History of Central finite difference coefficients

     constraints: $-r\leq j\leq r$ Comments:-  comment-step: 'The rows use spacing $h=1$. For a general mesh spacing $h$, the power-    $h^m$ is on the derivative side: $h^m f^{(m)}(x)\approx\sum_{j=-r}^{r}a_{m,r,j}f(x+jh)$.'+  comment-step: The coefficients do not depend on $h$, because $h^m$ is on the derivative+    side.   comment-ordering: 'Offsets are listed from $-r$ to $r$, the same order as the function     values in the formula. Symmetric duplicates are stored: $a_{m,r,-j}=(-1)^m a_{m,r,j}$,     and a reader may hold either sign.'   comment-zeroes: Rows with coefficient $0$ are omitted. Thus $a_{m,r,0}$ is absent-    for odd $m$, and any coefficient outside the listed stencil is also $0$.+    for odd $m$.   comment-neighbours: These coefficients differentiate the interpolating polynomial     on equally spaced nodes. HREF{Newton_Cotes_weights}[Newton-Cotes weights] integrate

   formula-moments: $\sum_{j=-r}^{r} a_{m,r,j}j^q=m!$ if $q=m$, and $\sum_{j=-r}^{r}     a_{m,r,j}j^q=0$ for $0\leq q\leq2r$ with $q\neq m$.-  formula-accuracy: The truncation error is $O(h^{2r+1-m})$ for odd $m$ and $O(h^{2r+2-m})$-    for even $m$.+  formula-accuracy: 'The approximation $f^{(m)}(x)\approx h^{-m}\sum_{j=-r}^{r}a_{m,r,j}f(x+jh)$+    has accuracy order $2r+1-m$ for odd $m$ and $2r+2-m$ for even $m$: its error is+    $O(h^{2r+1-m})$ or $O(h^{2r+2-m})$.'   formula-symmetry: $a_{m,r,-j}=(-1)^m a_{m,r,j}$.-  formula-lagrange: $a_{m,r,j}=\ell_{r,j}^{(m)}(0)$, where $\ell_{r,j}(x)=\prod_{\substack{-r\leq-    k\leq r\\ k\neq j}}\frac{x-k}{j-k}$ is the Lagrange basis polynomial for the offset-    $j$.+  formula-lagrange: $a_{m,r,j}=\ell_{2r,\,r+j}^{(m)}(r)$, where $\ell_{d,i}(x)=\prod_{\substack{0\leq+    k\leq d\\ k\neq i}}\frac{x-k}{i-k}$ is the Lagrange basis polynomial for the nodes+    $0,1,\ldots,d$. Programs:   program-sage:

       \ offsets] for q in range(2 * r + 1)]\n    rhs = [QQ(factorial(m)) if q == m\       \ else QQ(0) for q in range(2 * r + 1)]\n    return dict(zip(offsets, matrix(QQ,\-      \ rows).solve_right(vector(QQ, rhs))))\n\ncentral_difference_coefficients(2,\-      \ 2)"+      \ rows).solve_right(vector(QQ, rhs))))\n\nprint(central_difference_coefficients(2,\+      \ 2))" Similar tables: - table: HREF{Lagrange_basis_polynomials_for_equally_spaced_nodes}[Lagrange basis     polynomials for equally spaced nodes]-  relation: their derivatives at the centre give the finite difference coefficients+  relation: $a_{m,r,j}$ is the $m$-th derivative of $\ell_{2r,\,r+j}$ at the middle+    node $x=r$ - table: HREF{Newton_Cotes_weights}[Newton-Cotes weights]   relation: integrals of equally spaced Lagrange basis polynomials rather than derivatives

   number-header: $a_{m,r,j}$ Numbers:-  '1':-    '1':-      '-1':-        number: -1/2-        comment: 'Derivative order $m=1$, 2-order central formula: $\tfrac{1}{2}(-1,0,1)$.'-      '1': 1/2-    '2':-      '-2':-        number: 1/12-        comment: 'Derivative order $m=1$, 4-order central formula: $\tfrac{1}{12}(1,-8,0,8,-1)$.'-      '-1': -2/3-      '1': 2/3-      '2': -1/12-    '3':-      '-3':-        number: -1/60-        comment: 'Derivative order $m=1$, 6-order central formula: $\tfrac{1}{60}(-1,9,-45,0,45,-9,1)$.'-      '-2': 3/20-      '-1': -3/4-      '1': 3/4-      '2': -3/20-      '3': 1/60-    '4':-      '-4':-        number: 1/280-        comment: 'Derivative order $m=1$, 8-order central formula: $\tfrac{1}{840}(3,-32,168,-672,0,672,-168,32,-3)$.'-      '-3': -4/105-      '-2': 1/5-      '-1': -4/5-      '1': 4/5-      '2': -1/5-      '3': 4/105-      '4': -1/280-    '5':-      '-5':-        number: -1/1260-        comment: 'Derivative order $m=1$, 10-order central formula: $\tfrac{1}{2520}(-2,25,-150,600,-2100,0,2100,-600,150,-25,2)$.'-      '-4': 5/504-      '-3': -5/84-      '-2': 5/21-      '-1': -5/6-      '1': 5/6-      '2': -5/21-      '3': 5/84-      '4': -5/504-      '5': 1/1260-  '2':-    '1':-      '-1':-        number: '1'-        comment: 'Derivative order $m=2$, 2-order central formula: $1(1,-2,1)$.'-      '0': '-2'-      '1': '1'-    '2':-      '-2':-        number: -1/12-        comment: 'Derivative order $m=2$, 4-order central formula: $\tfrac{1}{12}(-1,16,-30,16,-1)$.'-      '-1': 4/3-      '0': -5/2-      '1': 4/3-      '2': -1/12-    '3':-      '-3':-        number: 1/90-        comment: 'Derivative order $m=2$, 6-order central formula: $\tfrac{1}{180}(2,-27,270,-490,270,-27,2)$.'-      '-2': -3/20-      '-1': 3/2-      '0': -49/18-      '1': 3/2-      '2': -3/20-      '3': 1/90-    '4':-      '-4':-        number: -1/560-        comment: 'Derivative order $m=2$, 8-order central formula: $\tfrac{1}{5040}(-9,128,-1008,8064,-14350,8064,-1008,128,-9)$.'-      '-3': 8/315-      '-2': -1/5-      '-1': 8/5-      '0': -205/72-      '1': 8/5-      '2': -1/5-      '3': 8/315-      '4': -1/560-    '5':-      '-5':-        number: 1/3150-        comment: 'Derivative order $m=2$, 10-order central formula: $\tfrac{1}{25200}(8,-125,1000,-6000,42000,-73766,42000,-6000,1000,-125,8)$.'-      '-4': -5/1008-      '-3': 5/126-      '-2': -5/21-      '-1': 5/3-      '0': -5269/1800-      '1': 5/3-      '2': -5/21-      '3': 5/126-      '4': -5/1008-      '5': 1/3150-  '3':-    '2':-      '-2':-        number: -1/2-        comment: 'Derivative order $m=3$, 2-order central formula: $\tfrac{1}{2}(-1,2,0,-2,1)$.'-      '-1': '1'-      '1': '-1'-      '2': 1/2-    '3':-      '-3':-        number: 1/8-        comment: 'Derivative order $m=3$, 4-order central formula: $\tfrac{1}{8}(1,-8,13,0,-13,8,-1)$.'-      '-2': '-1'-      '-1': 13/8-      '1': -13/8-      '2': '1'-      '3': -1/8-    '4':-      '-4':-        number: -7/240-        comment: 'Derivative order $m=3$, 6-order central formula: $\tfrac{1}{240}(-7,72,-338,488,0,-488,338,-72,7)$.'-      '-3': 3/10-      '-2': -169/120-      '-1': 61/30-      '1': -61/30-      '2': 169/120-      '3': -3/10-      '4': 7/240-    '5':-      '-5':-        number: 41/6048-        comment: 'Derivative order $m=3$, 8-order central formula: $\tfrac{1}{30240}(205,-2522,14607,-52428,70098,0,-70098,52428,-14607,2522,-205)$.'-      '-4': -1261/15120-      '-3': 541/1120-      '-2': -4369/2520-      '-1': 1669/720-      '1': -1669/720-      '2': 4369/2520-      '3': -541/1120-      '4': 1261/15120-      '5': -41/6048-  '4':-    '2':-      '-2':-        number: '1'-        comment: 'Derivative order $m=4$, 2-order central formula: $1(1,-4,6,-4,1)$.'-      '-1': '-4'-      '0': '6'-      '1': '-4'-      '2': '1'-    '3':-      '-3':-        number: -1/6-        comment: 'Derivative order $m=4$, 4-order central formula: $\tfrac{1}{6}(-1,12,-39,56,-39,12,-1)$.'-      '-2': '2'-      '-1': -13/2-      '0': 28/3-      '1': -13/2-      '2': '2'-      '3': -1/6-    '4':-      '-4':-        number: 7/240-        comment: 'Derivative order $m=4$, 6-order central formula: $\tfrac{1}{240}(7,-96,676,-1952,2730,-1952,676,-96,7)$.'-      '-3': -2/5-      '-2': 169/60-      '-1': -122/15-      '0': 91/8-      '1': -122/15-      '2': 169/60-      '3': -2/5-      '4': 7/240-    '5':-      '-5':-        number: -41/7560-        comment: 'Derivative order $m=4$, 8-order central formula: $\tfrac{1}{15120}(-82,1261,-9738,52428,-140196,192654,-140196,52428,-9738,1261,-82)$.'-      '-4': 1261/15120-      '-3': -541/840-      '-2': 4369/1260-      '-1': -1669/180-      '0': 1529/120-      '1': -1669/180-      '2': 4369/1260-      '3': -541/840-      '4': 1261/15120-      '5': -41/7560-  '5':-    '3':-      '-3':-        number: -1/2-        comment: 'Derivative order $m=5$, 2-order central formula: $\tfrac{1}{2}(-1,4,-5,0,5,-4,1)$.'-      '-2': '2'-      '-1': -5/2-      '1': 5/2-      '2': '-2'-      '3': 1/2-    '4':-      '-4':-        number: 1/6-        comment: 'Derivative order $m=5$, 4-order central formula: $\tfrac{1}{6}(1,-9,26,-29,0,29,-26,9,-1)$.'-      '-3': -3/2-      '-2': 13/3-      '-1': -29/6-      '1': 29/6-      '2': -13/3-      '3': 3/2-      '4': -1/6-    '5':-      '-5':-        number: -13/288-        comment: 'Derivative order $m=5$, 6-order central formula: $\tfrac{1}{288}(-13,152,-783,1872,-1938,0,1938,-1872,783,-152,13)$.'-      '-4': 19/36-      '-3': -87/32-      '-2': 13/2-      '-1': -323/48-      '1': 323/48-      '2': -13/2-      '3': 87/32-      '4': -19/36-      '5': 13/288-  '6':-    '3':-      '-3':-        number: '1'-        comment: 'Derivative order $m=6$, 2-order central formula: $1(1,-6,15,-20,15,-6,1)$.'-      '-2': '-6'-      '-1': '15'-      '0': '-20'-      '1': '15'-      '2': '-6'-      '3': '1'-    '4':-      '-4':-        number: -1/4-        comment: 'Derivative order $m=6$, 4-order central formula: $\tfrac{1}{4}(-1,12,-52,116,-150,116,-52,12,-1)$.'-      '-3': '3'-      '-2': '-13'-      '-1': '29'-      '0': -75/2-      '1': '29'-      '2': '-13'-      '3': '3'-      '4': -1/4-    '5':-      '-5':-        number: 13/240-        comment: 'Derivative order $m=6$, 6-order central formula: $\tfrac{1}{240}(13,-190,1305,-4680,9690,-12276,9690,-4680,1305,-190,13)$.'-      '-4': -19/24-      '-3': 87/16-      '-2': -39/2-      '-1': 323/8-      '0': -1023/20-      '1': 323/8-      '2': -39/2-      '3': 87/16-      '4': -19/24-      '5': 13/240+- comment: 'Accuracy order 2: $\tfrac{1}{2}(-1,0,1)$.'+  params:+    m: '1'+    r: '1'+    j: '-1'+  number: -1/2+- params:+    m: '1'+    r: '1'+    j: '1'+  number: 1/2+- comment: 'Accuracy order 4: $\tfrac{1}{12}(1,-8,0,8,-1)$.'+  params:+    m: '1'+    r: '2'+    j: '-2'+  number: 1/12+- params:+    m: '1'+    r: '2'+    j: '-1'+  number: -2/3+- params:+    m: '1'+    r: '2'+    j: '1'+  number: 2/3+- params:+    m: '1'+    r: '2'+    j: '2'+  number: -1/12+- comment: 'Accuracy order 6: $\tfrac{1}{60}(-1,9,-45,0,45,-9,1)$.'+  params:+    m: '1'+    r: '3'+    j: '-3'+  number: -1/60+- params:+    m: '1'+    r: '3'+    j: '-2'+  number: 3/20+- params:+    m: '1'+    r: '3'+    j: '-1'+  number: -3/4+- params:+    m: '1'+    r: '3'+    j: '1'+  number: 3/4+- params:+    m: '1'+    r: '3'+    j: '2'+  number: -3/20+- params:+    m: '1'+    r: '3'+    j: '3'+  number: 1/60+- comment: 'Accuracy order 8: $\tfrac{1}{840}(3,-32,168,-672,0,672,-168,32,-3)$.'+  params:+    m: '1'+    r: '4'+    j: '-4'+  number: 1/280+- params:+    m: '1'+    r: '4'+    j: '-3'+  number: -4/105+- params:+    m: '1'+    r: '4'+    j: '-2'+  number: 1/5+- params:+    m: '1'+    r: '4'+    j: '-1'+  number: -4/5+- params:+    m: '1'+    r: '4'+    j: '1'+  number: 4/5+- params:+    m: '1'+    r: '4'+    j: '2'+  number: -1/5+- params:+    m: '1'+    r: '4'+    j: '3'+  number: 4/105+- params:+    m: '1'+    r: '4'+    j: '4'+  number: -1/280+- comment: 'Accuracy order 10: $\tfrac{1}{2520}(-2,25,-150,600,-2100,0,2100,-600,150,-25,2)$.'+  params:+    m: '1'+    r: '5'+    j: '-5'+  number: -1/1260+- params:+    m: '1'+    r: '5'+    j: '-4'+  number: 5/504+- params:+    m: '1'+    r: '5'+    j: '-3'+  number: -5/84+- params:+    m: '1'+    r: '5'+    j: '-2'+  number: 5/21+- params:+    m: '1'+    r: '5'+    j: '-1'+  number: -5/6+- params:+    m: '1'+    r: '5'+    j: '1'+  number: 5/6+- params:+    m: '1'+    r: '5'+    j: '2'+  number: -5/21+- params:+    m: '1'+    r: '5'+    j: '3'+  number: 5/84+- params:+    m: '1'+    r: '5'+    j: '4'+  number: -5/504+- params:+    m: '1'+    r: '5'+    j: '5'+  number: 1/1260+- comment: 'Accuracy order 2: $(1,-2,1)$.'+  params:+    m: '2'+    r: '1'+    j: '-1'+  number: '1'+- params:+    m: '2'+    r: '1'+    j: '0'+  number: '-2'+- params:+    m: '2'+    r: '1'+    j: '1'+  number: '1'+- comment: 'Accuracy order 4: $\tfrac{1}{12}(-1,16,-30,16,-1)$.'+  params:+    m: '2'+    r: '2'+    j: '-2'+  number: -1/12+- params:+    m: '2'+    r: '2'+    j: '-1'+  number: 4/3+- params:+    m: '2'+    r: '2'+    j: '0'+  number: -5/2+- params:+    m: '2'+    r: '2'+    j: '1'+  number: 4/3+- params:+    m: '2'+    r: '2'+    j: '2'+  number: -1/12+- comment: 'Accuracy order 6: $\tfrac{1}{180}(2,-27,270,-490,270,-27,2)$.'+  params:+    m: '2'+    r: '3'+    j: '-3'+  number: 1/90+- params:+    m: '2'+    r: '3'+    j: '-2'+  number: -3/20+- params:+    m: '2'+    r: '3'+    j: '-1'+  number: 3/2+- params:+    m: '2'+    r: '3'+    j: '0'+  number: -49/18+- params:+    m: '2'+    r: '3'+    j: '1'+  number: 3/2+- params:+    m: '2'+    r: '3'+    j: '2'+  number: -3/20+- params:+    m: '2'+    r: '3'+    j: '3'+  number: 1/90+- comment: 'Accuracy order 8: $\tfrac{1}{5040}(-9,128,-1008,8064,-14350,8064,-1008,128,-9)$.'+  params:+    m: '2'+    r: '4'+    j: '-4'+  number: -1/560+- params:+    m: '2'+    r: '4'+    j: '-3'+  number: 8/315+- params:+    m: '2'+    r: '4'+    j: '-2'+  number: -1/5+- params:+    m: '2'+    r: '4'+    j: '-1'+  number: 8/5+- params:+    m: '2'+    r: '4'+    j: '0'+  number: -205/72+- params:+    m: '2'+    r: '4'+    j: '1'+  number: 8/5+- params:+    m: '2'+    r: '4'+    j: '2'+  number: -1/5+- params:+    m: '2'+    r: '4'+    j: '3'+  number: 8/315+- params:+    m: '2'+    r: '4'+    j: '4'+  number: -1/560+- comment: 'Accuracy order 10: $\tfrac{1}{25200}(8,-125,1000,-6000,42000,-73766,42000,-6000,1000,-125,8)$.'+  params:+    m: '2'+    r: '5'+    j: '-5'+  number: 1/3150+- params:+    m: '2'+    r: '5'+    j: '-4'+  number: -5/1008+- params:+    m: '2'+    r: '5'+    j: '-3'+  number: 5/126+- params:+    m: '2'+    r: '5'+    j: '-2'+  number: -5/21+- params:+    m: '2'+    r: '5'+    j: '-1'+  number: 5/3+- params:+    m: '2'+    r: '5'+    j: '0'+  number: -5269/1800+- params:+    m: '2'+    r: '5'+    j: '1'+  number: 5/3+- params:+    m: '2'+    r: '5'+    j: '2'+  number: -5/21+- params:+    m: '2'+    r: '5'+    j: '3'+  number: 5/126+- params:+    m: '2'+    r: '5'+    j: '4'+  number: -5/1008+- params:+    m: '2'+    r: '5'+    j: '5'+  number: 1/3150+- comment: 'Accuracy order 2: $\tfrac{1}{2}(-1,2,0,-2,1)$.'+  params:+    m: '3'+    r: '2'+    j: '-2'+  number: -1/2+- params:+    m: '3'+    r: '2'+    j: '-1'+  number: '1'+- params:+    m: '3'+    r: '2'+    j: '1'+  number: '-1'+- params:+    m: '3'+    r: '2'+    j: '2'+  number: 1/2+- comment: 'Accuracy order 4: $\tfrac{1}{8}(1,-8,13,0,-13,8,-1)$.'+  params:+    m: '3'+    r: '3'+    j: '-3'+  number: 1/8+- params:+    m: '3'+    r: '3'+    j: '-2'+  number: '-1'+- params:+    m: '3'+    r: '3'+    j: '-1'+  number: 13/8+- params:+    m: '3'+    r: '3'+    j: '1'+  number: -13/8+- params:+    m: '3'+    r: '3'+    j: '2'+  number: '1'+- params:+    m: '3'+    r: '3'+    j: '3'+  number: -1/8+- comment: 'Accuracy order 6: $\tfrac{1}{240}(-7,72,-338,488,0,-488,338,-72,7)$.'+  params:+    m: '3'+    r: '4'+    j: '-4'+  number: -7/240+- params:+    m: '3'+    r: '4'+    j: '-3'+  number: 3/10+- params:+    m: '3'+    r: '4'+    j: '-2'+  number: -169/120+- params:+    m: '3'+    r: '4'+    j: '-1'+  number: 61/30+- params:+    m: '3'+    r: '4'+    j: '1'+  number: -61/30+- params:+    m: '3'+    r: '4'+    j: '2'+  number: 169/120+- params:+    m: '3'+    r: '4'+    j: '3'+  number: -3/10+- params:+    m: '3'+    r: '4'+    j: '4'+  number: 7/240+- comment: 'Accuracy order 8: $\tfrac{1}{30240}(205,-2522,14607,-52428,70098,0,-70098,52428,-14607,2522,-205)$.'+  params:+    m: '3'+    r: '5'+    j: '-5'+  number: 41/6048+- params:+    m: '3'+    r: '5'+    j: '-4'+  number: -1261/15120+- params:+    m: '3'+    r: '5'+    j: '-3'+  number: 541/1120+- params:+    m: '3'+    r: '5'+    j: '-2'+  number: -4369/2520+- params:+    m: '3'+    r: '5'+    j: '-1'+  number: 1669/720+- params:+    m: '3'+    r: '5'+    j: '1'+  number: -1669/720+- params:+    m: '3'+    r: '5'+    j: '2'+  number: 4369/2520+- params:+    m: '3'+    r: '5'+    j: '3'+  number: -541/1120+- params:+    m: '3'+    r: '5'+    j: '4'+  number: 1261/15120+- params:+    m: '3'+    r: '5'+    j: '5'+  number: -41/6048+- comment: 'Accuracy order 2: $(1,-4,6,-4,1)$.'+  params:+    m: '4'+    r: '2'+    j: '-2'+  number: '1'+- params:+    m: '4'+    r: '2'+    j: '-1'+  number: '-4'+- params:+    m: '4'+    r: '2'+    j: '0'+  number: '6'+- params:+    m: '4'+    r: '2'+    j: '1'+  number: '-4'+- params:+    m: '4'+    r: '2'+    j: '2'+  number: '1'+- comment: 'Accuracy order 4: $\tfrac{1}{6}(-1,12,-39,56,-39,12,-1)$.'+  params:+    m: '4'+    r: '3'+    j: '-3'+  number: -1/6+- params:+    m: '4'+    r: '3'+    j: '-2'+  number: '2'+- params:+    m: '4'+    r: '3'+    j: '-1'+  number: -13/2+- params:+    m: '4'+    r: '3'+    j: '0'+  number: 28/3+- params:+    m: '4'+    r: '3'+    j: '1'+  number: -13/2+- params:+    m: '4'+    r: '3'+    j: '2'+  number: '2'+- params:+    m: '4'+    r: '3'+    j: '3'+  number: -1/6+- comment: 'Accuracy order 6: $\tfrac{1}{240}(7,-96,676,-1952,2730,-1952,676,-96,7)$.'+  params:+    m: '4'+    r: '4'+    j: '-4'+  number: 7/240+- params:+    m: '4'+    r: '4'+    j: '-3'+  number: -2/5+- params:+    m: '4'+    r: '4'+    j: '-2'+  number: 169/60+- params:+    m: '4'+    r: '4'+    j: '-1'+  number: -122/15+- params:+    m: '4'+    r: '4'+    j: '0'+  number: 91/8+- params:+    m: '4'+    r: '4'+    j: '1'+  number: -122/15+- params:+    m: '4'+    r: '4'+    j: '2'+  number: 169/60+- params:+    m: '4'+    r: '4'+    j: '3'+  number: -2/5+- params:+    m: '4'+    r: '4'+    j: '4'+  number: 7/240+- comment: 'Accuracy order 8: $\tfrac{1}{15120}(-82,1261,-9738,52428,-140196,192654,-140196,52428,-9738,1261,-82)$.'+  params:+    m: '4'+    r: '5'+    j: '-5'+  number: -41/7560+- params:+    m: '4'+    r: '5'+    j: '-4'+  number: 1261/15120+- params:+    m: '4'+    r: '5'+    j: '-3'+  number: -541/840+- params:+    m: '4'+    r: '5'+    j: '-2'+  number: 4369/1260+- params:+    m: '4'+    r: '5'+    j: '-1'+  number: -1669/180+- params:+    m: '4'+    r: '5'+    j: '0'+  number: 1529/120+- params:+    m: '4'+    r: '5'+    j: '1'+  number: -1669/180+- params:+    m: '4'+    r: '5'+    j: '2'+  number: 4369/1260+- params:+    m: '4'+    r: '5'+    j: '3'+  number: -541/840+- params:+    m: '4'+    r: '5'+    j: '4'+  number: 1261/15120+- params:+    m: '4'+    r: '5'+    j: '5'+  number: -41/7560+- comment: 'Accuracy order 2: $\tfrac{1}{2}(-1,4,-5,0,5,-4,1)$.'+  params:+    m: '5'+    r: '3'+    j: '-3'+  number: -1/2+- params:+    m: '5'+    r: '3'+    j: '-2'+  number: '2'+- params:+    m: '5'+    r: '3'+    j: '-1'+  number: -5/2+- params:+    m: '5'+    r: '3'+    j: '1'+  number: 5/2+- params:+    m: '5'+    r: '3'+    j: '2'+  number: '-2'+- params:+    m: '5'+    r: '3'+    j: '3'+  number: 1/2+- comment: 'Accuracy order 4: $\tfrac{1}{6}(1,-9,26,-29,0,29,-26,9,-1)$.'+  params:+    m: '5'+    r: '4'+    j: '-4'+  number: 1/6+- params:+    m: '5'+    r: '4'+    j: '-3'+  number: -3/2+- params:+    m: '5'+    r: '4'+    j: '-2'+  number: 13/3+- params:+    m: '5'+    r: '4'+    j: '-1'+  number: -29/6+- params:+    m: '5'+    r: '4'+    j: '1'+  number: 29/6+- params:+    m: '5'+    r: '4'+    j: '2'+  number: -13/3+- params:+    m: '5'+    r: '4'+    j: '3'+  number: 3/2+- params:+    m: '5'+    r: '4'+    j: '4'+  number: -1/6+- comment: 'Accuracy order 6: $\tfrac{1}{288}(-13,152,-783,1872,-1938,0,1938,-1872,783,-152,13)$.'+  params:+    m: '5'+    r: '5'+    j: '-5'+  number: -13/288+- params:+    m: '5'+    r: '5'+    j: '-4'+  number: 19/36+- params:+    m: '5'+    r: '5'+    j: '-3'+  number: -87/32+- params:+    m: '5'+    r: '5'+    j: '-2'+  number: 13/2+- params:+    m: '5'+    r: '5'+    j: '-1'+  number: -323/48+- params:+    m: '5'+    r: '5'+    j: '1'+  number: 323/48+- params:+    m: '5'+    r: '5'+    j: '2'+  number: -13/2+- params:+    m: '5'+    r: '5'+    j: '3'+  number: 87/32+- params:+    m: '5'+    r: '5'+    j: '4'+  number: -19/36+- params:+    m: '5'+    r: '5'+    j: '5'+  number: 13/288+- comment: 'Accuracy order 2: $(1,-6,15,-20,15,-6,1)$.'+  params:+    m: '6'+    r: '3'+    j: '-3'+  number: '1'+- params:+    m: '6'+    r: '3'+    j: '-2'+  number: '-6'+- params:+    m: '6'+    r: '3'+    j: '-1'+  number: '15'+- params:+    m: '6'+    r: '3'+    j: '0'+  number: '-20'+- params:+    m: '6'+    r: '3'+    j: '1'+  number: '15'+- params:+    m: '6'+    r: '3'+    j: '2'+  number: '-6'+- params:+    m: '6'+    r: '3'+    j: '3'+  number: '1'+- comment: 'Accuracy order 4: $\tfrac{1}{4}(-1,12,-52,116,-150,116,-52,12,-1)$.'+  params:+    m: '6'+    r: '4'+    j: '-4'+  number: -1/4+- params:+    m: '6'+    r: '4'+    j: '-3'+  number: '3'+- params:+    m: '6'+    r: '4'+    j: '-2'+  number: '-13'+- params:+    m: '6'+    r: '4'+    j: '-1'+  number: '29'+- params:+    m: '6'+    r: '4'+    j: '0'+  number: -75/2+- params:+    m: '6'+    r: '4'+    j: '1'+  number: '29'+- params:+    m: '6'+    r: '4'+    j: '2'+  number: '-13'+- params:+    m: '6'+    r: '4'+    j: '3'+  number: '3'+- params:+    m: '6'+    r: '4'+    j: '4'+  number: -1/4+- comment: 'Accuracy order 6: $\tfrac{1}{240}(13,-190,1305,-4680,9690,-12276,9690,-4680,1305,-190,13)$.'+  params:+    m: '6'+    r: '5'+    j: '-5'+  number: 13/240+- params:+    m: '6'+    r: '5'+    j: '-4'+  number: -19/24+- params:+    m: '6'+    r: '5'+    j: '-3'+  number: 87/16+- params:+    m: '6'+    r: '5'+    j: '-2'+  number: -39/2+- params:+    m: '6'+    r: '5'+    j: '-1'+  number: 323/8+- params:+    m: '6'+    r: '5'+    j: '0'+  number: -1023/20+- params:+    m: '6'+    r: '5'+    j: '1'+  number: 323/8+- params:+    m: '6'+    r: '5'+    j: '2'+  number: -39/2+- params:+    m: '6'+    r: '5'+    j: '3'+  number: 87/16+- params:+    m: '6'+    r: '5'+    j: '4'+  number: -19/24+- params:+    m: '6'+    r: '5'+    j: '5'+  number: 13/240