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Sparse Linear Systems in .NET Integrate pdf417 2d barcode in .NET Sparse Linear Systems

2.7 Sparse Linear Systems generate, create pdf417 2d barcode none for .net projects Microsoft SQL Server Matrix multip .NET barcode pdf417 ly A BT where A and B are two sparse matrices in row-index storage mode, and BT is the transpose of B. Here, sa and ija store the matrix A; sb and ijb store the matrix B.

This routine computes all components of the matrix product (which may be non-sparse!), but stores only those whose magnitude exceeds thresh. On output, the arrays sc and ijc (whose maximum size is input as nmax) give the product matrix in row-index storage mode. For sparse matrix multiplication, this routine will often be preceded by a call to sprstp, so as to construct the transpose of a known matrix into sb, ijb.

{ void nrerror(char error_text[]); unsigned long i,ijma,ijmb,j,k,ma,mb,mbb; float sum; if (ija[1] != ijb[1]) nrerror("sprstm: sizes do not match"); ijc[1]=k=ija[1]; for (i=1;i<=ija[1]-2;i++) { Loop over rows of A, for (j=1;j<=ijb[1]-2;j++) { and rows of B. if (i == j) sum=sa[i]*sb[j]; else sum=0.0e0; mb=ijb[j]; for (ma=ija[i];ma<=ija[i+1]-1;ma++) { Loop through elements in A s row.

Convoluted logic, following, accounts for the various combinations of diagonal and o -diagonal elements. ijma=ija[ma]; if (ijma == j) sum += sa[ma]*sb[j]; else { while (mb < ijb[j+1]) { ijmb=ijb[mb]; if (ijmb == i) { sum += sa[i]*sb[mb++]; continue; } else if (ijmb < ijma) { mb++; continue; } else if (ijmb == ijma) { sum += sa[ma]*sb[mb++]; continue; } break; } } } for (mbb=mb;mbb<=ijb[j+1]-1;mbb++) { Exhaust the remainder of B s row. if (ijb[mbb] == i) sum += sa[i]*sb[mbb]; } if (i == j) sc[i]=sum; Where to put the answer.

.. else if (fabs(sum) > thresh) { if (k > nmax) nrerror("sprstm: nmax too small"); sc[k]=sum; ijc[k++]=j; } } ijc[i+1]=k; } }.

Conjugate Gradient Method for a Sparse System So-called con Visual Studio .NET pdf417 2d barcode jugate gradient methods provide a quite general means for solving the N N linear system A x =b (2.7.

29) The attractiveness of these methods for large sparse systems is that they reference A only through its multiplication of a vector, or the multiplication of its transpose and a vector. As. 2. . Solution of Linear Algebraic Equations we have seen, these operations can be very ef cient for a properly stored sparse matrix. You, the owner of the matrix A, can be asked to provide functions that perform these sparse matrix multiplications as ef ciently as possible. We, the grand strategists supply the general routine, linbcg below, that solves the set of linear equations, (2.

7.29), using your functions. The simplest, ordinary conjugate gradient algorithm [11-13] solves (2.

7.29) only in the case that A is symmetric and positive de nite. It is based on the idea of minimizing the function 1 x A x b x 2 This function is minimized when its gradient f (x) = f = A x b (2.

7.30). (2.7.31).

is zero, whic h is equivalent to (2.7.29).

The minimization is carried out by generating a succession of search directions pk and improved minimizers xk . At each stage a quantity k is found that minimizes f (xk + k pk ), and xk+1 is set equal to the new point xk + k pk . The pk and xk are built up in such a way that xk+1 is also the minimizer of f over the whole vector space of directions already taken, {p1 , p2 , .

. . , pk }.

After N iterations you arrive at the minimizer over the entire vector space, i.e., the solution to (2.

7.29). Later, in 10.

6, we will generalize this ordinary conjugate gradient algorithm to the minimization of arbitrary nonlinear functions. Here, where our interest is in solving linear, but not necessarily positive de nite or symmetric, equations, a different generalization is important, the biconjugate gradient method. This method does not, in general, have a simple connection with function minimization.

It constructs four sequences of vectors, rk , rk , pk , pk , k = 1, 2, . . .

. You supply the initial vectors r1 and r1 , and set p1 = r1 , p1 = r1 . Then you carry out the following recurrence: k = rk+1 rk rk pk A pk = rk k A pk rk+1 rk+1 rk rk = rk+1 + k pk (2.

7.32).
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