**Input data**: , . |

**Result**: . |

(1) Check that matrix satisfies (5). |

(2) Check that matrices are singular, and check that the block matrix |

is regular. |

(3) Determine a number so that the matrix pencil is regular. |

(4) Determine matrices and defined by (10). |

(5) Determine matrices and defined by (11). |

(6) Consider the following cases: |

* *(i) *Case 1*. Condition (13) holds, that is, matrices and have a common eigenvector associated |

with eigenvalues and . In this case continue with *step *(*7*). |

* *(ii) *Case 2*. Condition (13) does not hold. In this case the algorithm stops because it is not possible to |

find the solution of (1)–(4) for the given data. |

(7) Determine , , and vector verifying |

such that: |

* *(i) Conditions (53) hold, that is: |

* *1.1: is an invariant subspace respect matrix . |

* *1.2: , . |

(ii) Conditions (14) hold, that is: |

* *1.3: , . |

(iii) The vectorial function satisfies (42), that is: |

* *1.4: . |

* *1.5: . |

* *1.6: . |

*If these conditions are not satisfied, return to step (6) of Algorithm 1 discarding the values * |

* * *taken for ** and **. * |

(8) Determine the positive solutions of (16) and determine defined by (27). |

(9) Determine degree of minimal polynomial of matrix . |

(10) Building block matrix defined by (31). |

(11) Determine so that rank . |

(12) Include the eigenvalue if . |

(13) Determine given by (44). |

(14) Determine vectors defined by (47). |

(15) Determine functions defined by (34). |

(16) Determine the series solution of problem (1)–(4) defined by (49). |