on if an element    of that element is required.    For example, one may have an expressions like:	; A[1,2][3].alpha[2];    if A[1,2][3].alpha is a list with at least three elements, A[1,2][3] is    an object of a type like  obj {alpha, beta}, A[1,2] is a matrix of    type mat[4] and A is a mat[2,3] matrix.  When an element of a matrix    is a matrix and the total number of indices does not exceed 4, the    indices can be combined into one list, e.g. the A[1,2][3] in the    above example can be shortened to A[1,2,3].	(Unlike C, A[1,2] cannot    be expressed as A[1][2].)    The function ismat(V) returns 1 if V is a matrix, 0 otherwise.    isident(V) returns 1 if V is a square matrix with diagonal elements 1,    off-diagonal elements zero, or a zero- or one-dimensional matrix with    every element 1; otherwise zero is returned.	 Thus  isident(V) = 1    indicates that for  V * A  and  A * V  where A is any matrix of    for which either product is defined and the elements of A are real    or complex numbers, that product will equal A.    If V is matrix-valued, test(V) returns 0 if every element of V tests    as zero; otherwise 1 is returned.    The dimension of a matrix A, i.e. the number of index-ranges in the    initial creation of the matrix, is returned by the function matdim(A).    For 1 <= i <= matdim(A), the minimum and maximum values for the i-th    index range are returned by matmin(A, i) and matmax(A,i), respectively.    The total number of elements in the matrix is returned by size(A).    The sum of the elements in the matrix is returned by matsum(A).    The default method of printing matrices is to give a line of information    about the matrix, and to list on separate lines up to 15 elements,    the indices and either the value (for numbers, strings, objects) or    some descriptive information for lists or matrices, etc.    Numbers are displayed in the current number-printing mode.    The maximum number of elements to be printed can be assigned    any nonnegative integer value m by config("maxprint", m).    Users may define another method of printing matrices by defining a    function mat_print(M); for example, for a not too big 2-dimensional    matrix A it is a common practice to use a loop like:	define mat_print(A) {		local i,j;		for (i = matmin(A,1); i <= matmax(A,1); i++) {			if (i != matmin(A,1))				printf("\t");			for (j = matmin(A,2); j <= matmax(A,2); j++)				printf(" [%d,%d]: %e", i, j, A[i,j]);			if (i != matmax(A,1))				printf("\n");	 	}	}    So that when one defines a 2D matrix such as:	; mat X[2,3] = {1,2,3,4,5,6}    then printing X results in:	[0,0]: 1 [0,1]: 2 [0,2]: 3	[1,0]: 4 [1,1]: 5 [1,2]: 6    The default printing may be restored by	; undefine mat_print;    The keyword "mat" followed by two or more index-range-lists returns a    matrix with indices specified by the first list, whose elements are    matrices as determined by the later index-range-lists.  For    example  mat[2][3]  is a 2-element matrix, each of whose elements has    as its value a 3-element matrix.  Values may be assigned to the    elements of the innermost matrices by nested = {...} operations as in	; mat [2][3] = {{1,2,3},{4,5,6}}    An example of the use of mat with a declarator is	; global mat A B [2,3], C [4]    This creates, if they do not already exist, three global variables with    names A, B, C, and assigns to A and B the value mat[2,3] and to C mat[4].    Some operations are defined for matrices.    A == B	Returns 1 if A and B are of the same "shape" and "corresponding"	elements are equal; otherwise 0 is returned.  Being of the same	shape means they have the same dimension d, and for each i <= d,	    matmax(A,i) - matmin(A,i) == matmax(B,i) - matmin(B,i),	One consequence of being the same shape is that the matrices will	have the same size.   Elements "correspond" if they have the same	double-bracket indices; thus A == B implies that A[[i]] == B[[i]]	for 0 <= i < size(A) == size(B).    A + B    A - B	These are defined A and B have the same shape, the element	with double-bracket index j being evaluated by A[[j]] + B[[j]] and	A[[j]] - B[[j]], respectively.	The index-ranges for the results	are those for the matrix A.    A[i,j]	If A is two-dimensional, it is customary to speak of the indices	i, j in A[i,j] as referring to rows and columns;  the number of