[100] 091 News, Tutorial by Dec. 10, 2012 =======================================

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Egbert Cross
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1 [100] 091 revised on cemmath The Simple is the Best News Dec. 10, 2012 ======================================= Cemmath 2.22 (a new name of Msharpmath) is newly upgraded. indefinite integrals are treated by a dot function '.integ(a,b,f(x))'. For example, #> double f(x) = exp(-x*x);.integ(-inf,inf,f); sqrt(pi); ans = ans = #> double f(x) = 1/sqrt(1-x^2);.integ(0,1,f); asin(1); ans = ans = For discrete data, either '.spline(x,y).splint', 'trapz' or 'cumtrapz' can be employed. Nov. 26, 2012 ======================================= Cemmath 2.20 (a new name of Msharpmath) has a few revisions. Three different grammars for an iterative command 'for'. The following two commands in C-language, for(i = a; i <= b; i += c ) stmt; // if a < b and c > 0 for(i = a; i >= b; i -= c ) stmt; // if a > b and c > 0 are working in cemm# as it is. Also, an alternative expression for.i (a,b,c=1) stmt; for[10] stmt; works in cemm#. For example, 1

2 #> s = 0; for.n(1,10) s += n; // = 55 s = 0 s = 1 s = 3... s = 55 // Pibonachi sequence, #> (a,b)=(1,1); for[10] (a,b) = (b,a+b);; b; b = 144 Note that exchanging values among variables are done by tuples such that #> (a,b,c) = (1,2,3); (b,c,a) = (a,b,c); a = 1 b = 2 c = 3 b = 1 c = 2 a = 3 The old command "A..B" for dot product is changed to "A**B". A.wide(n) = A._n ; // a new operator "._" replaces old one ".." #> (1:12)._4; ans = [ ] [ ] [ ] Nov. 19, 2012 ======================================= Cemmath 2.19 (a new name of Msharpmath) handles stiff ODEs (ordinary differential equations) by Spoke '.stiff(tol=0.01)'. An example is from the van der Pol equations #> ode.x[15000](0,300) ( y'' = 100*(1-y^2)*y'-y, y=2, y'=0).stiff.plot; 2

3 Oct. 22, 2012 ======================================= In declaring a polynomial, Cemmath 2.16 announces the most simplest method. It is just by using a leading dot! [ list ] is a matrix.[ list ] is a polynomial!!! #>.[ 1,2,3 ]; // old version [ 1,2,3 ].poly can be used alternatively y = x^2 + 2x + 3 roots [ -1 - i ] [ -1 + i ] vertex at ( -1, 2 ) y_minimum = 2 y-axis intersect ( 0, 3 ) symmetry at x = -1 directrix at y = 1.75 focus at ( -1, 2.25 ) #>.[1,3]*.[1,9,7]*.[1,-5,3,4] ; // (x+3)(x^2+9x+7)(x^3-5x^2+3x+4) ans = poly( ) = x^6 +7x^5-23x^4-109x^3 +45x^2 +199x +84 dot-only function '.tdma(a,b,c,d, na,nb)' is implemented for tridiagonal matrix algorithm. 0 x 0 matrix [] is available now. dot functions ".horzcat", ".vertcat ", ".diagcat " arguments..horzcat(a1,a2,a3,...) // horizontal concatenation.vertcat(a1,a2,a3,... ) // vertical concatenation, stack now handle multiple 3

4 .diagcat(a1,a2,a3,...) // diagonal concatenation, blkdiag in matlab When two arguments are involved, binary operators work also A B //.horzcat(a,b) A _ B //.vertcat(a,b) A \_ B //.diagcat(a,b) A = B // A = A B A _= B // A = A _ B A \_= B // A = A \_ B Oct. 04, 2012 ======================================= A special variable ans is implemented in Cemmath V2.14 as in other softwares such as Matlab. All other variables are of fixed data type, but the special variable ans can be changed each time it is defined. For example, #> 3+4 ; ans = 7 #> ans / 10 ; ans = 0.7 #> [ 1,2,3 ] ; ans = [ ] #> ans.* ans ; ans = [ ] A few dot-only-functions are implemented.bisect(a,b, f(x)).secant(a,b,f(x)).spline(x,f).spline1(x,f).polyfit(x,f, kth) // to find a root of f(x)=0 // root of f(x)=0 via the secant method // 3 rd -order spline interpolation // 1 st -order spline interpolation // kth-degree polynomial regression.regress(x,f,mftn(x)) // regression curve by matrix mftn(double x) Data set can be connected by splines and represented by piecewise continuous polynomials as follows. In a spline matrix, the first two columns indicate both ends of each interval. #> x = [ ];; #> f = [ ];; 4

5 #> P =.spline(x,f); // 1st and 2nd columns for interval P = [ ] [ ] [ ] #> P.splplot; //.spline(x,f).splplot; #> [ x', f' ].plot+ ; #>.regress(x,f,1).plot+(1,5); #> P.spldiff.splplot; //.spline(x,f).spldiff.splplot; differentiate #> P.splint.splplot; //.spline(x,f).splint.splplot; integrate Sep. 23, 2012 ======================================= Operators for set are available from Cemmath beta V2.13. We believe that eduction of set concept in High-Schools can be particularly aided by these operators. A ++ B // set union, [2,3,4]++[3,4,5] = [2,3,4,5] A -- B // set difference, [2,3,4]- -[3,4,5] = [2] A /\ B // set intersection, [2,3,4] /\ [3,4,5] = [3,4] ++A // ++A = A++A, remove duplicate elements, 5

6 A ++= B // A = A ++ B A --= B // A = A -- B A /\= B // A = A /\ B Example usages are listed below. #> matrix.format("%3g"); #> A = [2,3,4,3,4,6]; B = [1,2,3,2,3,8,8,9]; A = [ ] B = [ ] #> ++[1,2,3,1,2] ; ans = [1,2,3] #> A++B; ans = [ ] #> A--B; B--A; ans = [ 4 6 ] ans = [ ] #> A /\ B; ans = [ 2 3 ] #> (0:2:30) /\ (0:3:30); ans = [ ] Sep. 18, 2012 ======================================= Cemmath beta Version V2.12 is released on the homepage. In this new version,dot functions are refined to cover the Bessel functions and special polynomials..i_nu(x).j_nu(x).k_nu(x).y_nu(x).p_n(x) // Legendre polynomial.t_n(x) // Tchebyshev polynomial.h_n(x) // Hermite polynomial.l_n(x) // Laguerre polynomial // Bessel functions #> nu = 1;; x = 0.3;; #>.J_nu(x); ans = #>.I_nu+1/2(x); ans = #>.K_nu*nu(x); ans = #>.Y_nu+2/3(x); ans = #>.J_nu( 0.3:0.1:0.6 ); ans = [ ] Therefore, math.j_nu(x) in old version is no more recommended. A member function of a matrix 'A.rowskip(k)' is newly added. 6

7 // // end of file //

Homework #6 Brief Solutions 2012

Homework #6 Brief Solutions 2012 Homework #6 Brief Solutions %page 95 problem 4 data=[-,;-,;,;4,] data = - - 4 xk=data(:,);yk=data(:,);s=csfit(xk,yk,-,) %Using the program to find the coefficients S =.456 -.456 -.. -.5.9 -.5484. -.58.87.