Class 9. Root Finding in Multi-D, and Numerical Dif-ferentiationNonlinear Systems of Equations• Consider the system f(x, y) = 0, g(x, y) = 0. Plot zero contours of f and g:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 f = 0 f = 0g = 0g = 0• No information about f in g, and vice versa.– In general, no good method for finding roots.• If you are near root, best bet is NR.E.g., For F(x) = 0, choose xi+1= xi+ δ, where F0(x)δ = −F(x).– This is a matrix equation: F0(x) is a matrix with elements ∂Fi/∂xj. Thematrix is called the Jacobian.• Written out (2-D example):∂f∂xδx+∂f∂yδy= −f(x, y),∂g∂xδx+∂g∂yδy= −g(x, y).• Given initial guess, must evaluate matrix elements a nd RHS, solve system for δ, andcompute next iteration xi+1. Then repeat (must solve 2 × 2 linear system each time).• Essentially the non-linear system has been linearized to make it easier to work with.• NRiC §9.7 discusses a global convergence strategy that combines multi-D NR with“backtracking” to improve chances of finding solutions.Example: Interstellar Chemistry• ISM is multiphase plasma consisting of electrons, ions, atoms, and molecules.• Originally, the ISM was thought to be too hostile for molecules.1• But in 1968- 69, radio observations discovered absorption/emission lines of NH3, H2CO,H2O, ...• Lots of organic molecules, e.g., CH3CH2OH (ethanol), etc.• In some places, all atoms have been incorporated into molecules.• E.g., molecular clouds: dense, cold clouds of gas composed primarily of molecules.(T ∼ 30 K, n ∼ 106cm−3, M ∼ 105–6M, R ∼ 10–100 pc.)• How do we predict what the abundances of different molecules should be, given n andT ?• Need to solve a chemical reaction network.• Consider reaction between two species A and B:A + B → AB (reaction rate = nAnBRAB).• Reverse also possible:AB → A + B (reaction rate = nABR0AB).• In equilibrium:nAnBRAB= nABR0AB;nA+ nAB= n0A;nB+ nAB= n0B.where n0Aand n0Bare normalizations so that A and B are conserved.• Substitute normalization equations into reaction equation to get quadratic in nAB,easily solved.• However, many more possible reactions:AC + B ←→ AB + C (exchang e reaction);ABC ←→ AB + C (dissociation reaction).• Wind up with large nonlinear system describing all forward/reverse reactions, involvingknown reaction rates R, plus normalizations. Must solve given fixed n0and T .2Numerical Derivatives• For NR and function minimization, often need derivatives of functions. It’s alwaysbetter to use an analytical derivative if it’s available.• If you’re stuck, could try:f0(x) 'f(x + h) − f(x)h,where |h| is small.• However, this is verysusceptible to RE. Better:f0(x) 'f(x + h) − f(x − h)2h.(This version cancels the second-derivative term in the Taylor series expansion of f(x+h) − f(x − h), leaving just the third- and higher-order terms.)• Read NRiC §5.7 before trying