Successive Gaussian elimination

The original determinant M(A) can be easily brought to a diblock-diagonal form again with the help of successive Gaussian elimination. In this way one obtains for its diagonal blocks, in completely analogy to equation (4a),... 

Here a, and fiy are the usual Huckel parameters and A denotes the unknown root (energy eigenvalue) of the determinant. In a disordered chain the values of a, and are different from each other. The determinant I H(A) can be easily transformed into a didiagonal form with the help of successive Gaussian eliminations. To achieve this we must subtract from the second row of the determinant the first row multiplied by This will eliminate the element 2- Continuing this... 

The most widely used algorithm is the systematic elimination or Gaussian elimination by partial pivoting. The success of this method is due to its stability, i.e. the algorithm produces small residuals r = Ax — b (x being the numerical solution of the system), despite round-off errors introduced by the computer during computations. The concept of stability of a numerical algorithm will be discussed in more detail in Sect. 4.5. 

Gaussian elimination Algorithm that solves a linear system of equations via successive elimination of its unknowns, or, equivalently, via decomposition of the input coefficient matrix into a product of two triangular matrices. 

Figure 26. (A-G) Seven consecutive images recorded from the marked square section of Figure 24. The time interval of successive records is 13 min. In order to eliminate pixel-to-pixel counting statistical noise, the data have been smoothed with a Gaussian filter (half-width of 3 pixels) at the cost of spatial resolution. The intensity changes of individual molecules from image to image (e.g. molecules labeled 1 and 2) and their lateral displacement (molecules in 3) are striking. (H) Trajectories of the marked molecules. (Adopted from [92].)...

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