[100] 042 FAQ, Tutorial by www.msharpmath.com
[100] 042 revised on 2012.11.13 / cemmathThe Simple is the Best
FAQ
// (Q) What is unique in Cemmath ?
// (Q) Between C-language and Cemmath ?
// (Q) Between C++ and Cemmath ?
// (Q) Between MATLAB and Cemmath ?
// (Q) Between MATHEMATICA and Cemmath ?
// (Q) Pitfalls in Cemmath ?
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// (Q) What is unique in msharpmath ?
//------
(A) Several unique innovations are implemented in msharpmath.
■ various data types (matrix, poly, vertex, csys, complex, double, signal)
■ Umbrella grammar
■ dot function
■ tuple function
■ all-in-one file handling
Among the above, the most impressive one would be the 'Umbrella' which realizes 'expression-look-alike' treatment of equations. For example,
#> solve .x ( exp(x) = 5 ); // single nonlinear equation
#> solve .a.b ( a+b = 3, a*b = 2 ); // coupled nonlinear equations
#> plot .x( 0,pi ) ( |x|*sin(x) ); // plot 2d curve
#> int .x( 1,2 ) .y(x,2*x+1) ( exp(-x-y)*sin(x+y) ); // double integration
#> ode .t( 0,1 ) ( y' = t*y+1, y = 1 ).plot; // Ordinary Differential Equation
#> plot .r[21](0,1).t[61](0,6*pi) ( (r^(1/3)+t!/3).cyl .y ).cyl;
are almost similar to their relevant mathematical problems.
Another uniqueness could be the use of dot function. Most of built-in functions start with a dot (.) and thus can be distinguished from user-defined variables.
// 'I' for users' name space,
// while '.I' for name space of built-in functions
#> I = .I(2,3) ;
ans =
[ 1 0 0 ]
[ 0 1 0 ]
Also, it is possible to denote "expression-look-alike" functions such as the Bessel functions by dot functions
.J_nu(x) .I_nu(x) .K_n+1/2 (x) .Y_n(x)
The concept of 'tuple' is also useful to handle different types of data. For example,
#> (a,b,c) = ( 3 +2!, <1,2,3>, [1,2,3] );
a = 3 + 2!
b = < 1 2 3 >
c = [ 1 2 3 ]
As can be seen above, Cemmath employs a variety of data types including matrix, polynomial, double, complex, vertex, coordinate system, signal. Richness in data type however sacrifices the convenience in reference to data. For example, sqrt(-1) is NaN (not a number) in Cemmath, and thus sqrt(-1+0!) should be used instead.
From the standpoint of user-file management, no path is necessary to find a file. All functions can be included in one file and compiled for run.
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// (Q) Between C-language and Cemmath ?
//------
(A) In conclusion, Cemmath can be regarded as a mini-C. Some keywords of C are implemented in Cemmath.
■ if else do for while break continue goto
■ switch case default return void double
Cemmath does not use a pointer except for array pointer. Although array pointer is used in Cemmath, no operation is allowed except subscript. As a mini-C, Cemmath does not have so many functions in C. Since the primary objective is 'mathematics', math-related functions are of importance in Cemmath.
Nevertheless, some points are helpful in use. For example,
#> double f(x)= |x|*sin(x); // in Cemmath
#> f++( [1,2,3] ); // function upgrade for matrix
#> f ' (3) ; f '' (3); // extention to differentiation
#> double area(a, b=a) = a*b; // dynamic default assignment
#> s=0; for[10] s += 10; // simple iteration, no index
are implemented for easy use.
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// (Q) Between C++ and Cemmath ?
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(A) Absolutely, Cemmath cannot be compared with C++ in power and efficiency. Remember that Cemmath is just an easy-to-use interpreter not a powerful compiler. The size of Cemmath is only a few MB !
However, the concept of objective-oriented is somehow realized in Cemmath by member function. For example,
#> A.inv ; A.row(i) ; // inverse and i-th row vector of a matrix A
#> z.x ; z.r ; // elements of complex number
are of major use in Cemmath. All the data types are of built-in nature, and no new class can be defined.
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// (Q) Between MATLAB and Cemmath ?
//------
(A) The structure of matrix and data type are of major difference. At present, please do not mention the level of capabilities yet. However, Cemmath utilizes a variety of data such as polynomial, vertex including matrix. Richness in data type helps users work in 'expression-look-alike' manner. For example, a polynomial can be evaluated, differentiated, sumed or differenced just by
#> p = [1,0,0,0].poly ; // p(x) = x^3
#> p(2+3!); // p(z), z = 2+3!
#> p' ; // p'(x) = 3x^2, differentiation
#> p~ ; // int p(x) dx = x^4/4, integration
#> p` ; // p(x)-p(x-1) = 3x^2 -3x +1, finite difference
#> p.~; // sum k^3 = (1/4)n^2(n+1)^2, series sum
all of which are treated by functions in MATLAB such as polyval(p,z).
In Cemmath, data type must be explicitly or implicitly defined and cannot be changed during run until cleared. In this regard, 'double' and 'complex' is different. This means that sqrt(-1) = NaN in Cemmath but sqrt(-1) = i in MATLAB. Since -1 is 'double', sqrt(-1) is treated as a 'double' data with no value. The correct usage in Cemmath is sqrt( -1 + 0! ) where ! is employed to denote a pure imaginary number rather than a factorial in mathematics.
Matrix dimension can be modified by some operations, but it cannot be changed by 'subscript' operation in Cemmath. Thus, when a variable A is not a priori defined,
#> A(10) = 3;
line 1 : error #10970: undefined 'A'
; ??? A(10)=3
// no pre-defined 'A' in Cemmath, while MATLAB creates a matrix A
Also, for a matrix A=[1,2], A = 0 becomes [0,0] in M# but '0' in MATLAB.
The main cause of different treatment comes from the fact that Cemmath strives to be consistent with C-language. A unique advantage of using Cemmath is the approach of 'expression-look-alike' as can be seen in 'Umbrella' grammar.
From the aspect of function definition, *.msm file of Cemmath includes multiple functions in one file. But *.m file of MATLAB defines one function for each file.
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// (Q) Between MATHEMATICA and Cemmath ?
//------
(A) Cemmath does not provide 'symbolic operation'. Of course, grammars are very different between the two.
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// (Q) Pitfalls in Cemmath ?
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(A) Most frequent errors in using Cemmath arise from confusion between 'double' and 'complex' data type. Since (-1) is 'double' data, sqrt(-1) tries to return 'double' data. In general, the function 'sqrt' returns the same date type with the input. This means that
■ sqrt(double)=double, sqrt(-1) = NaN // not a number
■ sqrt(complex)=complex, sqrt(-1+0!) = 1! // complex number
Distinction between 'double' and 'complex' may seem redundant, but the main reason inherits from our desire to keep other valuable data type such as polynomial, vertex, csys (coordinate system). Otherwise, we have stick to matrix only, and treat all other data by matrices.
Similar to the above, subscript for a matrix can be misused.
■ A(i,j) for real part
■ A{i,j} for imaginary part
■ A[i,j] for complex element (real is extended)
This could be the most annoying feature of Cemmath. By sacrificing this instead, we are successful in handling various data types including polynomials and vertices.
As for the Umbrella innovated in Cemmath, care should be taken for nesting of Umbrella. Nested Umbrella causes a serous error at present. For example,
#> double f(x) = int.t(0,x) ( t );
#> double g(x) = int.t(0,x) ( f(t) ); // f(x) includes Umbrella
leads to the most dangerous pitfall in Cemmath at this stage.
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