Less-Numerical Algorithms in Software Assign pdf417 in Software Less-Numerical Algorithms

22. Less-Numerical Algorithms use software pdf-417 2d barcode generator todraw pdf417 in software iOS void mpneg(VecUch Software PDF 417 ar_IO &u) { Ones-complement negate the unsigned radix 256 number u. Int j,n=u.size(); Uint ireg=256; for (j=n-1;j>=0;j--) { ireg=255-u[j]+hibyte(ireg); u[j]=lobyte(ireg); } } void mpmov(VecUchar_O &u, VecUchar_I &v) { Move the unsigned radix 256 number v into u.

To achieve full accuracy, the array v must be as long as the array u. Int j,n=u.size(),m=v.

size(),n_min=MIN(n,m); for (j=0;j<n_min;j++) u[j]=v[j]; if (n > n_min) for(j=n_min;j<n-1;j++) u[j]=0; } void mplsh(VecUchar_IO &u) { Left-shift digits of unsigned radix 256 number u. The nal element of the array is set to 0. Int j,n=u.

size(); for (j=0;j<n-1;j++) u[j]=u[j+1]; u[n-1]=0; } Uchar lobyte(Uint x) {return (x & 0xff);} Uchar hibyte(Uint x) {return ((x >> 8) & 0xff);} The following, more complicated, methods have discussion and implementation below. void mpmul(VecUchar_O &w, VecUchar_I &u, VecUchar_I &v); void mpinv(VecUchar_O &u, VecUchar_I &v); void mpdiv(VecUchar_O &q, VecUchar_O &r, VecUchar_I &u, VecUchar_I &v); void mpsqrt(VecUchar_O &w, VecUchar_O &u, VecUchar_I &v); void mp2dfr(VecUchar_IO &a, string &s); string mppi(const Int np); };. Full multiplicati on of two strings of digits, if done by the traditional hand method, is not a fast operation: In multiplying two strings of length N , the multiplicand would be short-multiplied in turn by each byte of the multiplier, requiring O.N 2 / operations in all. We will see, however, that all the arithmetic operations on numbers of length N can in fact be done in O.

N log N log log N / operations. The trick is to recognize that multiplication is essentially a convolution ( 13.1) of the digits of the multiplicand and multiplier, followed by some kind of carry operation.

Consider, for example, two ways of writing the calculation 456 789: 456 789 4104 3648 3192 359784 4 7 36 40 42 118 7 5 6 8 9 45 54 48 93 54 8 4. 32 28 35 28 67 3 5 9. The tableau on th e left shows the conventional method of multiplication, in which three separate short multiplications of the full multiplicand (by 9, 8, and 7) are added. 22.7 Arithmetic at Arbitrary Precision to obtain the na l result. The tableau on the right shows a different method (sometimes taught for mental arithmetic), where the single-digit cross products are all computed (e.g.

8 6 D 48), then added in columns to obtain an incompletely carried result (here, the list 28; 67; 118; 93; 54). The nal step is a single pass from right to left, recording the single least-signi cant digit and carrying the higher digit or digits into the total to the left (e.g.

93 C 5 D 98, record the 8, carry 9). You can see immediately that the column sums in the right-hand method are components of the convolution of the digit strings, for example 118 D 4 9 C 5 8 C 6 7. In 13.

1, we learned how to compute the convolution of two vectors by the fast Fourier transform (FFT): Each vector is FFT d, the two complex transforms are multiplied, and the result is inverse-FFT d. Since the transforms are done with oating arithmetic, we need suf cient precision so that the exact integer value of each component of the result is discernible in the presence of roundoff error. We should therefore allow a (conservative) few times log2 .

log2 N / bits for roundoff in the FFT. A number of length N bytes in radix 256 can generate convolution components as large as the order of .256/2 N , thus requiring 16 C log2 N bits of precision for exact storage.

If it is the number of bits in the oating mantissa (cf. 22.2), we obtain the condition 16 C log2 N C few log2 log2 N < it (22.

7.3) We see that single precision, say with it D 24, is inadequate for any interesting value of N , while double precision, say with it D 53, allows N to be greater than 106 , corresponding to some millions of decimal digits. The use of Doub in the routines realft ( 12.

3) and four1 ( 12.2) is therefore a necessity, not merely a convenience, for this application..

void MParith::mpm PDF 417 for None ul(VecUchar_O &w, VecUchar_I &u, VecUchar_I &v) { Uses fast Fourier transform to multiply the unsigned radix 256 integers u[0..n-1] and v[0.

.m-1], yielding a product w[0..

n+m-1]. const Doub RX=256.0; Int j,nn=1,n=u.

size(),m=v.size(),p=w.size(),n_max=MAX(m,n); Doub cy,t; while (nn < n_max) nn <<= 1; Find the smallest usable power of 2 for the transform.

nn <<= 1; VecDoub a(nn,0.0),b(nn,0.0); for (j=0;j<n;j++) a[j]=u[j]; Move U and V to double precision oating arrays.

for (j=0;j<m;j++) b[j]=v[j]; realft(a,1); Perform the convolution: First, the two Fourier transrealft(b,1); forms. b[0] *= a[0]; Then multiply the complex results (real and imagib[1] *= a[1]; nary parts). for (j=2;j<nn;j+=2) { b[j]=(t=b[j])*a[j]-b[j+1]*a[j+1]; b[j+1]=t*a[j+1]+b[j+1]*a[j]; } realft(b,-1); Then do the inverse Fourier transform.

cy=0.0; Make a nal pass to do all the carries. for (j=nn-1;j>=0;j--) { t=b[j]/(nn >> 1)+cy+0.

5; The 0.5 allows for roundo error. cy=Uint(t/RX); b[j]=t-cy*RX; } if (cy >= RX) throw("cannot happen in mpmul"); for (j=0;j<p;j++) w[j]=0; w[0]=Uchar(cy); Copy answer to output.

for (j=1;j<MIN(n+m,p);j++) w[j]=Uchar(b[j-1]); } mparith.h.
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