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We choose the sign of in (11.2.14) to be the same as the sign of ai,i 1 to lessen roundoff error. using barcode implement for none control to generate, create none image in none qr code 11.2 Reduction of a Symmetric Matrix to Tridiagonal Form BIRT Reporting Tools Variables are thus com puted in the following order: , u, H, p, K, q, A . At any stage m, A is tridiagonal in its last m 1 rows and columns. If the eigenvectors of the nal tridiagonal matrix are found (for example, by the routine in the next section), then the eigenvectors of A can be obtained by applying the accumulated transformation Q = P1 P2 Pn 2 (11.

2.17). to those eigenvectors. none for none We therefore form Q by recursion after all the P s have been determined: Qn 2 = Pn 2 Qj = Pj Qj+1 , Q = Q1 Input for the routine below is the real, symmetric matrix a[1..


On output, a contains the elements of the orthogonal matrix q. The vector d[1..

n] is set to the diagonal elements of the tridiagonal matrix A , while the vector e[1..n] is set to the off-diagonal elements in its components 2 through n, with e[1]=0.

Note that since a is overwritten, you should copy it before calling the routine, if it is required for subsequent computations. No extra storage arrays are needed for the intermediate results. At stage m, the vectors p and q are nonzero only in elements 1, .

. . , i (recall that i = n m + 1), while u is nonzero only in elements 1, .

. . , i 1.

The elements of the vector e are being determined in the order n, n 1, . . .

, so we can store p in the elements of e not already determined. The vector q can overwrite p once p is no longer needed. We store u in the ith row of a and u/H in the ith column of a.

Once the reduction is complete, we compute the matrices Qj using the quantities u and u/H that have been stored in a. Since Qj is an identity matrix in the last n j + 1 rows and columns, we only need compute its elements up to row and column n j. These can overwrite the u s and u/H s in the corresponding rows and columns of a, which are no longer required for subsequent Q s.

The routine tred2, given below, includes one further re nement. If the quantity is zero or small at any stage, one can skip the corresponding transformation. A simple criterion, such as < smallest positive number representable on machine machine precision j = n 3, .

. . , 1 (11.

2.18). would be ne most of the time. A more careful criterion is actually used. De ne the quantity aik (11.2.19).

If = 0 to machine prec none none ision, we skip the transformation. Otherwise we rede ne aik becomes aik / (11.2.

20). 11. . Eigensystems and use the scaled var iables for the transformation. (A Householder transformation depends only on the ratios of the elements.) Note that when dealing with a matrix whose elements vary over many orders of magnitude, it is important that the matrix be permuted, insofar as possible, so that the smaller elements are in the top left-hand corner.

This is because the reduction is performed starting from the bottom right-hand corner, and a mixture of small and large elements there can lead to considerable rounding errors. The routine tred2 is designed for use with the routine tqli of the next section. tqli nds the eigenvalues and eigenvectors of a symmetric, tridiagonal matrix.

The combination of tred2 and tqli is the most ef cient known technique for nding all the eigenvalues and eigenvectors (or just all the eigenvalues) of a real, symmetric matrix. In the listing below, the statements indicated by comments are required only for subsequent computation of eigenvectors. If only eigenvalues are required, omission of the commented statements speeds up the execution time of tred2 by a factor of 2 for large n.

In the limit of large n, the operation count of the Householder reduction is 2n3 /3 for eigenvalues only, and 4n3 /3 for both eigenvalues and eigenvectors.. #include <math.h> none for none ; void tred2(float **a, int n, float d[], float e[]) Householder reduction of a real, symmetric matrix a[1..


On output, a is replaced by the orthogonal matrix Q e ecting the transformation. d[1..

n] returns the diagonal elements of the tridiagonal matrix, and e[1..n] the o -diagonal elements, with e[1]=0.

Several statements, as noted in comments, can be omitted if only eigenvalues are to be found, in which case a contains no useful information on output. Otherwise they are to be included. { int l,k,j,i; float scale,hh,h,g,f; for (i=n;i>=2;i--) { l=i-1; h=scale=0.

0; if (l > 1) { for (k=1;k<=l;k++) scale += fabs(a[i][k]); if (scale == 0.0) Skip transformation. e[i]=a[i][l]; else { for (k=1;k<=l;k++) { a[i][k] /= scale; Use scaled a s for transformation.

h += a[i][k]*a[i][k]; Form in h. } f=a[i][l]; g=(f >= 0.0 -sqrt(h) : sqrt(h)); e[i]=scale*g; h -= f*g; Now h is equation (11.

2.4). a[i][l]=f-g; Store u in the ith row of a.

f=0.0; for (j=1;j<=l;j++) { /* Next statement can be omitted if eigenvectors not wanted */ a[j][i]=a[i][j]/h; Store u/H in ith column of a. g=0.

0; Form an element of A u in g. for (k=1;k<=j;k++) g += a[j][k]*a[i][k]; for (k=j+1;k<=l;k++) g += a[k][j]*a[i][k]; e[j]=g/h; Form element of p in temporarily unused element of e..

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