List of Runge–Kutta methods

Runge–Kutta methods are methods for the numerical solution of the ordinary differential equation


 * $$\frac{d y}{d t} = f(t, y).$$

Explicit Runge–Kutta methods take the form


 * $$\begin{align}

y_{n+1} &= y_n + h \sum_{i=1}^s b_i k_i \\ k_1 &= f(t_n, y_n), \\ k_2 &= f(t_n+c_2h, y_n+h(a_{21}k_1)), \\ k_3 &= f(t_n+c_3h, y_n+h(a_{31}k_1+a_{32}k_2)), \\ &\;\;\vdots \\ k_i &= f\left(t_n + c_i h, y_n + h \sum_{j = 1}^{i-1} a_{ij} k_j\right). \end{align}$$

Stages for implicit methods of s stages take the more general form, with the solution to be found over all s


 * $$k_i = f\left(t_n + c_i h, y_n + h \sum_{j = 1}^{s} a_{ij} k_j\right). $$

Each method listed on this page is defined by its Butcher tableau, which puts the coefficients of the method in a table as follows:



\begin{array}{c|cccc} c_1   & a_{11} & a_{12}& \dots & a_{1s}\\ c_2   & a_{21} & a_{22}& \dots & a_{2s}\\ \vdots & \vdots & \vdots& \ddots& \vdots\\ c_s   & a_{s1} & a_{s2}& \dots & a_{ss} \\ \hline & b_1   & b_2   & \dots & b_s\\ \end{array} $$

For adaptive and implicit methods, the Butcher tableau is extended to give values of $$b^*_i$$, and the estimated error is then
 * $$ e_{n+1} = h\sum_{i=1}^s (b_i - b^*_i) k_i$$.

Explicit methods
The explicit methods are those where the matrix $$[a_{ij}]$$ is lower triangular.

Forward Euler
The Euler method is first order. The lack of stability and accuracy limits its popularity mainly to use as a simple introductory example of a numeric solution method.



\begin{array}{c|c} 0 & 0 \\ \hline & 1 \\ \end{array} $$

Explicit midpoint method
The (explicit) midpoint method is a second-order method with two stages (see also the implicit midpoint method below):



\begin{array}{c|cc} 0  & 0   & 0  \\ 1/2 & 1/2 & 0  \\ \hline & 0  & 1  \\ \end{array} $$

Heun's method
Heun's method is a second-order method with two stages. It is also known as the explicit trapezoid rule, improved Euler's method, or modified Euler's method:



\begin{array}{c|cc} 0  & 0   & 0  \\ 1 & 1 & 0  \\ \hline & 1/2 & 1/2 \\ \end{array} $$

Ralston's method
Ralston's method is a second-order method with two stages and a minimum local error bound:



\begin{array}{c|cc} 0  & 0   & 0  \\ 2/3 & 2/3 & 0  \\ \hline & 1/4  & 3/4  \\ \end{array} $$

Generic second-order method


\begin{array}{c|ccc} 0  & 0   & 0   \\ \alpha & \alpha & 0   \\ \hline & 1-\frac{1}{2\alpha} & \frac{1}{2\alpha} \\ \end{array} $$

Kutta's third-order method


\begin{array}{c|ccc} 0  & 0   & 0   & 0    \\ 1/2 & 1/2 & 0   & 0    \\ 1   & -1  & 2   & 0    \\ \hline & 1/6 & 2/3 & 1/6 \\ \end{array} $$

Generic third-order method
See Sanderse and Veldman (2019).

for α ≠ 0, $2/3$, 1:

\begin{array}{c|ccc} 0     & 0      & 0  & 0\\ \alpha & \alpha & 0  & 0\\ 1     &1+\frac{1- \alpha}{\alpha (3\alpha -2)} & -\frac{1- \alpha}{\alpha(3\alpha -2)} & 0\\ \hline & \frac{1}{2}-\frac{1}{6\alpha} & \frac{1}{6\alpha(1-\alpha)} & \frac{2-3\alpha}{6(1-\alpha)} \\ \end{array} $$

Heun's third-order method


\begin{array}{c|ccc} 0  & 0   & 0   & 0    \\ 1/3 & 1/3 & 0   & 0    \\ 2/3 & 0   & 2/3 & 0   \\ \hline & 1/4 & 0  & 3/4  \\ \end{array} $$

Van der Houwen's/Wray third-order method


\begin{array}{c|ccc} 0  & 0   & 0   & 0    \\ 8/15 & 8/15 & 0   & 0    \\ 2/3 & 1/4   & 5/12 & 0    \\ \hline & 1/4 & 0  & 3/4  \\ \end{array} $$

Ralston's third-order method
Ralston's third-order method is used in the embedded Bogacki–Shampine method.



\begin{array}{c|ccc} 0  & 0   & 0   & 0    \\ 1/2 & 1/2 & 0   & 0    \\ 3/4 & 0   & 3/4 & 0    \\ \hline & 2/9 & 1/3 & 4/9 \\ \end{array} $$

Third-order Strong Stability Preserving Runge-Kutta (SSPRK3)


\begin{array}{c|ccc} 0  & 0   & 0   & 0    \\ 1   & 1   & 0   & 0    \\ 1/2 & 1/4 & 1/4 & 0    \\ \hline & 1/6 & 1/6 & 2/3 \\ \end{array} $$

Classic fourth-order method
The "original" Runge–Kutta method.



\begin{array}{c|cccc} 0  & 0   & 0   & 0   & 0\\ 1/2 & 1/2 & 0   & 0   & 0\\ 1/2 & 0   & 1/2 & 0   & 0\\ 1   & 0   & 0   & 1   & 0\\ \hline & 1/6 & 1/3 & 1/3 & 1/6\\ \end{array} $$

3/8-rule fourth-order method
This method doesn't have as much notoriety as the "classic" method, but is just as classic because it was proposed in the same paper (Kutta, 1901).



\begin{array}{c|cccc} 0  & 0   & 0   & 0   & 0\\ 1/3 & 1/3 & 0   & 0   & 0\\ 2/3 & -1/3   & 1 & 0   & 0\\ 1   & 1   & -1   & 1   & 0\\                   \hline & 1/8 & 3/8 & 3/8 & 1/8\\ \end{array} $$

Ralston's fourth-order method
This fourth order method has minimum truncation error.

\begin{array}{c|cccc} 0  & 0   & 0   & 0   & 0\\ \frac{2}{5} & \frac{2}{5} & 0   & 0   & 0\\ \frac{14 - 3 \sqrt{5}}{16} & \frac{-2\,889 + 1\,428\sqrt{5}}{1\,024}  & \frac{3\,785 - 1\,620\sqrt{5}}{1\,024} & 0   & 0\\ 1  & \frac{-3\,365 + 2\,094\sqrt{5}}{6\,040}   & \frac{-975 - 3\,046\sqrt{5}}{2\,552}   & \frac{467\,040 + 203\,968\sqrt{5}}{240\,845}   & 0\\ \hline & \frac{263 + 24\sqrt{5}}{1\,812} & \frac{125 - 1000\sqrt{5}}{3\,828} & \frac{3\,426\,304 + 1\,661\,952\sqrt{5}}{5\,924\,787} & \frac{30 - 4\sqrt{5}}{123}\\ \end{array} $$

Embedded methods
The embedded methods are designed to produce an estimate of the local truncation error of a single Runge–Kutta step, and as result, allow to control the error with adaptive stepsize. This is done by having two methods in the tableau, one with order p and one with order p-1.

The lower-order step is given by


 * $$   y^*_{n+1} = y_n + h\sum_{i=1}^s b^*_i k_i, $$

where the $$k_i$$ are the same as for the higher order method. Then the error is


 * $$   e_{n+1} = y_{n+1} - y^*_{n+1} = h\sum_{i=1}^s (b_i - b^*_i) k_i, $$

which is $$O(h^p)$$. The Butcher Tableau for this kind of method is extended to give the values of $$b^*_i$$

\begin{array}{c|cccc} c_1   & a_{11} & a_{12}& \dots & a_{1s}\\ c_2   & a_{21} & a_{22}& \dots & a_{2s}\\ \vdots & \vdots & \vdots& \ddots& \vdots\\ c_s   & a_{s1} & a_{s2}& \dots & a_{ss} \\ \hline & b_1   & b_2   & \dots & b_s\\ & b_1^*   & b_2^*   & \dots & b_s^*\\ \end{array} $$

Heun–Euler
The simplest adaptive Runge–Kutta method involves combining Heun's method, which is order 2, with the Euler method, which is order 1. Its extended Butcher Tableau is:

\begin{array}{c|cc} 0&\\	1& 	1 \\ \hline &	1/2& 	1/2\\	&	1 &	0 \end{array} $$

The error estimate is used to control the stepsize.

Fehlberg RK1(2)
The Fehlberg method has two methods of orders 1 and 2. Its extended Butcher Tableau is:

The first row of b coefficients gives the second-order accurate solution, and the second row has order one.

Bogacki–Shampine
The Bogacki–Shampine method has two methods of orders 2 and 3. Its extended Butcher Tableau is:

The first row of b coefficients gives the third-order accurate solution, and the second row has order two.

Fehlberg
The Runge–Kutta–Fehlberg method has two methods of orders 5 and 4; it is sometimes dubbed RKF45. Its extended Butcher Tableau is:
 * $$\begin{array}{r|ccccc}

0 & & & & & \\ 1 / 4 & 1 / 4 & & & \\ 3 / 8 & 3 / 32 & 9 / 32 & & \\ 12 / 13 & 1932 / 2197 & -7200 / 2197 & 7296 / 2197 & \\ 1 & 439 / 216 & -8 & 3680 / 513 & -845 / 4104 & \\ 1 / 2 & -8 / 27 & 2 & -3544 / 2565 & 1859 / 4104 & -11 / 40 \\ \hline & 16 / 135 & 0 & 6656 / 12825 & 28561 / 56430 & -9 / 50 & 2 / 55 \\ & 25 / 216 & 0 & 1408 / 2565 & 2197 / 4104 & -1 / 5 & 0 \end{array}$$

The first row of b coefficients gives the fifth-order accurate solution, and the second row has order four. The coefficients here allow for an adaptive stepsize to be determined automatically.

Cash-Karp
Cash and Karp have modified Fehlberg's original idea. The extended tableau for the Cash–Karp method is

The first row of b coefficients gives the fifth-order accurate solution, and the second row has order four.

Dormand–Prince
The extended tableau for the Dormand–Prince method is

The first row of b coefficients gives the fifth-order accurate solution, and the second row gives the fourth-order accurate solution.

Backward Euler
The backward Euler method is first order. Unconditionally stable and non-oscillatory for linear diffusion problems.



\begin{array}{c|c} 1 & 1 \\ \hline & 1 \\ \end{array} $$

Implicit midpoint
The implicit midpoint method is of second order. It is the simplest method in the class of collocation methods known as the Gauss-Legendre methods. It is a symplectic integrator.



\begin{array}{c|c} 1/2 & 1/2 \\ \hline & 1 \end{array} $$

Crank-Nicolson method
The Crank–Nicolson method corresponds to the implicit trapezoidal rule and is a second-order accurate and A-stable method.



\begin{array}{c|cc} 0  & 0   & 0   \\ 1   & 1/2 & 1/2 \\ \hline & 1/2 & 1/2 \\ \end{array} $$

Gauss–Legendre methods
These methods are based on the points of Gauss–Legendre quadrature. The Gauss–Legendre method of order four has Butcher tableau:



\begin{array}{c|cc} \frac{1}{2}-\frac{\sqrt3}{6} & \frac{1}{4} & \frac{1}{4}-\frac{\sqrt3}{6} \\ \frac{1}{2}+\frac{\sqrt3}{6} & \frac{1}{4}+\frac{\sqrt3}{6} &\frac{1}{4} \\ \hline & \frac{1}{2} & \frac{1}{2}\\ & \frac12+\frac{\sqrt3}{2} & \frac12-\frac{\sqrt3}{2} \\ \end{array} $$

The Gauss–Legendre method of order six has Butcher tableau:



\begin{array}{c|ccc} \frac{1}{2} - \frac{\sqrt{15}}{10}  & \frac{5}{36} &  \frac{2}{9}- \frac{\sqrt{15}}{15} & \frac{5}{36} - \frac{\sqrt{15}}{30} \\ \frac{1}{2} & \frac{5}{36} + \frac{\sqrt{15}}{24} & \frac{2}{9} & \frac{5}{36} - \frac{\sqrt{15}}{24}\\ \frac{1}{2} + \frac{\sqrt{15}}{10} & \frac{5}{36} + \frac{\sqrt{15}}{30} & \frac{2}{9} + \frac{\sqrt{15}}{15} & \frac{5}{36} \\ \hline & \frac{5}{18} & \frac{4}{9} & \frac{5}{18} \\ & -\frac56 & \frac83 & -\frac56 \end{array} $$

Diagonally Implicit Runge–Kutta methods
Diagonally Implicit Runge–Kutta (DIRK) formulae have been widely used for the numerical solution of stiff initial value problems; the advantage of this approach is that here the solution may be found sequentially as opposed to simultaneously.

The simplest method from this class is the order 2 implicit midpoint method.

Kraaijevanger and Spijker's two-stage Diagonally Implicit Runge–Kutta method:



\begin{array}{c|cc} 1/2 &  1/2   &  0    \\ 3/2  &  -1/2  &  2    \\ \hline & -1/2  &  3/2  \\ \end{array} $$

Qin and Zhang's two-stage, 2nd order, symplectic Diagonally Implicit Runge–Kutta method:



\begin{array}{c|cc} 1/4 &  1/4  &  0    \\ 3/4  &  1/2  &  1/4  \\ \hline & 1/2  &  1/2  \\ \end{array} $$

Pareschi and Russo's two-stage 2nd order Diagonally Implicit Runge–Kutta method:



\begin{array}{c|cc} x     &  x           &  0  \\ 1 - x &  1 - 2x      &  x  \\ \hline & \frac{1}{2} & \frac{1}{2}\\ \end{array} $$

This Diagonally Implicit Runge–Kutta method is A-stable if and only if $x \ge \frac{1}{4}$. Moreover, this method is L-stable if and only if $$x$$ equals one of the roots of the polynomial $x^2 - 2x + \frac{1}{2}$, i.e. if $x = 1 \pm \frac{\sqrt2}{2}$. Qin and Zhang's Diagonally Implicit Runge–Kutta method corresponds to Pareschi and Russo's Diagonally Implicit Runge–Kutta method with $$x = 1/4$$.

Two-stage 2nd order Diagonally Implicit Runge–Kutta method:



\begin{array}{c|cc} x  &  x           &  0  \\ 1  &  1 - x       &  x  \\ \hline & 1 - x & x\\ \end{array} $$

Again, this Diagonally Implicit Runge–Kutta method is A-stable if and only if $x \ge \frac{1}{4}$. As the previous method, this method is again L-stable if and only if $$x$$ equals one of the roots of the polynomial $x^2 - 2x + \frac{1}{2}$, i.e. if $x = 1 \pm \frac{\sqrt2}{2}$. This condition is also necessary for 2nd order accuracy.

Crouzeix's two-stage, 3rd order Diagonally Implicit Runge–Kutta method:



\begin{array}{c|cc} \frac{1}{2}+\frac{\sqrt3}{6} &  \frac{1}{2}+\frac{\sqrt3}{6}  &  0                             \\ \frac{1}{2}-\frac{\sqrt3}{6} &  -\frac{\sqrt3}{3}             &  \frac{1}{2}+\frac{\sqrt3}{6}  \\ \hline & \frac{1}{2}                   & \frac{1}{2}\\ \end{array} $$

Crouzeix's three-stage, 4th order Diagonally Implicit Runge–Kutta method:



\begin{array}{c|ccc} \frac{1+\alpha}{2} & \frac{1+\alpha}{2} & 0                 & 0  \\ \frac{1}{2}       & -\frac{\alpha}{2}  & \frac{1+\alpha}{2} & 0  \\ \frac{1-\alpha}{2} & 1+\alpha          & -(1+2\,\alpha)     & \frac{1+\alpha}{2} \\\hline & \frac{1}{6\alpha^2} & 1 - \frac{1}{3\alpha^2} & \frac{1}{6\alpha^2}\\ \end{array} $$ with $\alpha = \frac{2}{\sqrt3}\cos{\frac{\pi}{18}}$.

Three-stage, 3rd order, L-stable Diagonally Implicit Runge–Kutta method:



\begin{array}{c|ccc} x             &  x               &  0              &  0   \\ \frac{1+x}{2} &  \frac{1-x}{2}   &  x              &  0   \\ 1             &  -3x^2/2+4x-1/4  &  3x^2/2-5x+5/4  &  x   \\ \hline & -3x^2/2+4x-1/4  &  3x^2/2-5x+5/4  &  x   \\ \end{array} $$

with $$x = 0.4358665215$$

Nørsett's three-stage, 4th order Diagonally Implicit Runge–Kutta method has the following Butcher tableau:



\begin{array}{c|ccc} x   &  x      &  0     &  0   \\ 1/2 &  1/2-x  &  x     &  0   \\ 1-x &  2x     &  1-4x  &  x   \\ \hline & \frac{1}{6(1-2x)^2} & \frac{3(1-2x)^2 - 1}{3(1-2x)^2} & \frac{1}{6(1-2x)^2} \\ \end{array} $$

with $$x$$ one of the three roots of the cubic equation $$x^3 -3x^2/2 + x/2 - 1/24 = 0$$. The three roots of this cubic equation are approximately $$x_1 = 1.06858$$, $$x_2 = 0.30254$$, and $$x_3 = 0.12889$$. The root $$x_1$$ gives the best stability properties for initial value problems.

Four-stage, 3rd order, L-stable Diagonally Implicit Runge–Kutta method



\begin{array}{c|cccc} 1/2 &  1/2   &  0     &  0    &  0    \\ 2/3  &  1/6   &  1/2   &  0    &  0    \\ 1/2  &  -1/2  &  1/2   &  1/2  &  0    \\ 1    &  3/2   &  -3/2  &  1/2  &  1/2  \\ \hline & 3/2   &  -3/2  &  1/2  &  1/2  \\ \end{array} $$

Lobatto methods
There are three main families of Lobatto methods, called IIIA, IIIB and IIIC (in classical mathematical literature, the symbols I and II are reserved for two types of Radau methods). These are named after Rehuel Lobatto as a reference to the Lobatto quadrature rule, but were introduced by Byron L. Ehle in his thesis. All are implicit methods, have order 2s &minus; 2 and they all have c1 = 0 and cs = 1. Unlike any explicit method, it's possible for these methods to have the order greater than the number of stages. Lobatto lived before the classic fourth-order method was popularized by Runge and Kutta.

Lobatto IIIA methods
The Lobatto IIIA methods are collocation methods. The second-order method is known as the trapezoidal rule:



\begin{array}{c|cc} 0  & 0   & 0  \\ 1   & 1/2 & 1/2\\ \hline & 1/2 & 1/2\\ & 1 & 0 \\ \end{array} $$

The fourth-order method is given by



\begin{array}{c|ccc} 0  & 0   & 0   & 0    \\ 1/2 & 5/24& 1/3 & -1/24\\ 1   & 1/6 & 2/3 & 1/6  \\ \hline & 1/6 & 2/3 & 1/6 \\ & -\frac12 & 2 & -\frac12 \\ \end{array} $$

These methods are A-stable, but not L-stable and B-stable.

Lobatto IIIB methods
The Lobatto IIIB methods are not collocation methods, but they can be viewed as discontinuous collocation methods. The second-order method is given by



\begin{array}{c|cc} 0  & 1/2 & 0  \\ 1   & 1/2 & 0  \\

\hline & 1/2 & 1/2\\ & 1 & 0 \\

\end{array} $$

The fourth-order method is given by



\begin{array}{c|ccc} 0  & 1/6 & -1/6& 0    \\ 1/2 & 1/6 & 1/3 & 0    \\ 1   & 1/6 & 5/6 & 0    \\ \hline & 1/6 & 2/3 & 1/6 \\ & -\frac12 & 2 & -\frac12 \\ \end{array} $$

Lobatto IIIB methods are A-stable, but not L-stable and B-stable.

Lobatto IIIC methods
The Lobatto IIIC methods also are discontinuous collocation methods. The second-order method is given by



\begin{array}{c|cc} 0  & 1/2 & -1/2\\ 1   & 1/2 & 1/2 \\ \hline & 1/2 & 1/2 \\ & 1 & 0 \\ \end{array} $$

The fourth-order method is given by



\begin{array}{c|ccc} 0  & 1/6 & -1/3& 1/6  \\ 1/2 & 1/6 & 5/12& -1/12\\ 1   & 1/6 & 2/3 & 1/6  \\ \hline & 1/6 & 2/3 & 1/6 \\ & -\frac12 & 2 & -\frac12 \\ \end{array} $$

They are L-stable. They are also algebraically stable and thus B-stable, that makes them suitable for stiff problems.

Lobatto IIIC* methods
The Lobatto IIIC* methods are also known as Lobatto III methods (Butcher, 2008), Butcher's Lobatto methods (Hairer et al., 1993), and Lobatto IIIC methods (Sun, 2000) in the literature. The second-order method is given by



\begin{array}{c|cc} 0  & 0 & 0\\ 1   & 1 & 0 \\ \hline & 1/2 & 1/2 \\ \end{array} $$

Butcher's three-stage, fourth-order method is given by



\begin{array}{c|ccc} 0  & 0 & 0 & 0  \\ 1/2 & 1/4 & 1/4 & 0\\ 1   & 0 & 1 & 0  \\ \hline & 1/6 & 2/3 & 1/6 \\ \end{array} $$

These methods are not A-stable, B-stable or L-stable. The Lobatto IIIC* method for $$s = 2$$ is sometimes called the explicit trapezoidal rule.

Generalized Lobatto methods
One can consider a very general family of methods with three real parameters $$ (\alpha_{A},\alpha_{B},\alpha_{C}) $$ by considering Lobatto coefficients of the form
 * $$a_{i,j}(\alpha_{A},\alpha_{B},\alpha_{C}) = \alpha_{A}a_{i,j}^A + \alpha_{B}a_{i,j}^B + \alpha_{C}a_{i,j}^C + \alpha_{C*}a_{i,j}^{C*} $$,

where
 * $$\alpha_{C*} = 1 - \alpha_{A} - \alpha_{B} - \alpha_{C}$$.

For example, Lobatto IIID family introduced in (Nørsett and Wanner, 1981), also called Lobatto IIINW, are given by

\begin{array}{c|cc} 0  & 1/2 & 1/2\\ 1   & -1/2 & 1/2 \\ \hline & 1/2 & 1/2 \\ \end{array} $$

and



\begin{array}{c|ccc} 0  & 1/6 & 0 & -1/6  \\ 1/2 & 1/12 & 5/12 & 0\\ 1   & 1/2 & 1/3 & 1/6  \\ \hline & 1/6 & 2/3 & 1/6 \\ \end{array} $$

These methods correspond to $$\alpha_{A} = 2$$, $$\alpha_{B} = 2$$, $$\alpha_{C} = -1$$, and $$\alpha_{C*} = -2$$. The methods are L-stable. They are algebraically stable and thus B-stable.

Radau methods
Radau methods are fully implicit methods (matrix A of such methods can have any structure). Radau methods attain order 2s &minus; 1 for s stages. Radau methods are A-stable, but expensive to implement. Also they can suffer from order reduction. The first order Radau method is similar to backward Euler method.

Radau IA methods
The third-order method is given by



\begin{array}{c|cc} 0  & 1/4  & -1/4  \\ 2/3 & 1/4  &  5/12 \\ \hline & 1/4 & 3/4 \\ \end{array} $$

The fifth-order method is given by



\begin{array}{c|ccc} 0  & \frac{1}{9} & \frac{-1 - \sqrt{6}}{18} & \frac{-1 + \sqrt{6}}{18} \\ \frac{3}{5} - \frac{\sqrt{6}}{10} & \frac{1}{9} & \frac{11}{45} + \frac{7\sqrt{6}}{360} & \frac{11}{45} - \frac{43\sqrt{6}}{360}\\ \frac{3}{5} + \frac{\sqrt{6}}{10} & \frac{1}{9} & \frac{11}{45} + \frac{43\sqrt{6}}{360} & \frac{11}{45} - \frac{7\sqrt{6}}{360} \\ \hline & \frac{1}{9} & \frac{4}{9} + \frac{\sqrt{6}}{36} & \frac{4}{9} - \frac{\sqrt{6}}{36} \\ \end{array} $$

Radau IIA methods
The ci of this method are zeros of
 * $$\frac{d^{s-1}}{dx^{s-1}}(x^{s-1}(x-1)^s)$$.

The third-order method is given by



\begin{array}{c|cc} 1/3 & 5/12 & -1/12\\ 1  & 3/4  &  1/4 \\ \hline & 3/4 & 1/4 \\ \end{array} $$

The fifth-order method is given by



\begin{array}{c|ccc} \frac{2}{5} - \frac{\sqrt{6}}{10}  & \frac{11}{45} - \frac{7\sqrt{6}}{360} & \frac{37}{225} - \frac{169\sqrt{6}}{1800} & -\frac{2}{225} + \frac{\sqrt{6}}{75} \\ \frac{2}{5} + \frac{\sqrt{6}}{10} & \frac{37}{225} + \frac{169\sqrt{6}}{1800} & \frac{11}{45} + \frac{7\sqrt{6}}{360} & -\frac{2}{225} - \frac{\sqrt{6}}{75}\\ 1                  & \frac{4}{9} - \frac{\sqrt{6}}{36} & \frac{4}{9} + \frac{\sqrt{6}}{36} & \frac{1}{9}  \\ \hline & \frac{4}{9} - \frac{\sqrt{6}}{36} & \frac{4}{9} + \frac{\sqrt{6}}{36} & \frac{1}{9} \\ \end{array} $$