Two thermistors are calibrated by finding the temperature at which a modelled…
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics
Computational Physics, Ch Where a function vanishes, and what lies between the points
Problem
Two thermistors are calibrated by finding the temperature T at which a modelled resistance function g(T) (in Ω) matches a measured value, using Newton-Raphson on g(T)=0. Near its own root, sensor A's calibration curve is steep, with slope gA′(TA∗)=42.0Ω∘C−1. Sensor B is operating near a shallow plateau in its resistance-temperature curve, with slope gB′(TB∗)=6.0×10−3Ω∘C−1. Both iterations are stopped using the identical residual criterion ∣g(T)∣<3.0×10−3Ω.An engineer writes: "Since both calibrations were stopped at the same residual tolerance, both temperatures are known to the same accuracy."
((a)) State the approximate relation between a residual ∣g(T)∣ and the distance ∣T−T∗∣ to the true root, valid near the root.
((b)) Use it to find the error in T that the stated stopping criterion actually guarantees for sensor A and for sensor B.
((c)) Say whether the engineer is right, and if not, by how many decades the two guaranteed errors differ.