# Atlas of general linear methods with inherent Runge-Kutta stability

We have generated the methods with a matlab function:
```irks(lambda,Cp+1,[c1,c2,...,cs],[betap-1,...,beta1],T)
```
Where
• lambda is the value in the diagonal of A,
• Cp+1 is the leading error constant (for explicit methods with lambda=0 only),
• c are the nodes,
• beta is the last column of X (without the first two elements),
• T is the matrix for which ``` inv(T)*V(3:end,3:end)*T``` is strictly upper triangular.

## Explicit methods

### Order 2

(big= 1.5) ``` irks(0,1/42,[1/3,2/3,1],[0.5],1) ``` (big=0.780777) ``` irks(0,1/42,[1/2,1,1],[0.640388],1) ```

### Order 3

(big=1.06962) ``` irks(0,1/312,[1/4,1/2,3/4,1],[0.2679 0.5731],[1 0; 0.2852 1]) ``` (big=1.15718) ``` irks(0,1/312,[1/3,2/3,1,1],[0.4494 0.6721],[1 0; 0.4742 1]) ```

### Order 4

(big=1.81176) ``` irks(0,1/3000,(1:5)/5,[0.0850, 0.3080, 0.9092], ... [1 0 0; 0.1503 0.54 1; 0.5476 1 0]) ``` (big=1.53777) ``` irks(0,1/3000,[(1:4),4]/4,[0.1700 0.0279 0.7497], ... [-0.3242 0.4359 1.0000 1.0000 0 0 0.8764 1.0000 0]) ```

### Order 5

(big=14.8117) ``` irks(0,1/52560,[1:6]/6,... [0.0074 0.0674 0.3361 0.8734], ... [ -0.2290 1.0000 0 0 0.3423 0.1148 0.5911 1.0000 0.7792 0.8617 1.0000 0 1.0000 0 0 0]) ``` (big=6.03705) ``` irks(0,1/52560,[(1:5),5]/5,... [-0.0695 0.1387 0.0194 0.5997],... [ 0.2994 -0.0577 -0.2206 1.0000 -0.1518 -0.2921 1.0000 0 1.0000 0 0 0 0.5493 1.0000 0 0]) ```

### Order 6

(big=325.94) ``` irks(0,1/640080,(1:7)/7,... [-0.0150 0.0256 0.0356 0.1423 0.5063],... [ 0.1303 -0.8593 0.3041 -0.3410 0.1896 -0.3520 -0.1616 0.1979 0.7368 0.5177 0.7867 -0.2037 -0.3170 0.4804 -0.0912 -0.1179 -0.1374 0.5089 0.3229 -0.7772 0.4758 0.4186 0.7134 -0.0759 0.2894]) ```

## Implicit methods

### Order 2

(big=1.03752) ``` irks(4/9,0,[1:3]/3,[0.33852],1) ``` (big=0.885561) ``` irks(4/9,0,[1:2,2]/2,[0.27861],1) ```

### Order 3

(big=1.0116) ``` irks(9/40,0,[1:4]/4,[0.3312 1.0050],[0.49541 1;1 0]) ``` (big=0.893763) ``` irks(9/40,0,[1:3,3]/3, [0.2463,0.5101],[ 1 0; 0.13519 1]) ```

### Order 4

(big=2.22612) ``` irks(3/11,0,[1:5]/5, [0.1715 -0.1238 0.6617],... [ -0.0812 0.4079 1.0000 1.0000 0 0 0.8270 1.0000 0]) ``` (big=1.97884) ``` irks(3/11,0,[1:4,4]/4,[0.0234 0.1864 0.7030],... [-0.0829 0.2541 1.0000 -0.1968 1.0000 0 1.0000 0 0]) ```

### Order 5

(big=15.3351) ``` irks(1/3,0,[1:6]/6,... [-0.0489 0.4228 -0.8814 0.9021],... [-0.3474 -0.6617 0.6294 0.2129 0.0044 -0.4256 -0.1427 -0.8936 -0.8267 0.4821 0.1371 -0.2557 -0.4426 -0.3855 -0.7514 0.3014]) ``` (big=9.45014) ``` irks(1/3,0,[1:5,5]/5,... [-0.0575 0.0472 -0.1525 -0.1387],... [ 0.0624 0.1208 0.9906 -0.0161 0.7625 0.6344 -0.1251 0.0220 -0.2316 0.3029 -0.0374 -0.9237 -0.6009 0.7008 -0.0414 0.3822]) ```

## Other methods

http://www.math.auckland.ac.nz/~hpod/atlas
last change: March 2006