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Release calc version 2.11.0t10.3

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Landon Curt Noll
1999-11-16 03:44:46 -08:00
parent 160f4102ab
commit 025b5e58d6
20 changed files with 929 additions and 292 deletions
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20
help/Makefile
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@@ -104,16 +104,16 @@ BLT_HELP_FILES= ${BLT_HELP_FILES_3} ${BLT_HELP_FILES_5} \
# This list is prodiced by the detaillist rule when no WARNINGS are detected.
#
DETAIL_HELP= abs access acos acosh acot acoth acsc acsch address agd append \
appr arg arrow asec asech asin asinh assign atan atan2 atanh avg \
base bit blk blkcpy blkfree blocks bround btrunc calclevel ceil \
cfappr cfsim char cmdbuf cmp comb conj cos cosh cot coth count cp \
csc csch ctime delete den dereference det digit digits dp epsilon \
errcount errmax errno error eval exp fact factor fclose fcnt feof \
ferror fflush fgetc fgetfield fgetline fgets fgetstr fib files \
floor fopen forall fprintf fputc fputs fputstr frac free freeglobals \
freeredc freestatics frem freopen fscan fscanf fseek fsize ftell gcd \
gcdrem gd getenv hash head highbit hmean hnrmod hypot ilog ilog10 \
ilog2 im inputlevel insert int inverse iroot isassoc isatty isblk \
appr arg arrow asec asech asin asinh assign atan atan2 atanh avg base \
bit blk blkcpy blkfree blocks bround btrunc calclevel ceil cfappr \
cfsim char cmdbuf cmp comb conj cos cosh cot coth count cp csc csch \
ctime delete den dereference det digit digits dp epsilon errcount \
errmax errno error eval exp fact factor fclose fcnt feof ferror \
fflush fgetc fgetfield fgetline fgets fgetstr fib files floor fopen \
forall fprintf fputc fputs fputstr frac free freeglobals freeredc \
freestatics frem freopen fscan fscanf fseek fsize ftell gcd gcdrem \
gd getenv hash head highbit hmean hnrmod hypot ilog ilog10 ilog2 \
im indices inputlevel insert int inverse iroot isassoc isatty isblk \
isconfig isdefined iserror iseven isfile ishash isident isint islist \
ismat ismult isnull isnum isobj isobjtype isodd isprime isptr isqrt \
isrand israndom isreal isrel issimple issq isstr istype jacobi join \

58
help/indices Normal file
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@@ -0,0 +1,58 @@
NAME
indices - indices for specified matrix or association element
SYNOPSIS
indices(V, index)
TYPES
V matrix or association
index integer
return list with up to 4 elements
DESCRIPTION
For 0 <= index < size(V), indices(V, index) returns list(i_0, i_1, ...)
for which V[i_0, i_1, ...] is the same lvalue as V[[index]].
For other values of index, a null value is returned.
This function can be useful for determining those elements for which
the indices satisfy some condition. This is particularly so for
associations since these have no simple relation between the
double-bracket index and the single-bracket indices, which may be
non-integer numbers or strings or other types of value. The
information provided by indices() is often required after the use
of search() or rsearch() which, when successful, return the
double-bracket index of the item found.
EXAMPLE
> mat M[2,3,1:5]
> indices(M, 11)
list (3 elements, 2 nonzero):
[[0]] = 0
[[1]] = 2
[[2]] = 2
> A = assoc();
> A["cat", "dog"] = "fight";
> A[2,3,5,7] = "primes";
> A["square", 3] = 9
> indices(A, search(A, "primes"))
list (4 elements, 4 nonzero):
[[0]] = 2
[[1]] = 3
[[2]] = 5
[[3]] = 7
LIMITS
abs(index) < 2^31
LIBRARY
LIST* associndices(ASSOC *ap, long index)
LIST* matindices(MATRIX *mp, long index)
SEE ALSO
assoc, mat

457
help/mat
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@@ -1,102 +1,397 @@
Using matrices
NAME
mat - keyword to create a matrix value
Matrices can have from 1 to 4 dimensions, and are indexed by a
normal-sized integer. The lower and upper bounds of a matrix can
be specified at runtime. The elements of a matrix are defaulted
to zeroes, but can be assigned to be of any type. Thus matrices
can hold complex numbers, strings, objects, etc. Matrices are
stored in memory as an array so that random access to the elements
is easy.
SYNOPSIS
mat [index-range-list] [ = {value_0. ...} ]
mat [] [= {value_0, ...}]
mat variable_1 ... [index-range-list] [ = {value_0, ...} ]
mat variable_1 ... [] [ = {value_0, ...} ]
Matrices are normally indexed using square brackets. If the matrix
is multi-dimensional, then an element can be indexed either by
using multiple pairs of square brackets (as in C), or else by
separating the indexes by commas. Thus the following two statements
reference the same matrix element:
mat [index-range-list_1[index-ranges-list_2] ... [ = { { ...} ...} ]
x = name[3][5];
x = name[3,5];
decl id_1 id_2 ... [index-range-list] ...
The double-square bracket operator can be used on any matrix to
make references to the elements easy and efficient. This operator
bypasses the normal indexing mechanism, and treats the array as if
it was one-dimensional and with a lower bound of zero. In this
indexing mode, elements correspond to the normal indexing mode where
the rightmost index increases most frequently. For example, when
using double-square bracket indexing on a two-dimensional matrix,
increasing indexes will reference the matrix elements left to right,
row by row. Thus in the following example, 'x' and 'y' are copied
from the same matrix element:
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
mat m[1:2, 1:3];
x = m[2,1];
y = m[[3]];
DESCRIPTION
The expression mat [index-range-list] returns a matrix value.
This may be assigned to one or more lvalues A, B, ... by either
There are functions which return information about a matrix.
The 'size' functions returns the total number of elements.
The 'matdim', 'matmin', and 'matmax' functions return the number
of dimensions of a matrix, and the lower and upper index bounds
for a dimension of a matrix. For square matrices, the 'det'
function calculates the determinant of the matrix.
mat A B ... [index-range-list]
Some functions return matrices as their results. These functions
do not affect the original matrix argument, but instead return
new matrices. For example, the 'mattrans' function returns the
transpose of a matrix, and 'inverse' returns the inverse of a
matrix. So to invert a matrix called 'x', you could use:
or
x = inverse(x);
A = B = ... = mat[index-range-list]
The 'matfill' function fills all elements of a matrix with the
specified value, and optionally fills the diagonal elements of a
square matrix with a different value. For example:
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.
matfill(x,1);
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.
will fill any matrix with ones, and:
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.
matfill(x, 0, 1);
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, ...].
will create an identity matrix out of any square matrix. Note that
unlike most matrix functions, this function does not return a matrix
value, but manipulates the matrix argument itself.
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
Matrices can be multiplied by numbers, which multiplies each element
by the number. Matrices can also be negated, conjugated, shifted,
rounded, truncated, fractioned, and modulo'ed. Each of these
operations is applied to each element.
A[0,0], A[0,1], A[0,2], A[1,0], A[1,,1], A[1,2].
Matrices can be added or multiplied together if the operation is
legal. Note that even if the dimensions of matrices are compatible,
operations can still fail because of mismatched lower bounds. The
lower bounds of two matrices must either match, or else one of them
must have a lower bound of zero. Thus the following code:
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.
mat x[3:3];
mat y[4:4];
z = x + y;
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
fails because the calculator does not have a way of knowing what
the bounds should be on the resulting matrix. If the bounds match,
then the resulting matrix has the same bounds. If exactly one of
the lower bounds is zero, then the resulting matrix will have the
nonzero lower bounds. Thus means that the bounds of a matrix are
preserved when operated on by matrices with lower bounds of zero.
For example:
A = {value_0, value_1, ...}
mat x[3:7];
mat y[5];
z = x + y;
assigns the values value_0, value_1, ... to the elements A[[0]],
A[[1]], ... Any blank "value" is passed over. For example,
will succeed and assign the variable 'z' a matrix whose
bounds are 3-7.
A = {1, , 2}
Vectors are matrices of only a single dimension. The 'dp' and 'cp'
functions calculate the dot product and cross product of a vector
(cross product is only defined for vectors of size 3).
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;
Matrices can be searched for particular values by using the 'search'
and 'rsearch' functions. They return the element number of the
found value (zero based), or null if the value does not exist in the
matrix. Using the element number in double-bracket indexing will
then refer to the found element.
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:
for (i = matmin(A,1); i <= matmax(A,1); i++) {
for (j = matmin(A,2); j <= matmax(A,2); j++)
printf("%8d", A[i,j];
print;
}
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
rows is matmax(A,1) - matmin(A,1) + 1; the number of columns if
matmax(A,2) - matmin(A,2) + 1. A matrix is said to be square
if it is two-dimensional and the number of rows is equal to the
number of columns.
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
the row-index-list for B, C = A * B is defined in the usual
way so that for i in the row-index-list of A and j in the
column-index-list for B,
C[i,j] = Sum A[i,k] * B[k,j]
the sum being over k in the column-index-list of A. The same
formula is used so long as the number of columns in A is the same
as the number of rows in B and k is taken to refer to the offset
from matmin(A,2) and matmin(B,1), respectively, for A and B.
If the multiplications and additions required cannot be performed,
an execution error may occur or the result for C may contain
one or more error-values as elements.
If A or B has dimension zero, the result for A * B is simply
that of multiplying the elements of the other matrix on the
left by A[] or on the right by B[].
If both A and B have dimension 1, A * B is defined if A and B
have the same size; the result has the same index-list as A
and each element is the product of corresponding elements of
A and B. If A and B have the same index-list, this multiplication
is consistent with multiplication of 2D matrices if A and B are
taken to represent 2D matrices for which the off-diagonal elements
are zero and the diagonal elements are those of A and B.
the real and complex numbers.
If A is of dimension 1 and B is of dimension 2, A * B is defined
if the number of rows in B is the same as the size of A. The
result has the same index-lists as B; each row of B is multiplied
on the left by the corresponding element of A.
If A is of dimension 2 and B is of dimension 1, A * B is defined
if number of columns in A is the same as the size of A. The
result has the same index-lists as A; each column of A is
multiplied on the right by the corresponding element of B.
The algebra of additions and multiplications involving both one-
and two-dimensional matrices is particularly simple when all the
elements are real or complex numbers and all the index-lists are
the same, as occurs, for example, if for some positive integer n,
all the matrices start as mat [n] or mat [n,n].
det(A)
If A is a square, det(A) is evaluated by an algorithm that returns
the determinant of A if the elements of A are real or complex
numbers, and if such an A is non-singular, inverse(A) returns
the inverse of A indexed in the same way as A. For matrix A of
dimension 0 or 1, det(A) is defined as the product of the elements
of A in the order in which they occur in A, inverse(A) returns
a matrix indexed in the same way as A with each element inverted.
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
specifier integer, real a real number.
num * A
A * num
A / num
- A
conj(A)
A << int, A >> int
scale(A, int)
round(A, int, rnd)
bround(A, int, rnd)
appr(A, real, rnd)
int(A)
frac(A)
A // real
A % real
A ^ int
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
cross-product.
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
of A have been sorted.
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
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
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.
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.
EXAMPLE
> obj point {x,y}
> mat A[5] = {1, 2+3i, "ab", mat[2] = {4,5}. obj point = {6,7}}
> A
mat [5] (5 elements, 5 nonzero):
[0] = 1
[1] = 2+3i
[2] = "ab"
[3] = mat [2] (2 elements, 2 nonzero)
[4] = obj point {6, 7}
> print A[0], A[1], A[2], A[3][0], A[4].x
1 2+3i ab 4 6
> define point_add(a,b) = obj point = {a.x + b.x, a.y + b.y}
point_add(a,b) defined
> mat [B] = {8, , "cd", mat[2] = {9,10}, obj point = {11,12}}
> A + B
mat [5] (5 elements, 5 nonzero):
[0] = 9
[1] = 2+3i
[2] = "abcd"
[3] = mat [2] (2 elements, 2 nonzero)
[4] = obj point {17, 19}
> mat C[2,2] = {1,2,3,4}
> C^10
mat [2,2] (4 elements, 4 nonzero):
[0,0] = 4783807
[0,1] = 6972050
[1,0] = 10458075
[1,1] = 15241882
> C^-10
mat [2,2] (4 elements, 4 nonzero):
[0,0] = 14884.650390625
[0,1] = -6808.642578125
[1,0] = -10212.9638671875
[1,1] = 4671.6865234375
> mat A[4] = {1,2,3,4}, A * reverse(A);
mat [4] (4 elements, 4 nonzero):
[0] = 4
[1] = 6
[2] = 6
[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.
LIBRARY
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

1
help/todo
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@@ -48,7 +48,6 @@ Very High priority items:
history command history
interrupt how interrupts are handled
list using lists
mat using matrices
obj user defined data types
operator math, relational, logic and variable access ...
statement flow control and declaration statements

2
help/wishlist
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@@ -11,7 +11,7 @@ Calc Enhancement Wish List:
The following items are in the calc wish list. Programs like this
can be extended and improved forever.
Calc bug repoers, however, should be sent to:
Calc bug reports, however, should be sent to:
calc-bugs at postofc dot corp dot sgi dot com