In short, Regular NR - less iterations but each iteration is time consuming, Modified NR- more iterations but each iteration is faster. solve the equations in FUN with user supplied initial relaxation factor. x NewtonRaphson (FUN,X0,lambda) starts at the initial guess X0 and tries to. Default values for solver and display setting. input x and return a vector of equation values F evaluated at x. But since you are computing stiffness matrix only once, each iteration is faster. FUN is a function handle and has to accept. On the other hand for modified NR since the stiffness is not updated at each point, you don't know if you are moving in the "right direction" or not and hence more iterations are usually needed than regular NR to achieve convergence. But the "right direction" comes at a cost - you need to evaluate stiffness matrix at each point which is a computationally expensive. For regular NR since you calculate tangent at each point you know that you are moving in the "right direction" and hence only a few iterations are needed. In the regular NR method you can see that at each incremental displacement the tangent slope is calculated(slope is decreasing), while in the modified NR method the slope is calculated just once, at start and the same slope is used (all lines are parallel)to progress ahead till convergence. The first one is a regular newton raphson and the second one is modified newton raphson. The final choice of these two methods could be done by what means? (we can always proceed by scanning the cases and see the difference of the results, which can be long I think)Īre there some structures that require an update of the stiffness matrix at each iteration (I guess yes as for hyperelastic materials) and others not? If the stiffness matrix is not updated at each iteration the accuracy is not the same I think, so the last method of solving should not exist? The difference in its resolutions is that for the first two methods, the stiffness matrix is updated at each iteration contrary to the last one which updates the stiffness matrix only on the first iterations (so there is less inversion of the matrix and less formulation). Several resolution techniques exist, among them: the full, asymmetric and modified Newton-Raphson method (respectively NROPT,full NROPT,unsym NROPT,modi). In Ansys, the resolution of the equation K*U=F is done by the Newton-Raphson method. Any value in that interval is a valid root approximation.I would like to ask for your expertise on finite elements in order to know how to make the right choice on the different methods of solving the finite element calculation. So after a few iterations the multiplicity is correctly detected, and one step of the modified method gets as close to the root as one can get, the next iterations will most likely oscillate around the interval $$. The fixed points are where the diagonal intersects the graph of the Newton step, the most massive part of it is in the segment $$. However getting close to the root the function value gets fuzzy over a rather long stretch of arguments, the Newton step takes rather random values. One sees that well away from the root one gets geometric convergence with factor $0.8=1-\frac15$ towards the center of the cluster at $5/7=0.7143$. In the first row the graph of the floating point evaluation of the polynomial, then the unmodified Newton step, the quotient of the step sizes of two steps and lastly the modified Newton step, in blue with the computed multiplicity, in red with fixed multiplicity $5$. In accordance with the prediction of a root cluster of radius $\sqrt$ around the real root location. One finds the coefficient sequence Īnd with a supplied root-finding method the roots One example is to take the expansion of $(x-5/7)^5$ in floating point coefficients and compute the roots of it. As also $f'(x)$ converges to $0$ at the multiple root, floating point errors will contribute a substantial distortion so that the computed Newton iterates can behave chaotically if the method is continued after reaching the theoretically possible maximum precision $\sqrt\mu$. Note that due to floating point errors a multiple root of $f(x)$ will most likely manifest as a root cluster of size $\sqrt\mu$ where $\mu$ is the machine constant. So if after say 5 or 10 iterations you detect that the reduction in step size is by a factor less than $1/2$, you can compute $m$ from the factor and apply the modified Newton method. Thus you can both detect the slow convergence and test for the behavior at a multiple root, and also speed up the computation of the remaining digits with the modified method. Newton-Raphson Method in Java Ask Question Asked 8 years, 3 months ago Modified 1 month ago Viewed 11k times 2 I am making a program to apply Newton-Raphson method in Java with an equation: f (x) 3x - ex sin (x) And g (x) f (x) 3- ex cos (x) The problem is when I tried to solve the equation in a paper to reach an error less than (0. This means that you need more than 3 iterations for each digit of the result. The convergence for multiplicity $m$ is geometric with factor $1-\frac1m$.
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