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x e in Software Include pdf417 2d barcode in Software x e

x e using software touse pdf417 2d barcode with asp.net web,windows application iPhone OS f .x/dx D N 1 X j D0 wj f .xj /. (4.6.18).

gauss wgts.h void gaulag(VecDoub_ O &x, VecDoub_O &w, const Doub alf) Given alf, the parameter of the Laguerre polynomials, this routine returns arrays x[0..n-1] and w[0.

.n-1] containing the abscissas and weights of the n-point Gauss-Laguerre quadrature formula. The smallest abscissa is returned in x[0], the largest in x[n-1].

{ const Int MAXIT=10; const Doub EPS=1.0e-14; EPS is the relative precision. Int i,its,j; Doub ai,p1,p2,p3,pp,z,z1; Int n=x.

size(); for (i=0;i<n;i++) { Loop over the desired roots. if (i == 0) { Initial guess for the smallest root. z=(1.

0+alf)*(3.0+0.92*alf)/(1.

0+2.4*n+1.8*alf); } else if (i == 1) { Initial guess for the second root.

z += (15.0+6.25*alf)/(1.

0+0.9*alf+2.5*n); } else { Initial guess for the other roots.

ai=i-1;. 4.6 Gaussian Quadratures and Orthogonal Polynomials z += ((1.0+2.55*ai)/ (1.

9*ai)+1.26*ai*alf/ (1.0+3.

5*ai))*(z-x[i-2])/(1.0+0.3*alf); } for (its=0;its<MAXIT;its++) { Re nement by Newton s method.

p1=1.0; p2=0.0; for (j=0;j<n;j++) { Loop up the recurrence relation to get the p3=p2; Laguerre polynomial evaluated at z.

p2=p1; p1=((2*j+1+alf-z)*p2-(j+alf)*p3)/(j+1); } p1 is now the desired Laguerre polynomial. We next compute pp, its derivative, by a standard relation involving also p2, the polynomial of one lower order. pp=(n*p1-(n+alf)*p2)/z; z1=z; z=z1-p1/pp; Newton s formula.

if (abs(z-z1) <= EPS) break; } if (its >= MAXIT) throw("too many iterations in gaulag"); x[i]=z; Store the root and the weight. w[i] = -exp(gammln(alf+n)-gammln(Doub(n)))/(pp*n*p2); } }. Next is a routine fo PDF-417 2d barcode for None r Gauss-Hermite abscissas and weights. If we use the standard normalization of these functions, as given in equation (4.6.

13), we nd that the computations over ow for large N because of various factorials that occur. We z can avoid this by using instead the orthonormal set of polynomials Hj . They are generated by the recurrence r r 2 z j z z z 1 D 0; H0 D 1 ; Hj C1 D x z H Hj Hj 1 (4.

6.19) 1=4 j C1 j C1 The formula for the weights becomes wj D 2 z0 HN .xj / 2 (4.

6.20). while the formula for the derivative with this normalization is p z z Hj0 D 2j Hj 1 (4.6.21).

The abscissas and weights returned by gauher are used with the integration formula Z f .x/dx D N 1 X j D0 wj f .xj /. (4.6.22).

void gauher(VecDoub_ pdf417 2d barcode for None O &x, VecDoub_O &w) This routine returns arrays x[0..n-1] and w[0.

.n-1] containing the abscissas and weights of the n-point Gauss-Hermite quadrature formula. The largest abscissa is returned in x[0], the most negative in x[n-1].

{ const Doub EPS=1.0e-14,PIM4=0.7511255444649425; Relative precision and 1= 1=4 .

const Int MAXIT=10; Maximum iterations. Int i,its,j,m;. gauss wgts.h 4. Integration of Functions Doub p1,p2,p3,pp,z,z 1; Int n=x.size(); m=(n+1)/2; The roots are symmetric about the origin, so we have to nd only half of them. for (i=0;i<m;i++) { Loop over the desired roots.

if (i == 0) { Initial guess for the largest root. z=sqrt(Doub(2*n+1))-1.85575*pow(Doub(2*n+1),-0.

16667); } else if (i == 1) { Initial guess for the second largest root. z -= 1.14*pow(Doub(n),0.

426)/z; } else if (i == 2) { Initial guess for the third largest root. z=1.86*z-0.

86*x[0]; } else if (i == 3) { Initial guess for the fourth largest root. z=1.91*z-0.

91*x[1]; } else { Initial guess for the other roots. z=2.0*z-x[i-2]; } for (its=0;its<MAXIT;its++) { Re nement by Newton s method.

p1=PIM4; p2=0.0; for (j=0;j<n;j++) { Loop up the recurrence relation to get p3=p2; the Hermite polynomial evaluated at p2=p1; z. p1=z*sqrt(2.

0/(j+1))*p2-sqrt(Doub(j)/(j+1))*p3; } p1 is now the desired Hermite polynomial. We next compute pp, its derivative, by the relation (4.6.

21) using p2, the polynomial of one lower order. pp=sqrt(Doub(2*n))*p2; z1=z; z=z1-p1/pp; Newton s formula. if (abs(z-z1) <= EPS) break; } if (its >= MAXIT) throw("too many iterations in gauher"); x[i]=z; Store the root x[n-1-i] = -z; and its symmetric counterpart.

w[i]=2.0/(pp*pp); Compute the weight w[n-1-i]=w[i]; and its symmetric counterpart. } }.

Finally, here is a r barcode pdf417 for None outine for Gauss-Jacobi abscissas and weights, which implement the integration formula Z. x/ .1 C x/ f .x/dx D N 1 X j D0 wj f .xj /. (4.6.23).

gauss wgts.h void gaujac(VecDoub_ Software PDF-417 2d barcode O &x, VecDoub_O &w, const Doub alf, const Doub bet) Given alf and bet, the parameters and of the Jacobi polynomials, this routine returns arrays x[0..n-1] and w[0.

.n-1] containing the abscissas and weights of the n-point GaussJacobi quadrature formula. The largest abscissa is returned in x[0], the smallest in x[n-1].

{ const Int MAXIT=10; const Doub EPS=1.0e-14; Int i,its,j; Doub alfbet,an,bn,r1,r2,r3; Doub a,b,c,p1,p2,p3,pp,temp,z,z1; Int n=x.size(); for (i=0;i<n;i++) { if (i == 0) { an=alf/n; EPS is the relative precision.

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