Numerical Differentiation Methods
Learn about various numerical differentiation methods such as forward-difference and backward-difference formulas, as well as three-point and five-point approximation methods. Understand how to apply these formulas using given data and tables to compute results and actual errors accurately.
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Sec:4.1 Numerical Differentiation
Sec:4.1 Numerical Differentiation ? ?? Approximation Forward-difference Formula Backward-difference Formula ? ?? =? ??+ ? ?(??) ? ?? =? ?? ?(?? ?) + 2? (?) 2? (?) where ? between ??and ??+ ? where ? between ??and ?? ? ?? ?? ?? ? ??+ ? Two-Point Formula Example Use table 4.2 and forward-difference formula to approximate ? 2 with = 0.1 1 ? = ?.? ? (2) (0.1)? 2.1 ?(2) Use table 4.2 and backward-difference formula to approximate ? 2 with = 0.2 1 (0.2)? 2 ?(1.8) ? (2) ? = ?.?
Sec:4.1 Numerical Differentiation ? ?? Approximation Three-Point Midpoint Formula Three-Point Endpoint Formula ?? ??+ ?? ??+ ? ?? ? ?? ??+ ? Five-Point Midpoint Formula Five-Point Endpoint Formula ??+ ?? ?? ??+ ?? ??+ ?? ??+ ? ?? ?? ?? ? ?? ??+ ?? ??+ ?
Sec:4.1 Numerical Differentiation ? ?? Approximation Three-Point Midpoint Formula Three-Point Endpoint Formula ?? ??+ ?? ??+ ? ?? ? ?? ??+ ? Example Use table 4.2 and three-point endpoint formula to approximate ? 2 with = 0.1 ? (2) = = 22.032310
Sec:4.1 Numerical Differentiation ? ?? Approximation Five-Point Midpoint Formula Five-Point Endpoint Formula ??+ ?? ?? ??+ ?? ??+ ?? ?? ?? ??+ ? ?? ? ?? ??+ ?? ??+ ? Example Use table 4.2 and five-point midpoint formula to approximate ? 2 with = 0.1 ? (2) The only five-point formula for which the table gives sufficient data is the midpoint formula = 22.166999
Sec:4.1 Numerical Differentiation ? ?? Approximation Five-Point Midpoint Formula Five-Point Endpoint Formula ??+ ?? ?? ??+ ?? ??+ ?? ?? ?? ??+ ? ?? ? ?? ??+ ?? ??+ ? Example Use table 4.2 and five-point midpoint formula to approximate ? 2 with = 0.1 = 22.166999 ? (2) Compute the actual error The data given in Table 4.2 are taken from the function ? ? = ??? The actual error = 22.166999 ? (1.8) = 22.166999 16.9390129
Sec:4.1 Numerical Differentiation ? ?? Approximation Five-Point Midpoint Formula Five-Point Endpoint Formula ??+ ?? ?? ??+ ?? ??+ ?? ?? ?? ??+ ? ?? ? ?? ??+ ?? ??+ ? Example Use table 4.2 and five-point midpoint formula to approximate ? 2 with = 0.1 = 22.166999 ? (2) Compute the actual error The data given in Table 4.2 are taken from the function ? ? = ??? The actual error = 22.166999 ? (1.8) = 22.166999 16.9390129
Sec:4.1 Numerical Differentiation ? ?? Approximation Three-Point Midpoint Formula Three-Point Endpoint Formula ?? ??+ ?? ??+ ? ?? ? ?? ??+ ? Example Use table 4.2 and three-point midpoint formula to approximate ? 2 (consider all possible cases) 1 ? = ?.? ? (2) 2(0.1)? 2.1 ?(1.9) 1 2(0.2)? 2.2 ?(1.8) ? (2) ? = ?.?
Sec:4.1 Numerical Differentiation ? ?? Approximation Three-Point Midpoint Formula Three-Point Endpoint Formula ?? ??+ ?? ??+ ? ?? ? ?? ??+ ? Example Use table 4.2 and three-point endpoint formula to approximate ? 2.2 (consider all possible cases) ? = ?.? 1 ? (2.2) 2( 0.1) 3? 2.2 + 4? 2.1 ?(2) ? = ?.? 1 2( 0.2) 3? 2.2 + 4? 2 ?(1.8) ? (2.2)
Sec:4.1 Numerical Differentiation ? ?? Approximation Three-Point Midpoint Formula Three-Point Endpoint Formula ?? ??+ ?? ??+ ? ?? ? ?? ??+ ? Five-Point Midpoint Formula Five-Point Endpoint Formula ??+ ?? ?? ??+ ?? ??+ ?? ??+ ? ?? ?? ?? ? ?? ??+ ?? ??+ ?
Sec:4.1 Numerical Differentiation Comparison Study Example True Error ? (2.0) The true value 22.167168 Three-point endpoint with h = 0.1 22.03231 1.35 e 1 Three-point endpoint with h = 0.1 22.054525 1.13 e 1 Three-point midpoint with h = 0.1 22.22879 6.16 e-2 Three-point midpoint with h = 0.2 22.414163 2.47 e-1 Five-point midpoint with h = 0.1 22.166999 1.69 e-4 Table 4.2 represents values of ? ? = ??? If we had no other information we would accept the five-point midpoint approximation using h = 0.1 as the most accurate, and expect the true value to be between that approximation and the three-point mid-point approximation that is in the interval [22.166, 22.229].
Sec:4.1 Numerical Differentiation ? ?? Approximation Second Derivative Midpoint Formula ?? ? ?? ??+ ? Example Use table 4.2 and Second Derivative Midpoint Formula to approximate ? 2 with = 0.1,0.2 ? (2) ? (2) = = = 29.593200 = 29.704275 True Error ? (2.0) 29.556224 29.5932 29.704275 The true value midpoint with h = 0.1 midpoint with h = 0.2 3.70 e-2 1.48 e-1
Sec:4.1 Numerical Differentiation ??????????? ???????? ????????????? ?????????? ??????????? ????? ???????? ?????? ?????? Product of two functions ? ? + ?? ???? = ? ?? + ? ?? ?? ? ?? ? ?? = Product of three functions ? ?? + ?? ? + ??? ????? = ? ?? ? ?? + ? ?? ? ?? + ? ?? ? ?? ?? ? ?? ? ?? ? ?? =
Sec:4.1 Numerical Differentiation ??????????? Three-Point Midpoint Formula Three-Point Endpoint Formula ?? ??+ ?? ??+ ? ?? ? ?? ??+ ? the second Lagrange polynomial ? ?0 ? ?0 2 +? (?(?)) ? ?0+ ? ?0 ? ?0+ ? ?0 2 ? ? = ? ?0 + ? ?0 + ? ?0+ (? ??+ ?)(? ??)(? ?? ?) ?! Differentiate both sides ? ?0+ ? ?0 2 ? (? ??+ ?)(? ??)(? ?? ?) + ? ? ?0+ + ? ?0 ? ?0+ + ? ?0 2 (? ??+ ?)(? ??)(? ?? ?) ? ? = ? ?0 + ? ?0 + ? ?0+ ? (?(?)) +? (?(?)) ?! ?! ? (? ??+ ?)(? ??)(? ?? ?) = (? ??)(? ?? ?)+ (? ??+ ?)(? ?? ?)+ (? ??+ ?)(? ??) Set ? = ?0 + ? ?0 = ? ?0 2 + ? ?0 ( ??) + ? ?0+ 2 +? ? ? ?!
Sec:4.1 Numerical Differentiation Three-Point Midpoint Formula Three-Point Endpoint Formula ?? ??+ ?? ??+ ? ?? ? ?? ??+ ? the second Lagrange polynomial ? ?0 ? ?0 2 2 ? ?0 ? ?0 2 ? ?0 ? ?0 2 ? ? = ? ?0 + ? ?0+ + ? ?0+ 2 +? ? ? (? ??)(? ?? ?)(? ?? ??) ?! Differentiate both sides ? ?0 + ? ?0 2 2 ? ?0 + ? ?0 2 ? ? ? ? ?0 + ? ?0 2 ? ? = ? ?0 + ? ?0+ + ? ?0+ 2 ? ? ? ?! +? (? ??)(? ?? ?)(? ?? ??) + ? (? ??)(? ?? ?)(? ?? ??) ?! ? (? ??)(? ?? ?)(? ?? ??) = (? ?? ?)(? ?? ??) + (? ??)(? ?? ??) + (? ??)(? ?? ?) Set ? = ?0 + 2 2 2 2 ? ?0 = ? ?0 + ? ?0+ + ? ?0+ 2 ? ? ? ?! +? (? ??)(? ?? ?)(? ?? ??)
Sec:4.1 Numerical Differentiation ??????????? ???????? ????????????? ?????????? ??????????? ????? ???????? ?????? ?????? Use Lagrange polynomial and the given nodes to express ?(?) as ?(?) + ????? then differentiate both sides. Next set ? = ?0
Sec:4.1 Numerical Differentiation ?????? ?????? ? ?0+ ? = ? ?0 + ?? ?0 +1 2?2? ?0 +1 1 6?3? ?0 + 24?4? ?0 + ??? ? = ? ? ?0+ = ? ?0 + ? ?0 +1 2 2? ?0 +1 1 6 3? ?0 + 24 4? ?0 + ??? ? = ?? ? ?0+ 2 = ? ?0 + 2 ? ?0 + 2 2? ?0 +4 3 3? ?0 +4 3 4? ?0 + ??? ? = ? ? ?0 = ? ?0 ? ?0 +1 2 2? ?0 1 1 6 3? ?0 + 24 4? ?0 + ??? ? = ?? ? ?0 2 = ? ?0 2 ? ?0 + 2 2? ?0 4 3 3? ?0 +4 3 4? ?0 +
Sec:4.1 Numerical Differentiation Example Derive a three-point formula to approximate f (x0) that uses f (x0), f (x0 + h), f (x0 + 2h). Then find the error term [what is the order of the error O( ?) ] ? ?? = ? ? ?? + ?? ??+ ? + ?? ??+ ?? + ????? ? ?0 = ? ?0 ? ? +1 2 2? ?0 +1 ? ?0+ = ? ?0 + ? ?0 6 3? ?0 + ? ?0+ 2 = ? ?0 + 2 ? ?0 + 2 2? ?0 +4 ? 3 3? ?0 + + 21 ? ? ?? +?? ??+ ? +?? ??+ ?? ? + ? + ? ? ?0 + ? + 2? ? ?0 2? + 2? ? ?0 = +(1 6?? ?0 +4 3?? ?0) 3 We have 3 constants to determine (A, B, C). So we need three equations ? + ? + ? = 0 21 2? + 2? ? = ? ? =? ? = ? ?? ? ?? = 0 ? + 2? = 1 ??? ?? +? ? ? ? ?? ?? ??+ ? ??? ??+ ??
Sec:4.1 Numerical Differentiation Example Derive a three-point formula to approximate f (x0) that uses f (x0), f (x0 + h), f (x0 + 2h). Then find the error term [what is the order of the error O( ?) ] ? ?? = ? ? ?? + ?? ??+ ? + ?? ??+ ?? + ????? ? ?0 = ? ?0 ? ? ? +1 2 2? ?0 +1 ? ?0+ = ? ?0 + ? ?0 ? ?0+ 2 = ? ?0 + 2 ? ?0 + 2 2? ?0 +4 6 3? ?0 + 3 3? ?0 + + 21 ? ? ?? +?? ??+ ? +?? ??+ ?? 2? + 2? ? ?0 ? + ? + ? ? ?0 + ? + 2? ? ?0 = +(1 6?? ?0 +4 3?? ?0) 3 ??? ?? +? ? ? ? = ? ? =? ? = ? find the error term ? ?? ?? ??+ ? ??? ??+ ?? ?? ? ?? = (1 6?? ?0 +4 = (? ? = ? Leading error term 3?? ?0) 3 ????? = ?(??) ??? ?? ??? ??)?? ?? ???? ? ?? = ? ??? ?? +? ??? ??+ ?? ? ? ?? ? ?? ?? ??+ ?
Sec:4.1 Numerical Differentiation Example Derive a three-point formula to approximate f (x0) that uses f (x0), f (x0 + h), f (x0 + 2h). ? ?? = ? ? ?? + ?? ??+ ? + ?? ??+ ?? + ????? ? ?0 = ? ?0 ? ? +1 2 2? ?0 +1 ? ?0+ = ? ?0 + ? ?0 6 3? ?0 + ? ?0+ 2 = ? ?0 + 2 ? ?0 + 2 2? ?0 +4 ? 3 3? ?0 + + 21 ? ? ?? +?? ??+ ? +?? ??+ ?? 2? + 2? ? ?0 ? + ? + ? ? ?0 + ? + 2? ? ?0 = +(1 6?? ?0 +4 3?? ?0) 3 We have 3 constants to determine (A, B, C). So we need three equations ? + ? + ? = 0 21 2? + 2? = ? ? + 2? = ?
Sec:4.1 Numerical Differentiation ? ?0 = ?? ?0 + ? ? ?0 + ?? ?0+ + ?? ?0+ 2 + ?? ?0+ 3 + ????? ? ?0 = ? ?0 ? ?0 +1 2 2? ?0 1 1 6 3? ?0 + 24 4? ?0 + ? ? ? ?0 = ? ?0 ? ?0+ = ? ?0 + ? ?0 +1 2 2? ?0 +1 1 6 3? ?0 + 3 3? ?0 +4 3 3? ?0 +4 24 4? ?0 + ? ? ?0+ 2 = ? ?0 + 2 ? ?0 + 2 2? ?0 +4 3 4? ?0 + ? ? ? ?0+ 2 = ? ?0 + 2 ? ?0 + 2 2? ?0 +4 3 4? ?0 + ??? = ? + ? + ? + ? + ? ? ?0 + ? ?0 + ? ?0 + ? ?0 + ? ?0 +
