Calculators and numerical differentiation
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11-01-2020, 11:43 PM
Post: #6
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RE: Calculators and numerical differentiation
(11-01-2020 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 = (E-1/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) = (En - E-n)/2 → this explained why En 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 - 10-30-2020, 09:57 PM
RE: Calculators and numerical differentiation - Paul Dale - 10-30-2020, 11:41 PM
RE: Calculators and numerical differentiation - Albert Chan - 10-31-2020, 01:20 AM
RE: Calculators and numerical differentiation - Wes Loewer - 11-01-2020, 05:39 AM
RE: Calculators and numerical differentiation - Albert Chan - 11-01-2020, 05:39 PM
RE: Calculators and numerical differentiation - Albert Chan - 11-01-2020 11:43 PM
RE: Calculators and numerical differentiation - Wes Loewer - 11-03-2020, 06:09 PM
RE: Calculators and numerical differentiation - Albert Chan - 11-03-2020, 10:14 PM
RE: Calculators and numerical differentiation - Wes Loewer - 11-04-2020, 04:14 PM
RE: Calculators and numerical differentiation - CMarangon - 11-03-2020, 06:55 PM
RE: Calculators and numerical differentiation - Wes Loewer - 11-04-2020, 04:04 PM
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