[11.8] Try Solve() For Symbolic System Solutions; [11.9] Nsolve() May Return "Questionable Accuracy" Warning Even With Good Solutions - Texas Instruments TI-89 Tip List

[11.8] Try solve() for symbolic system solutions

Consider this system of equations:
2x - y + z = a
x + y - 3z = b
3x - 2z = c
The solution for x, y and z depends on the values of a, b and c. solve() can usually find solutions to
these kinds of systems. For example,
solve(2x-y+z=a and x+y-3z=b and 3x-2z=c,{x,y,z})
returns
c+2$@1
x =
and
3
The notation "@1" indicates an arbitrary constant in the solution. For example, let @1 = k, then
x = c+2$k
and
3
If we let k = 5, b = 3 and c = 7, then a = 4, x = 17/3, y = 37/3 and z = 5. These values are solutions to
the original system.
(Credit to Rick Homard)

[11.9] nSolve() may return "Questionable Accuracy" warning even with good solutions

nSolve() is a reasonably robust numeric solver, but some equations give it difficulty, even when there
are no obvious problems such as singularities. Futher, nSolve() may give the "Questionable Accuracy"
warning in the status line, for solutions that seem valid. In these cases, nSolve() will take longer to
return the answer, even when it returns nearby solutions quickly.
These two programs illustrate the problem:
solvebug(vf)
func
local vfs,tapx
©Find estimate tapx for solution
ln(1000*(vf-.015))→vfs
polyeval({3.618919682914,⁻31.003830334444,76.737472978603,⁻68.237201523917,262.4
6139741751,84.916629306139},vfs)+polyeval({⁻3.9287348339733⁻7,5.9179552041553⁻
5,⁻0.0036896155610467,0.12308990642018,⁻2.7560332337098,0},1/vfs)→tapx
©Find solution, with bounds from tapx
nsolve(vfts120(t)=vf,t)|t≥tapx-.4 and t≤min({tapx+.4,705.47})
Endfunc
vfts120(ts)
3$b−c+7$@1
y =
and
3
y = 3$b−c+7$k
z =k
and
3
z =@1 a = −(b − c)
and
a = −(b − c)
and
11 - 8