a1b23ff9-3f36-42fa-9bad-ffec98bb2143 16 4 5 decimal

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Solving an equalities system

system 2 x * y + 5 - 2 z 2^* + 0 ≡ y 3^4 z * + 4 - 0 ≡ x y * z + z exp - 0 ≡ 31 mat :

init x 1.422 ≡ y 0.975 ≡ z 0.015 ≡ 31 mat :

start x 1 ≡ y 1 ≡ z 0.001 ≡ 31 mat

refine x 1.422 ≡ y 0.975 ≡ z 0.015 ≡ 31 mat

x y z sol system init FindRoot :

x y z 31 mat x y z sol : 1.42246938960007 0.975388528056254 0.768008038417007 31 mat

sanity 2 x * y + 5 - 2 z 2^* + y 3^4 z * + 4 - x y * z + z exp - 31 mat : 0.000000001 0.000000004 0.000000002 - 31 mat

x y z 31 mat 1.42246939034512 0.975388528087757 0.768008037465102 31 mat :

2 x * y + 5 - 2 z 2^* + y 3^4 z * + 4 - x y * z + z exp - 31 mat 71015 -^* - 11015 -^* - 031 mat

Quasi-Newton & Conjugate Gradient have ~ same accuracy 0.0000001

Modifications to the Levenberg-Marquardt Method: To make the Levenberg-Marquardt method more effective on actual problems, Mathcad includes the following modifications to the basic method: The first time the solver stops at a point that is not a solution, Mathcad adds a small random amount to all the variables and tries again. This helps avoid getting stuck in local minima and other points from which there is no preferred direction. Mathcad does this only once per solve, the first time the solver stops on a point that is not a solution. If you include inequality constraints in a solve block, Mathcad solves the subsystem consisting of only the inequalities first before adding in the in the equality constraints and attempting a full solution. This is equivalent to moving the guesses into an area where the inequalities are all satisfied before starting the solver.