Newton's method uses function and Jacobi evaluations to find a root of a function at a quadratic rate of convergence. Householder's methods, Halley's method, and related methods would converge at a faster rate but require higher derivatives, rendering them impractical.
We propose a novel class of methods that use only function and Jacobi information but converge at higher rates than Newton. We provide one particular candidate of our method class that converges of fourth order.
Until now, Newton's method is the go-to approach for numerically solving systems of nonlinear equations in practice. Given are an initial guess x0 in R^n and a function F : R^n --> R^n with locally well-defined square regular Jacobi matrix J(x).
Given x0, one iteration of Newton's method computes an improved guess x in attempt to solve F(x)=0.
One iteration proceeds as follows:
- Evaluate A:=J(x0)
- linsolver.compute(A)
- Set x=x0
- Evaluate b:=F(x)
- x=x-linsolver.solve(b) // after the update, x satisfies that ||F(x)|| is in O( ||F(x0)||^2 )
We explain these steps: The first step evaluates the function J at x0 and writes the result in an unspecified format into the storage A. In the second step, a linear system solver compatible with the format of A prepares the solution of linear systems with system matrix A. The third step is for pleasing notation of follow-up analysis only. The fourth step evaluates F at x and writes the result in an unspecified format into the storage of b. The final step decrements x by the solution of the linear system with system matrix A and right-hand side vector b.
An iterative loop would eventually set x0:=x and then repeat the steps all over again until x is sufficiently accurate.
In practical applications, the computational cost of one iteration is typically dominated to >=90% by steps 1 and 2, particularly when n is not tiny.
Newton's method is known to converge quadratically. If J is local Lipschitz-continuous then F(x) converges locally quadratic. That is, in Landau-notation according to Knuth: ||F(x)|| is in O( ||F(x0)||^2 )
When J is regular in a local neighborhood of the exact root of F then F(x)~x are bijective and hence the order result implies convergence of the error at the same rate.
We developed a little symbolic computation code for some other application. We then wondered whether we could not use that code to also find a superior iteration over Newton's method. In particular, we sought for iteration schemes of the following form:
-A * v_k = J(x0+sum_{j=1}^{k-1} alpha_kj v_j) * (sum_{j=1}^{k-1} beta_kj v_j) + F(x0+sum_{j=1}^{k-1} gamma_kj v_j) for k=1,...,m x = x0 + sum_k delta_k v_k
This forms the class of DJN methods. Therein, m is an integer number, v_k are vectors that solve linear systems, x0 is the guess, and x is the improved guess after one iteration. The coefficients alpha, beta, gamma, delta define a particular candidate (aka method) of the method class. Newton is the trivial case m=1, delta_1=1. Certainly desirable are methods that use beta=0, thus the Jacobian needs only be evaluated once --which can be expensive in comparison to linsolver.compute(A) in some applications and is thus attractive to avoid.
A DJN method performs step 2 (and step 1 if beta=0) only once per iteration but replaces the steps 3 to 5 with some other computations of similarly negligible cost. We sought for a DJN method candidate that uses m=3 intermediate steps, uses beta=0, and converges of fourth order. We succeeded. We now show what we did.
For k=1,...,m you successively develop symbolic Taylor series of F(x0+sum_{j=1}^{k-1} gamma_kj v_j) around x0, and thus obtain an expression of Taylor series for each v_k. You thus obtain a symbolic expression for x0 + sum_k delta_k v_k. Finally, you develop the symbolic Taylor series of F(x0 + sum_k delta_k v_k) around x0 into which you insert that symbolic expression which you had just obtained for sum_k delta_k v_k.
In the resulting formula, you compare terms and try to eliminate as many low-order terms of the Taylor series as possible. The lowest order-term that remains is the order of the resulting method.
Example: For the scheme with m=1 you have -A * v_1 = F(x0) --> v1 = -inv(A) * F0 --> x = x0 - delta * inv(A) * F0 F(x) = F0 - A * delta_1 * inv(A) * F0 + O( ||delta_1 * inv(A) * F0||^2 ) = (1-delta_1) *F0 + O(||F0||^2).
This explains why Newton's method (aka, m=1, delta_1=1) converges locally quadratic. In the same way, you can develop Taylor series of higher orders for F(x0+sum gamma_kj v_j) and for F(x0+sum delta_k v_k).
The symbolic computation of the priorly described instructions (Taylor series of Taylor series) reveals the expressions given in Appendix A.
The expressions until before the fourh-order condition are set to zero when gamma_21=1, gamma_31=0, gamma_32=1, and delta_1=delta_2=delta_3=1. Hence, that ||F(x)|| is in O( ||F(x0)||^4 ) results from choosing the coefficients of gamma and delta like so. The analysis uses the fourth-order Taylor series of F. Thus, the result only holds when the third derivative of F is Lipschitz-continuous.
For the readers' convenience, the method with these coefficients inserted is now written out.
- Evaluate A:=J(x0)
- linsolver.compute(A)
- Set x=x0
- Evaluate b:=F(x)
- x=x-linsolver.solve(b) // after the update, x satisfies that ||F(x)|| is in O( ||F(x0)||^2 )
- Evaluate b:=F(x)
- x=x-linsolver.solve(b) // after the update, x satisfies that ||F(x)|| is in O( ||F(x0)||^3 )
- Evaluate b:=F(x)
- x=x-linsolver.solve(b) // after the update, x satisfies that ||F(x)|| is in O( ||F(x0)||^4 )
This is one iteration of our proposed DJN method candidate. By mere coincidence, one iteration of our proposed method coincides with exactly three iterations of Chord's method.
We have presented a class of methods that is roughly of the same cost as Newton's method per iteration. However, methods of our class can converge of much higher rate than Newton's method.
There was a time when people would solve differential equations numerically via Euler's method. These times are over since the advent of Runge-Kutta methods. Likewise, the time where people would solve nonlinear equation systems via Newton's method might be over now. And at the time of this publication there are a few hundreds of million devices out there that run Newton iterations every second.
For a JDN method that yields fifth order, there result 16 equations that must be solved by the chosen coefficients. We have been unable to solve these equations because they are 1.7e+5 lines of code and we fail at finding a compiler that compiles the file. We hence cannot even evaluate the residuals in order to check whether a trivial extension of the above method would be of fifth order. A trivial extension is a method with beta=0, delta=1, and gamma a lower-left-shifted identity matrix.
The following hash is the SHA256 digest of a text that states the discoverer's identity. aedea59b60118791775472f12293e1a70a7dcb9c39031a5a2e084bdc4c56e85f If you want to use any work from this page then cite this URL.
We actually developed the code to develop numerical schemes for linearly implicit ODEs of the form -M(y,t) * y'(t) = f(y,t) with M regular in a local neighborhood of the unique solution y(t).
The below single-step method is of third order, with the attractive feature of only requiring one matrix factorization per single-step:
- linsolver.compute( func_M(y0,t0) );
- v1 = -linsolver.solve( func_f(y0 ,t0 ) );
- v2 = -linsolver.solve( func_M(y0+h2/3v1 ,t0+h2/3) * v1 + func_f(y0+h2/3v1,t0+h2/3) );
- v3 = -linsolver.solve( func_M(y0+h2 v1+hv2,t0+h4/3) * 2 * v2 + func_f(y0+h4/3v2,t0 ) );
- t = t0 + h;
- y = y0 + h/16 * (13v1+18v2+3*v3);
The residuals y_ab are for the b-th condition of the a-th order. For instance, the first and second order have one conditional equation; whereas the third order has three conditional equations. The reader can verify that the above choice for gamma and delta yields all residuals to zero. While we publish the residuals, we do not publish our code because there is no reason or need. Readers may choose to reproduce and confirm all our results independently from the descriptions.
y11 = ...
+ ( 1 ) ...
- ( delta_1 ) ...
- ( delta_2 ) ...
+ ( delta_2 .* gamma21 ) ...
- ( delta_3 ) ...
+ ( delta_3 .* gamma31 ) ...
+ ( delta_3 .* gamma32 ) ...
- ( delta_3 .* gamma32 .* gamma21 ) ;
y21 = ...
+ ( delta_1 .* delta_1 ) ...
+ ( delta_1 .* delta_2 ) ...
- ( delta_1 .* delta_2 .* gamma21 ) ...
+ ( delta_1 .* delta_3 ) ...
- ( delta_1 .* delta_3 .* gamma31 ) ...
- ( delta_1 .* delta_3 .* gamma32 ) ...
+ ( delta_1 .* delta_3 .* gamma32 .* gamma21 ) ...
+ ( delta_2 .* delta_1 ) ...
+ ( delta_2 .* delta_2 ) ...
- ( delta_2 .* delta_2 .* gamma21 ) ...
+ ( delta_2 .* delta_3 ) ...
- ( delta_2 .* delta_3 .* gamma31 ) ...
- ( delta_2 .* delta_3 .* gamma32 ) ...
+ ( delta_2 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_2 .* gamma21 .* delta_1 ) ...
- ( delta_2 .* gamma21 .* delta_2 ) ...
+ ( delta_2 .* gamma21 .* delta_2 .* gamma21 ) ...
- ( delta_2 .* gamma21 .* delta_3 ) ...
+ ( delta_2 .* gamma21 .* delta_3 .* gamma31 ) ...
+ ( delta_2 .* gamma21 .* delta_3 .* gamma32 ) ...
- ( delta_2 .* gamma21 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_2 .* gamma21 .* gamma21 ) ...
+ ( delta_3 .* delta_1 ) ...
+ ( delta_3 .* delta_2 ) ...
- ( delta_3 .* delta_2 .* gamma21 ) ...
+ ( delta_3 .* delta_3 ) ...
- ( delta_3 .* delta_3 .* gamma31 ) ...
- ( delta_3 .* delta_3 .* gamma32 ) ...
+ ( delta_3 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* gamma31 .* delta_1 ) ...
- ( delta_3 .* gamma31 .* delta_2 ) ...
+ ( delta_3 .* gamma31 .* delta_2 .* gamma21 ) ...
- ( delta_3 .* gamma31 .* delta_3 ) ...
+ ( delta_3 .* gamma31 .* delta_3 .* gamma31 ) ...
+ ( delta_3 .* gamma31 .* delta_3 .* gamma32 ) ...
- ( delta_3 .* gamma31 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* gamma31 .* gamma31 ) ...
- ( delta_3 .* gamma31 .* gamma32 ) ...
+ ( delta_3 .* gamma31 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* delta_1 ) ...
- ( delta_3 .* gamma32 .* delta_2 ) ...
+ ( delta_3 .* gamma32 .* delta_2 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* delta_3 ) ...
+ ( delta_3 .* gamma32 .* delta_3 .* gamma31 ) ...
+ ( delta_3 .* gamma32 .* delta_3 .* gamma32 ) ...
- ( delta_3 .* gamma32 .* delta_3 .* gamma32 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_1 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_2 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* delta_2 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_3 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma31 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* gamma31 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* gamma32 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* gamma31 ) ...
- ( delta_3 .* gamma32 .* gamma32 ) ...
+ ( delta_3 .* gamma32 .* gamma32 .* gamma21 ) ;
y31 = ...
+ ( delta_1 .* delta_2 .* gamma21 .* gamma21 ) ...
+ ( delta_1 .* delta_3 .* gamma31 .* gamma31 ) ...
+ ( delta_1 .* delta_3 .* gamma31 .* gamma32 ) ...
- ( delta_1 .* delta_3 .* gamma31 .* gamma32 .* gamma21 ) ...
- ( delta_1 .* delta_3 .* gamma32 .* gamma21 .* gamma21 ) ...
- ( delta_1 .* delta_3 .* gamma32 .* gamma21 .* gamma31 ) ...
- ( delta_1 .* delta_3 .* gamma32 .* gamma21 .* gamma32 ) ...
+ ( delta_1 .* delta_3 .* gamma32 .* gamma21 .* gamma32 .* gamma21 ) ...
+ ( delta_1 .* delta_3 .* gamma32 .* gamma31 ) ...
+ ( delta_1 .* delta_3 .* gamma32 .* gamma32 ) ...
- ( delta_1 .* delta_3 .* gamma32 .* gamma32 .* gamma21 ) ...
+ ( delta_2 .* delta_2 .* gamma21 .* gamma21 ) ...
+ ( delta_2 .* delta_3 .* gamma31 .* gamma31 ) ...
+ ( delta_2 .* delta_3 .* gamma31 .* gamma32 ) ...
- ( delta_2 .* delta_3 .* gamma31 .* gamma32 .* gamma21 ) ...
- ( delta_2 .* delta_3 .* gamma32 .* gamma21 .* gamma21 ) ...
- ( delta_2 .* delta_3 .* gamma32 .* gamma21 .* gamma31 ) ...
- ( delta_2 .* delta_3 .* gamma32 .* gamma21 .* gamma32 ) ...
+ ( delta_2 .* delta_3 .* gamma32 .* gamma21 .* gamma32 .* gamma21 ) ...
+ ( delta_2 .* delta_3 .* gamma32 .* gamma31 ) ...
+ ( delta_2 .* delta_3 .* gamma32 .* gamma32 ) ...
- ( delta_2 .* delta_3 .* gamma32 .* gamma32 .* gamma21 ) ...
- ( delta_2 .* gamma21 .* delta_2 .* gamma21 .* gamma21 ) ...
- ( delta_2 .* gamma21 .* delta_3 .* gamma31 .* gamma31 ) ...
- ( delta_2 .* gamma21 .* delta_3 .* gamma31 .* gamma32 ) ...
+ ( delta_2 .* gamma21 .* delta_3 .* gamma31 .* gamma32 .* gamma21 ) ...
+ ( delta_2 .* gamma21 .* delta_3 .* gamma32 .* gamma21 .* gamma21 ) ...
+ ( delta_2 .* gamma21 .* delta_3 .* gamma32 .* gamma21 .* gamma31 ) ...
+ ( delta_2 .* gamma21 .* delta_3 .* gamma32 .* gamma21 .* gamma32 ) ...
- ( delta_2 .* gamma21 .* delta_3 .* gamma32 .* gamma21 .* gamma32 .* gamma21 ) ...
- ( delta_2 .* gamma21 .* delta_3 .* gamma32 .* gamma31 ) ...
- ( delta_2 .* gamma21 .* delta_3 .* gamma32 .* gamma32 ) ...
+ ( delta_2 .* gamma21 .* delta_3 .* gamma32 .* gamma32 .* gamma21 ) ...
+ ( delta_3 .* delta_2 .* gamma21 .* gamma21 ) ...
+ ( delta_3 .* delta_3 .* gamma31 .* gamma31 ) ...
+ ( delta_3 .* delta_3 .* gamma31 .* gamma32 ) ...
- ( delta_3 .* delta_3 .* gamma31 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* delta_3 .* gamma32 .* gamma21 .* gamma21 ) ...
- ( delta_3 .* delta_3 .* gamma32 .* gamma21 .* gamma31 ) ...
- ( delta_3 .* delta_3 .* gamma32 .* gamma21 .* gamma32 ) ...
+ ( delta_3 .* delta_3 .* gamma32 .* gamma21 .* gamma32 .* gamma21 ) ...
+ ( delta_3 .* delta_3 .* gamma32 .* gamma31 ) ...
+ ( delta_3 .* delta_3 .* gamma32 .* gamma32 ) ...
- ( delta_3 .* delta_3 .* gamma32 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* gamma31 .* delta_2 .* gamma21 .* gamma21 ) ...
- ( delta_3 .* gamma31 .* delta_3 .* gamma31 .* gamma31 ) ...
- ( delta_3 .* gamma31 .* delta_3 .* gamma31 .* gamma32 ) ...
+ ( delta_3 .* gamma31 .* delta_3 .* gamma31 .* gamma32 .* gamma21 ) ...
+ ( delta_3 .* gamma31 .* delta_3 .* gamma32 .* gamma21 .* gamma21 ) ...
+ ( delta_3 .* gamma31 .* delta_3 .* gamma32 .* gamma21 .* gamma31 ) ...
+ ( delta_3 .* gamma31 .* delta_3 .* gamma32 .* gamma21 .* gamma32 ) ...
- ( delta_3 .* gamma31 .* delta_3 .* gamma32 .* gamma21 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* gamma31 .* delta_3 .* gamma32 .* gamma31 ) ...
- ( delta_3 .* gamma31 .* delta_3 .* gamma32 .* gamma32 ) ...
+ ( delta_3 .* gamma31 .* delta_3 .* gamma32 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* gamma31 .* gamma32 .* gamma21 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* delta_2 .* gamma21 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* delta_3 .* gamma31 .* gamma31 ) ...
- ( delta_3 .* gamma32 .* delta_3 .* gamma31 .* gamma32 ) ...
+ ( delta_3 .* gamma32 .* delta_3 .* gamma31 .* gamma32 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* delta_3 .* gamma32 .* gamma21 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* delta_3 .* gamma32 .* gamma21 .* gamma31 ) ...
+ ( delta_3 .* gamma32 .* delta_3 .* gamma32 .* gamma21 .* gamma32 ) ...
- ( delta_3 .* gamma32 .* delta_3 .* gamma32 .* gamma21 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* delta_3 .* gamma32 .* gamma31 ) ...
- ( delta_3 .* gamma32 .* delta_3 .* gamma32 .* gamma32 ) ...
+ ( delta_3 .* gamma32 .* delta_3 .* gamma32 .* gamma32 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_2 .* gamma21 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma31 .* gamma31 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma31 .* gamma32 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma31 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 .* gamma21 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 .* gamma21 .* gamma31 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 .* gamma21 .* gamma32 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 .* gamma21 .* gamma32 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 .* gamma31 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 .* gamma32 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 .* gamma32 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* gamma32 .* gamma21 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* gamma32 .* gamma21 .* gamma21 ) ;
y32 = ...
+ ( delta_2 .* gamma21 .* gamma21 .* delta_1 ) ...
+ ( delta_2 .* gamma21 .* gamma21 .* delta_2 ) ...
- ( delta_2 .* gamma21 .* gamma21 .* delta_2 .* gamma21 ) ...
+ ( delta_2 .* gamma21 .* gamma21 .* delta_3 ) ...
- ( delta_2 .* gamma21 .* gamma21 .* delta_3 .* gamma31 ) ...
- ( delta_2 .* gamma21 .* gamma21 .* delta_3 .* gamma32 ) ...
+ ( delta_2 .* gamma21 .* gamma21 .* delta_3 .* gamma32 .* gamma21 ) ...
+ ( delta_3 .* gamma31 .* gamma31 .* delta_1 ) ...
+ ( delta_3 .* gamma31 .* gamma31 .* delta_2 ) ...
- ( delta_3 .* gamma31 .* gamma31 .* delta_2 .* gamma21 ) ...
+ ( delta_3 .* gamma31 .* gamma31 .* delta_3 ) ...
- ( delta_3 .* gamma31 .* gamma31 .* delta_3 .* gamma31 ) ...
- ( delta_3 .* gamma31 .* gamma31 .* delta_3 .* gamma32 ) ...
+ ( delta_3 .* gamma31 .* gamma31 .* delta_3 .* gamma32 .* gamma21 ) ...
+ ( delta_3 .* gamma31 .* gamma32 .* delta_1 ) ...
+ ( delta_3 .* gamma31 .* gamma32 .* delta_2 ) ...
- ( delta_3 .* gamma31 .* gamma32 .* delta_2 .* gamma21 ) ...
+ ( delta_3 .* gamma31 .* gamma32 .* delta_3 ) ...
- ( delta_3 .* gamma31 .* gamma32 .* delta_3 .* gamma31 ) ...
- ( delta_3 .* gamma31 .* gamma32 .* delta_3 .* gamma32 ) ...
+ ( delta_3 .* gamma31 .* gamma32 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* gamma31 .* gamma32 .* gamma21 .* delta_1 ) ...
- ( delta_3 .* gamma31 .* gamma32 .* gamma21 .* delta_2 ) ...
+ ( delta_3 .* gamma31 .* gamma32 .* gamma21 .* delta_2 .* gamma21 ) ...
- ( delta_3 .* gamma31 .* gamma32 .* gamma21 .* delta_3 ) ...
+ ( delta_3 .* gamma31 .* gamma32 .* gamma21 .* delta_3 .* gamma31 ) ...
+ ( delta_3 .* gamma31 .* gamma32 .* gamma21 .* delta_3 .* gamma32 ) ...
- ( delta_3 .* gamma31 .* gamma32 .* gamma21 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* gamma21 .* delta_1 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* gamma21 .* delta_2 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* gamma21 .* delta_2 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* gamma21 .* delta_3 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* gamma21 .* delta_3 .* gamma31 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* gamma21 .* delta_3 .* gamma32 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* gamma21 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* gamma21 .* gamma31 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* gamma21 .* gamma32 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* gamma21 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* gamma31 .* delta_1 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* gamma31 .* delta_2 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* gamma31 .* delta_2 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* gamma31 .* delta_3 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* gamma31 .* delta_3 .* gamma31 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* gamma31 .* delta_3 .* gamma32 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* gamma31 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* gamma32 .* delta_1 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* gamma32 .* delta_2 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* gamma32 .* delta_2 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* gamma32 .* delta_3 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* gamma32 .* delta_3 .* gamma31 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* gamma32 .* delta_3 .* gamma32 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* gamma32 .* delta_3 .* gamma32 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* gamma32 .* gamma21 .* delta_1 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* gamma32 .* gamma21 .* delta_2 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* gamma32 .* gamma21 .* delta_2 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* gamma32 .* gamma21 .* delta_3 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* gamma32 .* gamma21 .* delta_3 .* gamma31 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* gamma32 .* gamma21 .* delta_3 .* gamma32 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* gamma32 .* gamma21 .* delta_3 .* gamma32 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* gamma31 .* delta_1 ) ...
+ ( delta_3 .* gamma32 .* gamma31 .* delta_2 ) ...
- ( delta_3 .* gamma32 .* gamma31 .* delta_2 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* gamma31 .* delta_3 ) ...
- ( delta_3 .* gamma32 .* gamma31 .* delta_3 .* gamma31 ) ...
- ( delta_3 .* gamma32 .* gamma31 .* delta_3 .* gamma32 ) ...
+ ( delta_3 .* gamma32 .* gamma31 .* delta_3 .* gamma32 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* gamma32 .* delta_1 ) ...
+ ( delta_3 .* gamma32 .* gamma32 .* delta_2 ) ...
- ( delta_3 .* gamma32 .* gamma32 .* delta_2 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* gamma32 .* delta_3 ) ...
- ( delta_3 .* gamma32 .* gamma32 .* delta_3 .* gamma31 ) ...
- ( delta_3 .* gamma32 .* gamma32 .* delta_3 .* gamma32 ) ...
+ ( delta_3 .* gamma32 .* gamma32 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* gamma32 .* gamma21 .* delta_1 ) ...
- ( delta_3 .* gamma32 .* gamma32 .* gamma21 .* delta_2 ) ...
+ ( delta_3 .* gamma32 .* gamma32 .* gamma21 .* delta_2 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* gamma32 .* gamma21 .* delta_3 ) ...
+ ( delta_3 .* gamma32 .* gamma32 .* gamma21 .* delta_3 .* gamma31 ) ...
+ ( delta_3 .* gamma32 .* gamma32 .* gamma21 .* delta_3 .* gamma32 ) ...
- ( delta_3 .* gamma32 .* gamma32 .* gamma21 .* delta_3 .* gamma32 .* gamma21 ) ;
y33 = ...
+ ( delta_1 .* delta_1 .* delta_1 ) ...
+ ( delta_1 .* delta_1 .* delta_2 ) ...
- ( delta_1 .* delta_1 .* delta_2 .* gamma21 ) ...
+ ( delta_1 .* delta_1 .* delta_3 ) ...
- ( delta_1 .* delta_1 .* delta_3 .* gamma31 ) ...
- ( delta_1 .* delta_1 .* delta_3 .* gamma32 ) ...
+ ( delta_1 .* delta_1 .* delta_3 .* gamma32 .* gamma21 ) ...
+ ( delta_1 .* delta_2 .* delta_1 ) ...
+ ( delta_1 .* delta_2 .* delta_2 ) ...
- ( delta_1 .* delta_2 .* delta_2 .* gamma21 ) ...
+ ( delta_1 .* delta_2 .* delta_3 ) ...
- ( delta_1 .* delta_2 .* delta_3 .* gamma31 ) ...
- ( delta_1 .* delta_2 .* delta_3 .* gamma32 ) ...
+ ( delta_1 .* delta_2 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_1 .* delta_2 .* gamma21 .* delta_1 ) ...
- ( delta_1 .* delta_2 .* gamma21 .* delta_2 ) ...
+ ( delta_1 .* delta_2 .* gamma21 .* delta_2 .* gamma21 ) ...
- ( delta_1 .* delta_2 .* gamma21 .* delta_3 ) ...
+ ( delta_1 .* delta_2 .* gamma21 .* delta_3 .* gamma31 ) ...
+ ( delta_1 .* delta_2 .* gamma21 .* delta_3 .* gamma32 ) ...
- ( delta_1 .* delta_2 .* gamma21 .* delta_3 .* gamma32 .* gamma21 ) ...
+ ( delta_1 .* delta_3 .* delta_1 ) ...
+ ( delta_1 .* delta_3 .* delta_2 ) ...
- ( delta_1 .* delta_3 .* delta_2 .* gamma21 ) ...
+ ( delta_1 .* delta_3 .* delta_3 ) ...
- ( delta_1 .* delta_3 .* delta_3 .* gamma31 ) ...
- ( delta_1 .* delta_3 .* delta_3 .* gamma32 ) ...
+ ( delta_1 .* delta_3 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_1 .* delta_3 .* gamma31 .* delta_1 ) ...
- ( delta_1 .* delta_3 .* gamma31 .* delta_2 ) ...
+ ( delta_1 .* delta_3 .* gamma31 .* delta_2 .* gamma21 ) ...
- ( delta_1 .* delta_3 .* gamma31 .* delta_3 ) ...
+ ( delta_1 .* delta_3 .* gamma31 .* delta_3 .* gamma31 ) ...
+ ( delta_1 .* delta_3 .* gamma31 .* delta_3 .* gamma32 ) ...
- ( delta_1 .* delta_3 .* gamma31 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_1 .* delta_3 .* gamma32 .* delta_1 ) ...
- ( delta_1 .* delta_3 .* gamma32 .* delta_2 ) ...
+ ( delta_1 .* delta_3 .* gamma32 .* delta_2 .* gamma21 ) ...
- ( delta_1 .* delta_3 .* gamma32 .* delta_3 ) ...
+ ( delta_1 .* delta_3 .* gamma32 .* delta_3 .* gamma31 ) ...
+ ( delta_1 .* delta_3 .* gamma32 .* delta_3 .* gamma32 ) ...
- ( delta_1 .* delta_3 .* gamma32 .* delta_3 .* gamma32 .* gamma21 ) ...
+ ( delta_1 .* delta_3 .* gamma32 .* gamma21 .* delta_1 ) ...
+ ( delta_1 .* delta_3 .* gamma32 .* gamma21 .* delta_2 ) ...
- ( delta_1 .* delta_3 .* gamma32 .* gamma21 .* delta_2 .* gamma21 ) ...
+ ( delta_1 .* delta_3 .* gamma32 .* gamma21 .* delta_3 ) ...
- ( delta_1 .* delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma31 ) ...
- ( delta_1 .* delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 ) ...
+ ( delta_1 .* delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 .* gamma21 ) ...
+ ( delta_2 .* delta_1 .* delta_1 ) ...
+ ( delta_2 .* delta_1 .* delta_2 ) ...
- ( delta_2 .* delta_1 .* delta_2 .* gamma21 ) ...
+ ( delta_2 .* delta_1 .* delta_3 ) ...
- ( delta_2 .* delta_1 .* delta_3 .* gamma31 ) ...
- ( delta_2 .* delta_1 .* delta_3 .* gamma32 ) ...
+ ( delta_2 .* delta_1 .* delta_3 .* gamma32 .* gamma21 ) ...
+ ( delta_2 .* delta_2 .* delta_1 ) ...
+ ( delta_2 .* delta_2 .* delta_2 ) ...
- ( delta_2 .* delta_2 .* delta_2 .* gamma21 ) ...
+ ( delta_2 .* delta_2 .* delta_3 ) ...
- ( delta_2 .* delta_2 .* delta_3 .* gamma31 ) ...
- ( delta_2 .* delta_2 .* delta_3 .* gamma32 ) ...
+ ( delta_2 .* delta_2 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_2 .* delta_2 .* gamma21 .* delta_1 ) ...
- ( delta_2 .* delta_2 .* gamma21 .* delta_2 ) ...
+ ( delta_2 .* delta_2 .* gamma21 .* delta_2 .* gamma21 ) ...
- ( delta_2 .* delta_2 .* gamma21 .* delta_3 ) ...
+ ( delta_2 .* delta_2 .* gamma21 .* delta_3 .* gamma31 ) ...
+ ( delta_2 .* delta_2 .* gamma21 .* delta_3 .* gamma32 ) ...
- ( delta_2 .* delta_2 .* gamma21 .* delta_3 .* gamma32 .* gamma21 ) ...
+ ( delta_2 .* delta_3 .* delta_1 ) ...
+ ( delta_2 .* delta_3 .* delta_2 ) ...
- ( delta_2 .* delta_3 .* delta_2 .* gamma21 ) ...
+ ( delta_2 .* delta_3 .* delta_3 ) ...
- ( delta_2 .* delta_3 .* delta_3 .* gamma31 ) ...
- ( delta_2 .* delta_3 .* delta_3 .* gamma32 ) ...
+ ( delta_2 .* delta_3 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_2 .* delta_3 .* gamma31 .* delta_1 ) ...
- ( delta_2 .* delta_3 .* gamma31 .* delta_2 ) ...
+ ( delta_2 .* delta_3 .* gamma31 .* delta_2 .* gamma21 ) ...
- ( delta_2 .* delta_3 .* gamma31 .* delta_3 ) ...
+ ( delta_2 .* delta_3 .* gamma31 .* delta_3 .* gamma31 ) ...
+ ( delta_2 .* delta_3 .* gamma31 .* delta_3 .* gamma32 ) ...
- ( delta_2 .* delta_3 .* gamma31 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_2 .* delta_3 .* gamma32 .* delta_1 ) ...
- ( delta_2 .* delta_3 .* gamma32 .* delta_2 ) ...
+ ( delta_2 .* delta_3 .* gamma32 .* delta_2 .* gamma21 ) ...
- ( delta_2 .* delta_3 .* gamma32 .* delta_3 ) ...
+ ( delta_2 .* delta_3 .* gamma32 .* delta_3 .* gamma31 ) ...
+ ( delta_2 .* delta_3 .* gamma32 .* delta_3 .* gamma32 ) ...
- ( delta_2 .* delta_3 .* gamma32 .* delta_3 .* gamma32 .* gamma21 ) ...
+ ( delta_2 .* delta_3 .* gamma32 .* gamma21 .* delta_1 ) ...
+ ( delta_2 .* delta_3 .* gamma32 .* gamma21 .* delta_2 ) ...
- ( delta_2 .* delta_3 .* gamma32 .* gamma21 .* delta_2 .* gamma21 ) ...
+ ( delta_2 .* delta_3 .* gamma32 .* gamma21 .* delta_3 ) ...
- ( delta_2 .* delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma31 ) ...
- ( delta_2 .* delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 ) ...
+ ( delta_2 .* delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_2 .* gamma21 .* delta_1 .* delta_1 ) ...
- ( delta_2 .* gamma21 .* delta_1 .* delta_2 ) ...
+ ( delta_2 .* gamma21 .* delta_1 .* delta_2 .* gamma21 ) ...
- ( delta_2 .* gamma21 .* delta_1 .* delta_3 ) ...
+ ( delta_2 .* gamma21 .* delta_1 .* delta_3 .* gamma31 ) ...
+ ( delta_2 .* gamma21 .* delta_1 .* delta_3 .* gamma32 ) ...
- ( delta_2 .* gamma21 .* delta_1 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_2 .* gamma21 .* delta_2 .* delta_1 ) ...
- ( delta_2 .* gamma21 .* delta_2 .* delta_2 ) ...
+ ( delta_2 .* gamma21 .* delta_2 .* delta_2 .* gamma21 ) ...
- ( delta_2 .* gamma21 .* delta_2 .* delta_3 ) ...
+ ( delta_2 .* gamma21 .* delta_2 .* delta_3 .* gamma31 ) ...
+ ( delta_2 .* gamma21 .* delta_2 .* delta_3 .* gamma32 ) ...
- ( delta_2 .* gamma21 .* delta_2 .* delta_3 .* gamma32 .* gamma21 ) ...
+ ( delta_2 .* gamma21 .* delta_2 .* gamma21 .* delta_1 ) ...
+ ( delta_2 .* gamma21 .* delta_2 .* gamma21 .* delta_2 ) ...
- ( delta_2 .* gamma21 .* delta_2 .* gamma21 .* delta_2 .* gamma21 ) ...
+ ( delta_2 .* gamma21 .* delta_2 .* gamma21 .* delta_3 ) ...
- ( delta_2 .* gamma21 .* delta_2 .* gamma21 .* delta_3 .* gamma31 ) ...
- ( delta_2 .* gamma21 .* delta_2 .* gamma21 .* delta_3 .* gamma32 ) ...
+ ( delta_2 .* gamma21 .* delta_2 .* gamma21 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_2 .* gamma21 .* delta_3 .* delta_1 ) ...
- ( delta_2 .* gamma21 .* delta_3 .* delta_2 ) ...
+ ( delta_2 .* gamma21 .* delta_3 .* delta_2 .* gamma21 ) ...
- ( delta_2 .* gamma21 .* delta_3 .* delta_3 ) ...
+ ( delta_2 .* gamma21 .* delta_3 .* delta_3 .* gamma31 ) ...
+ ( delta_2 .* gamma21 .* delta_3 .* delta_3 .* gamma32 ) ...
- ( delta_2 .* gamma21 .* delta_3 .* delta_3 .* gamma32 .* gamma21 ) ...
+ ( delta_2 .* gamma21 .* delta_3 .* gamma31 .* delta_1 ) ...
+ ( delta_2 .* gamma21 .* delta_3 .* gamma31 .* delta_2 ) ...
- ( delta_2 .* gamma21 .* delta_3 .* gamma31 .* delta_2 .* gamma21 ) ...
+ ( delta_2 .* gamma21 .* delta_3 .* gamma31 .* delta_3 ) ...
- ( delta_2 .* gamma21 .* delta_3 .* gamma31 .* delta_3 .* gamma31 ) ...
- ( delta_2 .* gamma21 .* delta_3 .* gamma31 .* delta_3 .* gamma32 ) ...
+ ( delta_2 .* gamma21 .* delta_3 .* gamma31 .* delta_3 .* gamma32 .* gamma21 ) ...
+ ( delta_2 .* gamma21 .* delta_3 .* gamma32 .* delta_1 ) ...
+ ( delta_2 .* gamma21 .* delta_3 .* gamma32 .* delta_2 ) ...
- ( delta_2 .* gamma21 .* delta_3 .* gamma32 .* delta_2 .* gamma21 ) ...
+ ( delta_2 .* gamma21 .* delta_3 .* gamma32 .* delta_3 ) ...
- ( delta_2 .* gamma21 .* delta_3 .* gamma32 .* delta_3 .* gamma31 ) ...
- ( delta_2 .* gamma21 .* delta_3 .* gamma32 .* delta_3 .* gamma32 ) ...
+ ( delta_2 .* gamma21 .* delta_3 .* gamma32 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_2 .* gamma21 .* delta_3 .* gamma32 .* gamma21 .* delta_1 ) ...
- ( delta_2 .* gamma21 .* delta_3 .* gamma32 .* gamma21 .* delta_2 ) ...
+ ( delta_2 .* gamma21 .* delta_3 .* gamma32 .* gamma21 .* delta_2 .* gamma21 ) ...
- ( delta_2 .* gamma21 .* delta_3 .* gamma32 .* gamma21 .* delta_3 ) ...
+ ( delta_2 .* gamma21 .* delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma31 ) ...
+ ( delta_2 .* gamma21 .* delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 ) ...
- ( delta_2 .* gamma21 .* delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_2 .* gamma21 .* gamma21 .* gamma21 ) ...
+ ( delta_3 .* delta_1 .* delta_1 ) ...
+ ( delta_3 .* delta_1 .* delta_2 ) ...
- ( delta_3 .* delta_1 .* delta_2 .* gamma21 ) ...
+ ( delta_3 .* delta_1 .* delta_3 ) ...
- ( delta_3 .* delta_1 .* delta_3 .* gamma31 ) ...
- ( delta_3 .* delta_1 .* delta_3 .* gamma32 ) ...
+ ( delta_3 .* delta_1 .* delta_3 .* gamma32 .* gamma21 ) ...
+ ( delta_3 .* delta_2 .* delta_1 ) ...
+ ( delta_3 .* delta_2 .* delta_2 ) ...
- ( delta_3 .* delta_2 .* delta_2 .* gamma21 ) ...
+ ( delta_3 .* delta_2 .* delta_3 ) ...
- ( delta_3 .* delta_2 .* delta_3 .* gamma31 ) ...
- ( delta_3 .* delta_2 .* delta_3 .* gamma32 ) ...
+ ( delta_3 .* delta_2 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* delta_2 .* gamma21 .* delta_1 ) ...
- ( delta_3 .* delta_2 .* gamma21 .* delta_2 ) ...
+ ( delta_3 .* delta_2 .* gamma21 .* delta_2 .* gamma21 ) ...
- ( delta_3 .* delta_2 .* gamma21 .* delta_3 ) ...
+ ( delta_3 .* delta_2 .* gamma21 .* delta_3 .* gamma31 ) ...
+ ( delta_3 .* delta_2 .* gamma21 .* delta_3 .* gamma32 ) ...
- ( delta_3 .* delta_2 .* gamma21 .* delta_3 .* gamma32 .* gamma21 ) ...
+ ( delta_3 .* delta_3 .* delta_1 ) ...
+ ( delta_3 .* delta_3 .* delta_2 ) ...
- ( delta_3 .* delta_3 .* delta_2 .* gamma21 ) ...
+ ( delta_3 .* delta_3 .* delta_3 ) ...
- ( delta_3 .* delta_3 .* delta_3 .* gamma31 ) ...
- ( delta_3 .* delta_3 .* delta_3 .* gamma32 ) ...
+ ( delta_3 .* delta_3 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* delta_3 .* gamma31 .* delta_1 ) ...
- ( delta_3 .* delta_3 .* gamma31 .* delta_2 ) ...
+ ( delta_3 .* delta_3 .* gamma31 .* delta_2 .* gamma21 ) ...
- ( delta_3 .* delta_3 .* gamma31 .* delta_3 ) ...
+ ( delta_3 .* delta_3 .* gamma31 .* delta_3 .* gamma31 ) ...
+ ( delta_3 .* delta_3 .* gamma31 .* delta_3 .* gamma32 ) ...
- ( delta_3 .* delta_3 .* gamma31 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* delta_3 .* gamma32 .* delta_1 ) ...
- ( delta_3 .* delta_3 .* gamma32 .* delta_2 ) ...
+ ( delta_3 .* delta_3 .* gamma32 .* delta_2 .* gamma21 ) ...
- ( delta_3 .* delta_3 .* gamma32 .* delta_3 ) ...
+ ( delta_3 .* delta_3 .* gamma32 .* delta_3 .* gamma31 ) ...
+ ( delta_3 .* delta_3 .* gamma32 .* delta_3 .* gamma32 ) ...
- ( delta_3 .* delta_3 .* gamma32 .* delta_3 .* gamma32 .* gamma21 ) ...
+ ( delta_3 .* delta_3 .* gamma32 .* gamma21 .* delta_1 ) ...
+ ( delta_3 .* delta_3 .* gamma32 .* gamma21 .* delta_2 ) ...
- ( delta_3 .* delta_3 .* gamma32 .* gamma21 .* delta_2 .* gamma21 ) ...
+ ( delta_3 .* delta_3 .* gamma32 .* gamma21 .* delta_3 ) ...
- ( delta_3 .* delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma31 ) ...
- ( delta_3 .* delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 ) ...
+ ( delta_3 .* delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* gamma31 .* delta_1 .* delta_1 ) ...
- ( delta_3 .* gamma31 .* delta_1 .* delta_2 ) ...
+ ( delta_3 .* gamma31 .* delta_1 .* delta_2 .* gamma21 ) ...
- ( delta_3 .* gamma31 .* delta_1 .* delta_3 ) ...
+ ( delta_3 .* gamma31 .* delta_1 .* delta_3 .* gamma31 ) ...
+ ( delta_3 .* gamma31 .* delta_1 .* delta_3 .* gamma32 ) ...
- ( delta_3 .* gamma31 .* delta_1 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* gamma31 .* delta_2 .* delta_1 ) ...
- ( delta_3 .* gamma31 .* delta_2 .* delta_2 ) ...
+ ( delta_3 .* gamma31 .* delta_2 .* delta_2 .* gamma21 ) ...
- ( delta_3 .* gamma31 .* delta_2 .* delta_3 ) ...
+ ( delta_3 .* gamma31 .* delta_2 .* delta_3 .* gamma31 ) ...
+ ( delta_3 .* gamma31 .* delta_2 .* delta_3 .* gamma32 ) ...
- ( delta_3 .* gamma31 .* delta_2 .* delta_3 .* gamma32 .* gamma21 ) ...
+ ( delta_3 .* gamma31 .* delta_2 .* gamma21 .* delta_1 ) ...
+ ( delta_3 .* gamma31 .* delta_2 .* gamma21 .* delta_2 ) ...
- ( delta_3 .* gamma31 .* delta_2 .* gamma21 .* delta_2 .* gamma21 ) ...
+ ( delta_3 .* gamma31 .* delta_2 .* gamma21 .* delta_3 ) ...
- ( delta_3 .* gamma31 .* delta_2 .* gamma21 .* delta_3 .* gamma31 ) ...
- ( delta_3 .* gamma31 .* delta_2 .* gamma21 .* delta_3 .* gamma32 ) ...
+ ( delta_3 .* gamma31 .* delta_2 .* gamma21 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* gamma31 .* delta_3 .* delta_1 ) ...
- ( delta_3 .* gamma31 .* delta_3 .* delta_2 ) ...
+ ( delta_3 .* gamma31 .* delta_3 .* delta_2 .* gamma21 ) ...
- ( delta_3 .* gamma31 .* delta_3 .* delta_3 ) ...
+ ( delta_3 .* gamma31 .* delta_3 .* delta_3 .* gamma31 ) ...
+ ( delta_3 .* gamma31 .* delta_3 .* delta_3 .* gamma32 ) ...
- ( delta_3 .* gamma31 .* delta_3 .* delta_3 .* gamma32 .* gamma21 ) ...
+ ( delta_3 .* gamma31 .* delta_3 .* gamma31 .* delta_1 ) ...
+ ( delta_3 .* gamma31 .* delta_3 .* gamma31 .* delta_2 ) ...
- ( delta_3 .* gamma31 .* delta_3 .* gamma31 .* delta_2 .* gamma21 ) ...
+ ( delta_3 .* gamma31 .* delta_3 .* gamma31 .* delta_3 ) ...
- ( delta_3 .* gamma31 .* delta_3 .* gamma31 .* delta_3 .* gamma31 ) ...
- ( delta_3 .* gamma31 .* delta_3 .* gamma31 .* delta_3 .* gamma32 ) ...
+ ( delta_3 .* gamma31 .* delta_3 .* gamma31 .* delta_3 .* gamma32 .* gamma21 ) ...
+ ( delta_3 .* gamma31 .* delta_3 .* gamma32 .* delta_1 ) ...
+ ( delta_3 .* gamma31 .* delta_3 .* gamma32 .* delta_2 ) ...
- ( delta_3 .* gamma31 .* delta_3 .* gamma32 .* delta_2 .* gamma21 ) ...
+ ( delta_3 .* gamma31 .* delta_3 .* gamma32 .* delta_3 ) ...
- ( delta_3 .* gamma31 .* delta_3 .* gamma32 .* delta_3 .* gamma31 ) ...
- ( delta_3 .* gamma31 .* delta_3 .* gamma32 .* delta_3 .* gamma32 ) ...
+ ( delta_3 .* gamma31 .* delta_3 .* gamma32 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* gamma31 .* delta_3 .* gamma32 .* gamma21 .* delta_1 ) ...
- ( delta_3 .* gamma31 .* delta_3 .* gamma32 .* gamma21 .* delta_2 ) ...
+ ( delta_3 .* gamma31 .* delta_3 .* gamma32 .* gamma21 .* delta_2 .* gamma21 ) ...
- ( delta_3 .* gamma31 .* delta_3 .* gamma32 .* gamma21 .* delta_3 ) ...
+ ( delta_3 .* gamma31 .* delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma31 ) ...
+ ( delta_3 .* gamma31 .* delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 ) ...
- ( delta_3 .* gamma31 .* delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* gamma31 .* gamma31 .* gamma31 ) ...
- ( delta_3 .* gamma31 .* gamma31 .* gamma32 ) ...
+ ( delta_3 .* gamma31 .* gamma31 .* gamma32 .* gamma21 ) ...
+ ( delta_3 .* gamma31 .* gamma32 .* gamma21 .* gamma31 ) ...
+ ( delta_3 .* gamma31 .* gamma32 .* gamma21 .* gamma32 ) ...
- ( delta_3 .* gamma31 .* gamma32 .* gamma21 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* gamma31 .* gamma32 .* gamma31 ) ...
- ( delta_3 .* gamma31 .* gamma32 .* gamma32 ) ...
+ ( delta_3 .* gamma31 .* gamma32 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* delta_1 .* delta_1 ) ...
- ( delta_3 .* gamma32 .* delta_1 .* delta_2 ) ...
+ ( delta_3 .* gamma32 .* delta_1 .* delta_2 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* delta_1 .* delta_3 ) ...
+ ( delta_3 .* gamma32 .* delta_1 .* delta_3 .* gamma31 ) ...
+ ( delta_3 .* gamma32 .* delta_1 .* delta_3 .* gamma32 ) ...
- ( delta_3 .* gamma32 .* delta_1 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* delta_2 .* delta_1 ) ...
- ( delta_3 .* gamma32 .* delta_2 .* delta_2 ) ...
+ ( delta_3 .* gamma32 .* delta_2 .* delta_2 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* delta_2 .* delta_3 ) ...
+ ( delta_3 .* gamma32 .* delta_2 .* delta_3 .* gamma31 ) ...
+ ( delta_3 .* gamma32 .* delta_2 .* delta_3 .* gamma32 ) ...
- ( delta_3 .* gamma32 .* delta_2 .* delta_3 .* gamma32 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* delta_2 .* gamma21 .* delta_1 ) ...
+ ( delta_3 .* gamma32 .* delta_2 .* gamma21 .* delta_2 ) ...
- ( delta_3 .* gamma32 .* delta_2 .* gamma21 .* delta_2 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* delta_2 .* gamma21 .* delta_3 ) ...
- ( delta_3 .* gamma32 .* delta_2 .* gamma21 .* delta_3 .* gamma31 ) ...
- ( delta_3 .* gamma32 .* delta_2 .* gamma21 .* delta_3 .* gamma32 ) ...
+ ( delta_3 .* gamma32 .* delta_2 .* gamma21 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* delta_3 .* delta_1 ) ...
- ( delta_3 .* gamma32 .* delta_3 .* delta_2 ) ...
+ ( delta_3 .* gamma32 .* delta_3 .* delta_2 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* delta_3 .* delta_3 ) ...
+ ( delta_3 .* gamma32 .* delta_3 .* delta_3 .* gamma31 ) ...
+ ( delta_3 .* gamma32 .* delta_3 .* delta_3 .* gamma32 ) ...
- ( delta_3 .* gamma32 .* delta_3 .* delta_3 .* gamma32 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* delta_3 .* gamma31 .* delta_1 ) ...
+ ( delta_3 .* gamma32 .* delta_3 .* gamma31 .* delta_2 ) ...
- ( delta_3 .* gamma32 .* delta_3 .* gamma31 .* delta_2 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* delta_3 .* gamma31 .* delta_3 ) ...
- ( delta_3 .* gamma32 .* delta_3 .* gamma31 .* delta_3 .* gamma31 ) ...
- ( delta_3 .* gamma32 .* delta_3 .* gamma31 .* delta_3 .* gamma32 ) ...
+ ( delta_3 .* gamma32 .* delta_3 .* gamma31 .* delta_3 .* gamma32 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* delta_3 .* gamma32 .* delta_1 ) ...
+ ( delta_3 .* gamma32 .* delta_3 .* gamma32 .* delta_2 ) ...
- ( delta_3 .* gamma32 .* delta_3 .* gamma32 .* delta_2 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* delta_3 .* gamma32 .* delta_3 ) ...
- ( delta_3 .* gamma32 .* delta_3 .* gamma32 .* delta_3 .* gamma31 ) ...
- ( delta_3 .* gamma32 .* delta_3 .* gamma32 .* delta_3 .* gamma32 ) ...
+ ( delta_3 .* gamma32 .* delta_3 .* gamma32 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* delta_3 .* gamma32 .* gamma21 .* delta_1 ) ...
- ( delta_3 .* gamma32 .* delta_3 .* gamma32 .* gamma21 .* delta_2 ) ...
+ ( delta_3 .* gamma32 .* delta_3 .* gamma32 .* gamma21 .* delta_2 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* delta_3 .* gamma32 .* gamma21 .* delta_3 ) ...
+ ( delta_3 .* gamma32 .* delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma31 ) ...
+ ( delta_3 .* gamma32 .* delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 ) ...
- ( delta_3 .* gamma32 .* delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_1 .* delta_1 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_1 .* delta_2 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* delta_1 .* delta_2 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_1 .* delta_3 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* delta_1 .* delta_3 .* gamma31 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* delta_1 .* delta_3 .* gamma32 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_1 .* delta_3 .* gamma32 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_2 .* delta_1 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_2 .* delta_2 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* delta_2 .* delta_2 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_2 .* delta_3 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* delta_2 .* delta_3 .* gamma31 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* delta_2 .* delta_3 .* gamma32 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_2 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* delta_2 .* gamma21 .* delta_1 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* delta_2 .* gamma21 .* delta_2 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_2 .* gamma21 .* delta_2 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* delta_2 .* gamma21 .* delta_3 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_2 .* gamma21 .* delta_3 .* gamma31 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_2 .* gamma21 .* delta_3 .* gamma32 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* delta_2 .* gamma21 .* delta_3 .* gamma32 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* delta_1 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* delta_2 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* delta_2 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* delta_3 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* delta_3 .* gamma31 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* delta_3 .* gamma32 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma31 .* delta_1 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma31 .* delta_2 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma31 .* delta_2 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma31 .* delta_3 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma31 .* delta_3 .* gamma31 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma31 .* delta_3 .* gamma32 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma31 .* delta_3 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 .* delta_1 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 .* delta_2 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 .* delta_2 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 .* delta_3 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 .* delta_3 .* gamma31 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 .* delta_3 .* gamma32 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 .* delta_3 .* gamma32 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 .* gamma21 .* delta_1 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 .* gamma21 .* delta_2 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 .* gamma21 .* delta_2 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 .* gamma21 .* delta_3 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma31 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 .* gamma21 .* delta_3 .* gamma32 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* gamma21 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* gamma31 .* gamma31 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* gamma31 .* gamma32 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* gamma31 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* gamma32 .* gamma21 .* gamma31 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* gamma32 .* gamma21 .* gamma32 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* gamma32 .* gamma21 .* gamma32 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* gamma32 .* gamma31 ) ...
+ ( delta_3 .* gamma32 .* gamma21 .* gamma32 .* gamma32 ) ...
- ( delta_3 .* gamma32 .* gamma21 .* gamma32 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* gamma31 .* gamma31 ) ...
- ( delta_3 .* gamma32 .* gamma31 .* gamma32 ) ...
+ ( delta_3 .* gamma32 .* gamma31 .* gamma32 .* gamma21 ) ...
+ ( delta_3 .* gamma32 .* gamma32 .* gamma21 .* gamma31 ) ...
+ ( delta_3 .* gamma32 .* gamma32 .* gamma21 .* gamma32 ) ...
- ( delta_3 .* gamma32 .* gamma32 .* gamma21 .* gamma32 .* gamma21 ) ...
- ( delta_3 .* gamma32 .* gamma32 .* gamma31 ) ...
- ( delta_3 .* gamma32 .* gamma32 .* gamma32 ) ...
+ ( delta_3 .* gamma32 .* gamma32 .* gamma32 .* gamma21 ) ;