Calculators and numerical differentiation

11012020, 11:43 PM
Post: #6




RE: Calculators and numerical differentiation
(11012020 05:39 PM)Albert Chan Wrote: For more accuracy, we can add more terms: (note, there is no even powers of δ ) We can show Df(0) = f'(0) does not require calculating f(0). In other words, operator form will not have a constant term, the "1" operator. From previous post, we have µδ = (E1/E)/2, δδ = (E+1/E)  2 Doing "operator" mathematics, with x = log(E), we have: µδ = sinh(x) δδ = 2*cosh(x)  2 Hyperbolics identities: (1): cosh(z1)*cosh(z2) = (cosh(z1  z2) + cosh(z1 + z2)) / 2 (2): sinh(z1)*cosh(z2) = (sinh(z1  z2) + sinh(z1 + z2)) / 2 hD = sinh(x) * (k1 + k2*cosh(x) + k3*cosh(x)^2 + k4*cosh(x)^3 + ...) // apply (1) = sinh(x) * (k1' + k2'*cosh(x) + k3'*cosh(2x) + k4'*cosh(3x) + ... ) // apply (2) = k1''*sinh(x) + k2''*sinh(2x) + k3''*sinh(3x) + k4''*sinh(4x) + ... sinh(nx) = (E^{n}  E^{n})/2 → this explained why E^{n} coefs = negative of E^{n} coefs. → RHS terms will not generate constant term (i.e., no "1" operator) → D does not require calculating f(0) → same for D^odd_powers, since RHS is still linear combinations of sinh's. 

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Messages In This Thread 
Calculators and numerical differentiation  robve  10302020, 09:57 PM
RE: Calculators and numerical differentiation  Paul Dale  10302020, 11:41 PM
RE: Calculators and numerical differentiation  Albert Chan  10312020, 01:20 AM
RE: Calculators and numerical differentiation  Wes Loewer  11012020, 05:39 AM
RE: Calculators and numerical differentiation  Albert Chan  11012020, 05:39 PM
RE: Calculators and numerical differentiation  Albert Chan  11012020 11:43 PM
RE: Calculators and numerical differentiation  Wes Loewer  11032020, 06:09 PM
RE: Calculators and numerical differentiation  Albert Chan  11032020, 10:14 PM
RE: Calculators and numerical differentiation  Wes Loewer  11042020, 04:14 PM
RE: Calculators and numerical differentiation  CMarangon  11032020, 06:55 PM
RE: Calculators and numerical differentiation  Wes Loewer  11042020, 04:04 PM

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