`Solve` Performance Tuning

Question

I'm facing a Problem in Mathematica, where I have to solve a large number of equations generated by an AppendTo.

n = 200;
dz = 1000/n;
RollLM[l_, EM_, Ixx_, p_] := {{1, l, Power[l, 2] / (2 EM Ixx), Power[l, 3] / (6 EM Ixx), p Power[l, 4] / (24 EM Ixx)}, {0, 1, l / (EM Ixx), Power[l, 2] / (2 EM Ixx), p Power[l, 3] / (6 EM Ixx)}, {0, 0, 1, l , p Power[l, 2] / 2}, {0, 0, 0, 1, p l}, {0, 0, 0, 0, 1}};
BearLM[d_, c_] := {{1, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, d, 1, 0, 0}, {-c, 0, 0, 1, 0}, {0, 0, 0, 0, 1}};
pBF = Function[z, Piecewise[{{989000/500, 250 <= z <= 750}}]];

Vec1A = {vA, PhiA, 0, 0, 1};
Vec2A = BearLM[810000, 850000].Vec1A;
VecAList = {Vec2A};
Do[AppendTo[VecAList, RollLM[dz, 210000, 262440000 \[Pi], pBF[(i-0.5)dz]] .VecAList[[i]]], {i, n}];
Vec3A = BearLM[810000, 850000].Last[VecAList];
SolA = Solve[{Vec3A[[3]] == 0, Vec3A[[4]] == 0},{vA,PhiA}][[1]]

I'm now trying to tune this code performancewise and first thought about switching from AppendTotoo Reap & Sowbut the most time consuming factor is the solving. Is there any option how to speed this up?

Also I already asked a similar question here: Switching from AppendTo to Reap & Sow

Answer 1

If I understand your computations, this can be done with Fold:

Fold[RollLM[dz, 210000, 262440000 π, pBF[(#2 - 0.5) dz]].#1 &, Vec2A, Range[n]] // Simplify;

This gives the same result as your Do[...] // Last // Simplify. Next let's compute Vec3A and solve:

Vec3A = BearLM[810000, 850000].% // Simplify
Solve[{%[[3]] == 0, %[[4]] == 0}, {vA, PhiA}][[1]]

{vA -> 989/1700, PhiA -> 271975/(1944 (1 + 136080 π))}

If you need all parts of VecAList, you can use FoldList.