matrix triangularization routine that uses full row and column pivoting has been tried with 

 no resultant increase in accuracy. 



FINDING EIGENVALUES 



There are versions of this program under development that will locate the eigenvalues 

 and do the entire computation without user intervention. Currently, however, these versions 

 are reliable only for the simpler types of profiles — usually those with only one duct — and 

 are not ready to be reported. Locating eigenvalues with the standard version of the program 

 is discussed here. 



The standard version of the program requires the user to find the eigenvalues. In this 

 version, each time an eigenvalue is determined by iteration, the resulting value is stored and 

 counted as an eigenvalue. Therefore, the user must ensure that all iterations result in good 

 roots, that all required modes have been determined, and that no modes are present more 

 than once. In most cases the user must expect to make more than one computer run to 

 obtain this result. 



CONTROL CARDS 



The user controls the eigenvalue determination by using any of four different types 

 of control cards. The first type specifies an initial value for v and an initial step size, Av. 

 These are both complex numbers with a real and an imaginary part. G is then evaluated at v 

 and at V + Av to start the iteration. These are essentially the v; and vj+j of eq (15). If these 

 two trial eigenvalues are in the vicinity of a root, the iteration will converge to that root. 



The second type of card specifies a line segment in the complex plane, along which a 

 search for eigenvalues, v, is made. The end points of the line are given along with the number 

 of equally spaced points at which the line is to be divided. G is then evaluated at each suc- 

 cessive division point along the line until a relative minimum in IG^I is found, indicating that 

 a root is nearby. The iterative process is applied to find the root. The initial step size, Av, is 

 first computed to bring the second evaluation at v + Av as close as possible to the true root. 

 This is done by using the point which resulted in minimum iG2| and the points on either side 

 of it to determine the minimum of the parabola passing through them. If v - h, v, and v + h 

 are the three points at which G was evaluated, it follows that the distance from v to the mini- 

 mum of the parabola 



Av = h[G(v + h) - G(v - h)] /2[2G(v) - G(v + h) - G(v - h)] . (26) 



When the iteration is complete, the eigenvalue is recorded and the program continues 

 to step along in the direction of the given line, checking again for a minimum. However, the 

 stepping is resumed from the newly located root rather than from the approximate location 

 where the minimum was detected. With this correction in position, the designated Une does 

 not have to hug the curve on which the eigenvalues are located because it is corrected at each 

 eigenvalue. 



This method of finding eigenvalues has proven very successful. Its main utility arises, 

 though, because the eigenvalues of the trapped modes have negligible imaginary parts and the 



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