Richardson Extrapolation

$A\; -\; A(h)\; =\; a\_0h^\{k\_0\}\; +\; a\_1h^\{k\_1\}\; +\; a\_2h^\{k\_2\}\; +\; \backslash cdots$

$A\; =\; A(h)\; +\; a\_0h^\{k\_0\}\; +\; a\_1h^\{k\_1\}\; +\; a\_2h^\{k\_2\}\; +\; \backslash cdots$

$A\; =\; A(h)+\; a\_0h^\{k\_0\}\; +\; O(h^\{k\_1\}).\; \backslash ,\backslash !$

$A\; =\; A(h)+\; a\_0h^\{k\_0\}\; +\; O(h^\{k\_1\})\; \backslash ,\backslash !$

$A\; =\; A\backslash left(\backslash frac\{h\}\{t\}\backslash right)\; +\; a\_0\backslash left(\backslash frac\{h\}\{t\}\backslash right)^\{k\_0\}\; +\; O(h^\{k\_1\})\; .$

$(t^\{k\_0\}-1)A\; =\; t^\{k\_0\}A\backslash left(\backslash frac\{h\}\{t\}\backslash right)\; -\; A(h)\; +\; O(h^\{k\_1\})$

$A\; =\; \backslash frac\{t^\{k\_0\}A\backslash left(\backslash frac\{h\}\{t\}\backslash right)\; -\; A(h)\}\{t^\{k\_0\}-1\}\; +\; O(h^\{k\_1\})\; .$

$A\_\{i+1\}(h)\; =\; \backslash frac\{t^\{k\_i\}A\_i\backslash left(\backslash frac\{h\}\{t\}\backslash right)\; -\; A\_i(h)\}\{t^\{k\_i\}-1\}$

$A\; =\; A\_\{i+1\}(h)\; +\; O(h^\{k\_\{i+1\}\})\; .$

Example

$f(x+h)\; =\; f(x)\; +\; f\text{'}(x)h\; +\; \backslash frac\{f\text{'}\text{'}(x)\}\{2\}h^2\; +\; \backslash cdots$

$f\text{'}(x)\; =\; \backslash frac\{f(x+h)\; -\; f(x)\}\{h\}\; -\; \backslash frac\{f\text{'}\text{'}(x)\}\{2\}h\; +\; \backslash cdots.$

$A\_0(h)\; =\; \backslash frac\{f(x+h)\; -\; f(x)\}\{h\}$

$A\; =\; 2A\_0\backslash left(\backslash frac\{h\}\{2\}\backslash right)\; -\; A\_0(h)\; +\; O(h^2)\; .$

$A\_1(h)\; =\; 2A\_0\backslash left(\backslash frac\{h\}\{2\}\backslash right)\; -\; A\_0(h)$

$A\; =\; \backslash frac\{4A\_1\backslash left(\backslash frac\{h\}\{2\}\backslash right)\; -\; A\_1(h)\}\{3\}\; +\; O(h^3)\; .$

<< Previous Word Browser Next >>
prosper garnot kenneth j. hsu mycin stu irwin mall of arabia tobas hernndez two factor theory margaret d. tutwiler parmenio fifa series johnny paredes	ticto barb redhouse quilombo cory hobbs anquan boldin yuan cheng fung protected areas of the united states dixie cup scott simms everett, ontario christian century	claudio ranieri gallery paget's disease of bone francis kenrick poor management the willows at chillico woods, guelph, ontario south pointe, innisfil, ontario paget's disease sunflowers, barrie, ontario central league (football) creditvalley, mississauga, ontario	the call the call (television) lakelands village, brampton, ontario al pedrique gro par bahia de todos os santos vanguard progressive unionist party loch lubnaig katie leung viceroyalty of brazil contour line