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Release calc version 2.11.11

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Landon Curt Noll
2005-12-11 22:58:55 -08:00
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NAME
mat - keyword to create a matrix value
mat - keyword to create a matrix value
SYNOPSIS
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] [ = {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] ... [ = { { ...} ...} ]
mat [index-range-list_1[index-ranges-list_2] ... [ = { { ...} ...} ]
decl id_1 id_2 ... [index-range-list] ...
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, ... identifier
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, ... identifier
DESCRIPTION
The expression mat [index-range-list] returns a matrix value.
This may be assigned to one or more lvalues A, B, ... by either
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
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.
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.
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 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, ...].
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
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.
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
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,
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;
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
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].
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}.
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
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].)
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.
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.
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.
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 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).
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:
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("%8d", A[i,j];
print;
}
printf(" [%d,%d]: %e", i, j, A[i,j]);
if (i != matmax(A,1))
printf("\n");
}
}
The default printing may be restored by
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
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
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].
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.
Some operations are defined for matrices.
A == B
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,
@@ -191,14 +206,14 @@ DESCRIPTION
double-bracket indices; thus A == B implies that A[[i]] == B[[i]]
for 0 <= i < size(A) == size(B).
A + B
A - 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]
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
rows is matmax(A,1) - matmin(A,1) + 1; the number of columns if
@@ -206,7 +221,7 @@ DESCRIPTION
if it is two-dimensional and the number of rows is equal to the
number of columns.
A * B
A * B
Multiplication is defined provided certain conditions by the
dimensions and shapes of A and B are satisfied. If both have
dimension 2 and the column-index-list for A is the same as
@@ -263,7 +278,7 @@ DESCRIPTION
a matrix indexed in the same way as A with each element inverted.
The following functions are defined to return matrices with the same
The following functions are defined to return matrices with the same
index-ranges as A and the specified operations performed on all
elements of A. Here num is an arbitrary complex number (nonzero
when it is a divisor), int an integer, rnd a rounding-type
@@ -285,52 +300,52 @@ DESCRIPTION
A % real
A ^ int
If A and B are one-dimensional of the same size dp(A, B) returns
If A and B are one-dimensional of the same size dp(A, B) returns
their dot-product, i.e. the sum of the products of corresponding
elements.
If A and B are one-dimension and of size 3, cp(A, B) returns their
If A and B are one-dimension and of size 3, cp(A, B) returns their
cross-product.
randperm(A) returns a matrix indexed the same as A in which the elements
randperm(A) returns a matrix indexed the same as A in which the elements
of A have been randomly permuted.
sort(A) returns a matrix indexed the same as A in which the elements
sort(A) returns a matrix indexed the same as A in which the elements
of A have been sorted.
If A is an lvalue whose current value is a matrix, matfill(A, v)
If A is an lvalue whose current value is a matrix, matfill(A, v)
assigns the value v to every element of A, and if also, A is
square, matfill(A, v1, v2) assigns v1 to the off-diagonal elements,
v2 to the diagonal elements. To create and assign to A the unit
n * n matrix, one may use matfill(mat A[n,n], 0, 1).
For a square matrix A, mattrace(A) returns the trace of A, i.e. the
For a square matrix A, mattrace(A) returns the trace of A, i.e. the
sum of the diagonal elements. For zero- or one-dimensional A,
mattrace(A) returns the sum of the elements of A.
For a two-dimensional matrix A, mattrans(A) returns the transpose
For a two-dimensional matrix A, mattrans(A) returns the transpose
of A, i.e. if A is mat[m,n], it returns a mat[n,m] matrix with
[i,j] element equal to A[j,i]. For zero- or one-dimensional A,
mattrace(A) returns a matrix with the same value as A.
The functions search(A, value, start, end]) and
rsearch(A, value, start, end]) return the first or last index i
for which A[[i]] == value and start <= i < end, or if there is
no such index, the null value. For further information on default
values and the use of an "accept" function, see the help files for
search and rsearch.
The functions search(A, value, start, end]) and
rsearch(A, value, start, end]) return the first or last index i
for which A[[i]] == value and start <= i < end, or if there is
no such index, the null value. For further information on default
values and the use of an "accept" function, see the help files for
search and rsearch.
reverse(A) returns a matrix with the same index-lists as A but the
elements in reversed order.
reverse(A) returns a matrix with the same index-lists as A but the
elements in reversed order.
The copy and blkcpy functions may be used to copy data to a matrix from
a matrix or list, or from a matrix to a list. In copying from a
matrix to a matrix the matrices need not have the same dimension;
in effect they are treated as linear arrays.
The copy and blkcpy functions may be used to copy data to a matrix from
a matrix or list, or from a matrix to a list. In copying from a
matrix to a matrix the matrices need not have the same dimension;
in effect they are treated as linear arrays.
EXAMPLE
> obj point {x,y}
> mat A[5] = {1, 2+3i, "ab", mat[2] = {4,5}. obj point = {6,7}}
> mat A[5] = {1, 2+3i, "ab", mat[2] = {4,5}, obj point = {6,7}}
> A
mat [5] (5 elements, 5 nonzero):
[0] = 1
@@ -381,20 +396,20 @@ EXAMPLE
[3] = 4
LIMITS
The theoretical upper bound for the absolute values of indices is
2^31 - 1, but the size of matrices that can be handled in practice will
be limited by the availability of memory and what is an acceptable
runtime. For example, although it may take only a fraction of a
second to invert a 10 * 10 matrix, it will probably take about 1000
times as long to invert a 100 * 100 matrix.
The theoretical upper bound for the absolute values of indices is
2^31 - 1, but the size of matrices that can be handled in practice will
be limited by the availability of memory and what is an acceptable
runtime. For example, although it may take only a fraction of a
second to invert a 10 * 10 matrix, it will probably take about 1000
times as long to invert a 100 * 100 matrix.
LINK LIBRARY
n/a
n/a
SEE ALSO
ismat, matdim, matmax, matmin, mattrans, mattrace, matsum, det, inverse,
isident, test, config, search, rsearch, reverse, copy, blkcpy, dp, cp,
randperm, sort
ismat, matdim, matmax, matmin, mattrans, mattrace, matsum, det, inverse,
isident, test, config, search, rsearch, reverse, copy, blkcpy, dp, cp,
randperm, sort
## Copyright (C) 1999 Landon Curt Noll
##
@@ -412,8 +427,8 @@ SEE ALSO
## received a copy with calc; if not, write to Free Software Foundation, Inc.
## 59 Temple Place, Suite 330, Boston, MA 02111-1307, USA.
##
## @(#) $Revision: 29.2 $
## @(#) $Id: mat,v 29.2 2000/06/07 14:02:33 chongo Exp $
## @(#) $Revision: 29.4 $
## @(#) $Id: mat,v 29.4 2005/10/18 10:08:45 chongo Exp $
## @(#) $Source: /usr/local/src/cmd/calc/help/RCS/mat,v $
##
## Under source code control: 1991/07/21 04:37:22