Over [0,1]
Here are the approximate maximum errors (read off of the plots) of the various methods. Unless otherwise stated, 7 d.f. are used in each case.
Over [0,1]
These require evaluating a sqrt
.| Method | Max Error |
|---|---|
| Taylor series about the origin | 0.016 |
| Pade quotient about the origin | 0.009 |
| Taylor series about the center | 0.0032 |
| Chebyshev series | 0.00025 |
| Minimax polynomial | 0.00015 |
| Chebyshev-Pade quotient | 0.00004 |
| Pade quotient about the center | singularity |
| [5,1] Minimax quotient | 0.00012 |
| [4,2] Minimax quotient | 6e-5 |
| [2,4] Minimax quotient | 5e-5 |
These use only elementary operations.
| Method | Max Error |
|---|---|
| Taylor series about the origin | 0.33 |
| Pade quotient about the origin | 0.27 |
| Taylor series about the center | 0.22 |
| Taylor series with 7 non-zero terms, about the origin | 0.21 |
| Pade quotient about the center | 0.16 |
| Pade quotient with 7 non-zero terms, about the origin | 0.13 |
| Chebyshev series | 0.06 |
| Chebyshev-Pade quotient | 0.035 |
| Minimax polynomial | 0.033 |
| [1,5] minimax quotient | 0.026 |
| [3,3] minimax quotient | "Error, (in numapprox/remez) error curve fails to oscillate sufficiently; try different degrees" |
| [5,1] minimax quotient | "Error, (in numapprox/remez) error curve fails to oscillate sufficiently; try different degrees" |
| [4,2] minimax quotient | "Error, (in numapprox/remez) error curve fails to oscillate sufficiently; try different degrees" |
| [2,4] minimax quotient | "Error, (in numapprox/remez) error curve fails to oscillate sufficiently; try different degrees" |
Over [0,0.7071]
These require 1 or 2 sqrts, depending on x.
| Method | Max Error |
|---|---|
| Taylor series about the origin | 0.0023 |
| Chebyshev-Pade quotient | 0.00012 |
| Taylor series about the center | 2e-5 |
| Chebyshev series | 1.4e-5 |
| Minimax polynomial | 2.8e-7 |
These require 0 or 1 sqrts, depending on x.
| Method | Max Error |
|---|---|
| Taylor series about the origin | 0.006 |
| Pade quotient about the origin | 0.0023 |
| Taylor series about the center | 0.0005 |
| Taylor series with 7 non-zero terms, about the origin | 0.00014 |
| Pade quotient about the center | 7e-5 |
| Chebyshev series | 8e-6 |
| Minimax polynomial | 7e-6 |
| Pade quotient with 7 d.f., about the origin | 2e-6 |
| Chebyshev-Pade quotient | 8e-7 |
| [5,1] minimax quotient | 7e-7 |
| [3,3] minimax quotient | 4.8e-7 |