1 Engineering Computation Part 4

2 Enrique Castillo University of Cantabria An algorithm that permits solving many problems in Algebra. Applications to Engineering by

3 The Jubete Algorithm 1.Obtain the orthogonal subespace to a given subspace and its complement. 2.Obtain the inverse of a matrix 3.Update the inverse of a matrix after changing one row or column. 4.Obtain the determinant of a matrix. 5.Update the determinant of a matrix after changing one of its rows or columns. 6.Obtain the rank of a matrix. 7.Determine if a vector belongs to a given subspace. 8.Obtain the intersection of two given subspaces. 9.Solve an homogeneous linear system of equations. 10.Solve a complete linear system of equations. 11.Study the compatibility of a linear system of equations.

4 The Jubete Algorithm

5 Inverse and determinant of a matrix

6 Simultaneous inverses of submatrices of a matrix

7 Inverses after modifying rows of a matrix Inverse updating

8 Inverses after modifying rows of a matrix Inverse updating

9 Orthogonal subspaces and complements

10 Orthogonal subspaces and complements

11 Rank of a matrix In addition it gives the coefficients of the linear combination

12 Vector in a subspace

13 Intersection of two subspaces

14 Solving a homogeneous linear system of equations

15 Solving a homogeneous linear system of equations. Example

16 Solving a complete linear system of equations

17 Solving a complete linear system of equations. Example

18 Solving several complete linear systems of equations.

19 Compatibility

20 Compatibility. Example

21 Connection model-reality Mathematics is a fundamental tool in Engineering

22 Connection model-reality The student must know the connection between engineering and mathematical concepts. The student must know how to update solutions.

23 Connection model-reality The student must know when one element is redundant from the engineering and the mathematical points of view, and of its implications in the service reliability, together with the number of degrees of freedom of the general solution.

24 Connection model-reality The student must relate the topology of the network with the number of unknowns and mathematical equations defining the engineering problem. The student must know how to state the problem in different forms (mathematical and engineering).

25 Connection model-reality The student must know how to state problem with inequalities. The student must know how to state alternative hypotheses.

26 The water supply problem The student must identify the problem unknowns and equations.

27 The water supply problem Number of equations Number of unknowns Numbering the nodes Constraints. ¿What the data are? ¿What the equations are?

28 Statement of the problem The student must know how the state the problem as a system of equations and specially in matrix form

29 Statement of the problem The student must know how to number the nodes and distinguish between correct and incorrect numberings.

30 Analysis of the solution Has the problem a solution? Is it unique?

31 Compatibility condition Flow entering = Flow leaving

32 Compatibility condition

33 Set of all solutions (without capacity limits) The student must know how to obtain the set of all possible solutions. There are an infinite number of solutions. (Affine space associated with a linear space of dimension 4).

34 Interpretation of the different basic solutions Particular solution. It can be replaced by any other one.

35 Interpretation of solutions Internal flow solution without inputs or outputs of fluid

36 Interpretation of solutions Internal flow solution without inputs or outputs of fluid

37 Interpretation of solutions Internal flow solution without inputs or outputs of fluid

38 Interpretation of solutions Internal flow solution without inputs or outputs of fluid

39 Statement of the problem The student must identify non-adequate models and identify lacking constraints.

40 Cones

41 Linear space as a cone

42 Cone and dual of a cone

43 Dual of a cone. Gamma algorithm

44 Dual of a cone. Gamma algorithm

45 Some problems solved by the gamma algorithm 1.Obtain the dual cone of a given cone 2.Obtain the minimum representation of a cone. 3.Obtain the facets of any given dimension (vertices, edges, faces, etc.) of a cone or polytope. 4.Determine of a given vector belongs to a given cone. 5.Check if two cones are identical. 6.Obtain the intersection of two cones. 7.Obtain the reciprocal image of a cone by a linear transformation. 8.Decide if a linear system of inequalities is compatible. 9.Solve an homogeneous system of linear inequalities. 10.Solve a complete system of linear inequalities.

46 Cone associated with a polytope

47 Facets and vertices of a polytope

48 Facets and vertices of a polytope

49 Facets and vertices of a polytope

50 Facets and vertices of a polytope

51 Facets and vertices of a polytope

52 Solving an homogeneous system of linear inequalities

53 Solving a complete system of linear inequalities

54 Solving a complete system of linear inequalities

55 Compatibility of systems of linear inequalities

56 Statement of the Problem. The student must identify non-adequate models and identify lacking constraints.

57 Compatibility conditions It is necessary to interpret them physically to see If they represent the adequate model.

58 Compatibility conditions

59 Set of all solutions (with capacity constraints) The set of all solutions allows answering many interesting questions from a mathematical and engineering point of view. The solution is a polytope Capacity of each link = 6

60 Set of all solutions (Search of overdimensioned links) Capacity of each link = 6 Capacity is not attained in all its solution components. The capacity could be limited to the maximum values attained in its solution components.

61 Set of all solutions (Search of links that cannot fail) The take the same sign (all positive or all negative) in all solution components. Capacity of each link = 6

62 Set of all solutions (Search of pairs of link that cannot fail simultaneously) This condition implies that all lambda values must be null. Capacity of each link = 6

63 Set of all solutions (Search of links with constant flow) They take identical values in all solution components. Capacity of each link = 6

64 Set of all solutions (Link 10 failed) In order link 10 to have no flow, the first four lambdas must be null. Capacity of each link = 6

65 Set of all solutions (Link 10 failed) Link 7 can fail because it has positive and negative components. ¿Can link 7 fail?

66 Set of all solutions (Links 7 and 10 failed) Link 4 fails if lambda 2 is null. ¿Can link 4 fail?

67 Set of all solutions (Links 4,7 and 10 failed) ¿Can any other link fail? No link can fail, because the solution is unique (bad engineering solution, because of the lack of flexibility).

68 Retention valves in links 2 and 15 It is the sum of an affine space of dimension 2 and a cone generated by two vectors.

69 Evaluation example

70 Evaluation example

71 Evaluation example

72 Evaluation example

73 BIBLIOGRAPHY

74 INTERNET With the collaboration of Elena Alvarez Sáiz the orthogonalization algorithm has been implemented in an application of computer aided instruction, which is accessible throughout INTERNET :

75 LABORATORY 1.Design a water supply system with two deposits and several nodes containing redundant links. 2.Determine the dimension of the linear space which appears in the general solution of the resulting system of equations. 3.Obtain the general solution of this system manually, based on the physical interpretation of the general solution. 4.State an optimization programming problem leading to a unique solution.