Theory Gauss_Jordan.Examples_Gauss_Jordan_Abstract
section‹Examples of computations over abstract matrices›
theory Examples_Gauss_Jordan_Abstract
imports
Determinants2
Inverse
System_Of_Equations
Code_Z2
"HOL-Library.Code_Target_Numeral"
begin
subsection‹Transforming a list of lists to an abstract matrix›
text‹Definitions to transform a matrix to a list of list and vice versa›
definition vec_to_list :: "'a^'n::{finite, enum} => 'a list"
where "vec_to_list A = map (($) A) (enum_class.enum::'n list)"
definition matrix_to_list_of_list :: "'a^'n::{finite, enum}^'m::{finite, enum} => 'a list list"
where "matrix_to_list_of_list A = map (vec_to_list) (map (($) A) (enum_class.enum::'m list))"
text‹This definition should be equivalent to ‹vector_def› (in suitable types)›
definition list_to_vec :: "'a list => 'a^'n::{finite, enum, mod_type}"
where "list_to_vec xs = vec_lambda (% i. xs ! (to_nat i))"
lemma [code abstract]: "vec_nth (list_to_vec xs) = (%i. xs ! (to_nat i))"
unfolding list_to_vec_def by fastforce
definition list_of_list_to_matrix :: "'a list list => 'a^'n::{finite, enum, mod_type}^'m::{finite, enum, mod_type}"
where "list_of_list_to_matrix xs = vec_lambda (%i. list_to_vec (xs ! (to_nat i)))"
lemma [code abstract]: "vec_nth (list_of_list_to_matrix xs) = (%i. list_to_vec (xs ! (to_nat i)))"
unfolding list_of_list_to_matrix_def by auto
subsection‹Examples›
text‹The following three lemmas are presented in both this file and in the
‹Examples_Gauss_Jordan_IArrays› one. They allow a more convenient printing of rational and
real numbers after evaluation. They have already been added to the repository version of Isabelle,
so after Isabelle2014 they should be removed from here.›
lemma [code_post]:
"int_of_integer (- 1) = - 1"
by simp
lemma [code_abbrev]:
"(of_rat (- 1) :: real) = - 1"
by simp
lemma [code_post]:
"(of_rat (- (1 / numeral k)) :: real) = - 1 / numeral k"
"(of_rat (- (numeral k / numeral l)) :: real) = - numeral k / numeral l"
by (simp_all add: of_rat_divide of_rat_minus)
subsubsection‹Ranks and dimensions›
text‹Examples on computing ranks, dimensions of row space, null space and col space and the Gauss Jordan algorithm›
value "matrix_to_list_of_list (Gauss_Jordan (list_of_list_to_matrix ([[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,0],[0,0,0,0,8,2]])::real^6^4))"
value "matrix_to_list_of_list (Gauss_Jordan (list_of_list_to_matrix ([[1,-2,1,-3,0],[3,-6,2,-7,0]])::rat^5^2))"
value "matrix_to_list_of_list (Gauss_Jordan (list_of_list_to_matrix ([[1,0,0,1,1],[1,0,1,1,1]])::bit^5^2))"
value "(reduced_row_echelon_form_upt_k (list_of_list_to_matrix ([[1,0,8],[0,1,9],[0,0,0]])::real^3^3)) 3"
value "matrix_to_list_of_list (Gauss_Jordan (list_of_list_to_matrix [[Complex 1 1,Complex 1 (- 1), Complex 0 0],[Complex 2 (- 1),Complex 1 3, Complex 7 3]]::complex^3^2))"
value "DIM(real^5)"
value "vec.dimension (TYPE(bit)) (TYPE(5))"
value "vec.dimension (TYPE(real)) (TYPE(2))"
value "DIM(real^5^4)"
value "row_rank (list_of_list_to_matrix [[1,0,0,7,5],[1,0,4,8,-1],[1,0,0,9,8],[1,2,3,6,5]]::real^5^4)"
value "vec.dim (row_space (list_of_list_to_matrix [[1,0,0,7,5],[1,0,4,8,-1],[1,0,0,9,8],[1,2,3,6,5]]::real^5^4))"
value "col_rank (list_of_list_to_matrix [[1,0,0,7,5],[1,0,4,8,-1],[1,0,0,9,8],[1,2,3,6,5]]::real^5^4)"
value "vec.dim (col_space (list_of_list_to_matrix [[1,0,0,7,5],[1,0,4,8,-1],[1,0,0,9,8],[1,2,3,6,5]]::real^5^4))"
value "rank (list_of_list_to_matrix [[1,0,0,7,5],[1,0,4,8,-1],[1,0,0,9,8],[1,2,3,6,5]]::real^5^4)"
value "vec.dim (null_space (list_of_list_to_matrix [[1,0,0,7,5],[1,0,4,8,-1],[1,0,0,9,8],[1,2,3,6,5]]::real^5^4))"
value "rank (list_of_list_to_matrix [[Complex 1 1,Complex 1 (- 1), Complex 0 0],[Complex 2 (- 1),Complex 1 3, Complex 7 3]]::complex^3^2)"
subsubsection‹Inverse of a matrix›
text‹Examples on computing the inverse of matrices›
value "let A=(list_of_list_to_matrix [[1,1,2,4,5,9,8],[3,0,8,4,5,0,8],[3,2,0,4,5,9,8], [3,2,8,0,5,9,8] ,[3,2,8,4,0,9,8] ,[3,2,8,4,5,0,8], [3,2,8,4,5,9,0]]::real^7^7)
in matrix_to_list_of_list (P_Gauss_Jordan A)"
value "let A=(list_of_list_to_matrix [[1,1,2,4,5,9,8],[3,0,8,4,5,0,8],[3,2,0,4,5,9,8], [3,2,8,0,5,9,8] ,[3,2,8,4,0,9,8] ,[3,2,8,4,5,0,8], [3,2,8,4,5,9,0]]::real^7^7) in
matrix_to_list_of_list (A ** (P_Gauss_Jordan A))"
value "let A=(list_of_list_to_matrix [[1,1,2,4,5,9,8],[3,0,8,4,5,0,8],[3,2,0,4,5,9,8], [3,2,8,0,5,9,8] ,[3,2,8,4,0,9,8] ,[3,2,8,4,5,0,8], [3,2,8,4,5,9,0]]::real^7^7)
in (inverse_matrix A)"
value "let A=(list_of_list_to_matrix [[1,1,2,4,5,9,8],[3,0,8,4,5,0,8],[3,2,0,4,5,9,8], [3,2,8,0,5,9,8] ,[3,2,8,4,0,9,8] ,[3,2,8,4,5,0,8], [3,2,8,4,5,9,0]]::real^7^7)
in matrix_to_list_of_list (the (inverse_matrix A))"
value "let A=(list_of_list_to_matrix [[1,1,1,1,1,1,1],[2,2,2,2,2,2,2],[3,2,0,4,5,9,8], [3,2,8,0,5,9,8] ,[3,2,8,4,0,9,8] ,[3,2,8,4,5,0,8], [3,2,8,4,5,9,0]]::real^7^7)
in (inverse_matrix A)"
value "let A=(list_of_list_to_matrix [[Complex 1 1,Complex 1 (- 1), Complex 0 0],[Complex 1 1,Complex 1 (- 1), Complex 8 0],[Complex 2 (- 1),Complex 1 3, Complex 7 3]]::complex^3^3)
in matrix_to_list_of_list (the (inverse_matrix A))"
subsubsection‹Determinant of a matrix›
text‹Examples on computing determinants of matrices›
value "(let A = list_of_list_to_matrix[[1,2,7,8,9],[3,4,12,10,7],[-5,4,8,7,4],[0,1,2,4,8],[9,8,7,13,11]]::real^5^5 in det A)"
value "det (list_of_list_to_matrix ([[1,0,0],[0,1,0],[0,0,1]])::real^3^3)"
value "det (list_of_list_to_matrix ([[1,8,9,1,47],[7,2,2,5,9],[3,2,7,7,4],[9,8,7,5,1],[1,2,6,4,5]])::rat^5^5)"
subsubsection‹Bases of the fundamental subspaces›
text‹Examples on computing basis for null space, row space, column space and left null space›
value "let A = (list_of_list_to_matrix ([[1,3,-2,0,2,0],[2,6,-5,-2,4,-3],[0,0,5,10,0,15],[2,6,0,8,4,18]])::real^6^4)
in vec_to_list` (basis_null_space A)"
value "let A = (list_of_list_to_matrix ([[3,4,0,7],[1,-5,2,-2],[-1,4,0,3],[1,-1,2,2]])::real^4^4)
in vec_to_list` (basis_null_space A)"
value "let A = (list_of_list_to_matrix ([[1,3,-2,0,2,0],[2,6,-5,-2,4,-3],[0,0,5,10,0,15],[2,6,0,8,4,18]])::real^6^4)
in vec_to_list` (basis_row_space A)"
value "let A = (list_of_list_to_matrix ([[3,4,0,7],[1,-5,2,-2],[-1,4,0,3],[1,-1,2,2]])::real^4^4)
in vec_to_list` (basis_row_space A)"
value "let A = (list_of_list_to_matrix ([[1,3,-2,0,2,0],[2,6,-5,-2,4,-3],[0,0,5,10,0,15],[2,6,0,8,4,18]])::real^6^4)
in vec_to_list` (basis_col_space A)"
value "let A = (list_of_list_to_matrix ([[3,4,0,7],[1,-5,2,-2],[-1,4,0,3],[1,-1,2,2]])::real^4^4)
in vec_to_list` (basis_col_space A)"
value "let A = (list_of_list_to_matrix ([[1,3,-2,0,2,0],[2,6,-5,-2,4,-3],[0,0,5,10,0,15],[2,6,0,8,4,18]])::real^6^4)
in vec_to_list` (basis_left_null_space A)"
value "let A = (list_of_list_to_matrix ([[3,4,0,7],[1,-5,2,-2],[-1,4,0,3],[1,-1,2,2]])::real^4^4)
in vec_to_list` (basis_left_null_space A)"
subsubsection‹Consistency and inconsistency›
text‹Examples on checking the consistency/inconsistency of a system of equations›
value "independent_and_consistent (list_of_list_to_matrix ([[1,0,0],[0,1,0],[0,0,1],[0,0,0],[0,0,0]])::real^3^5) (list_to_vec([2,3,4,0,0])::real^5)"
value "consistent (list_of_list_to_matrix ([[1,0,0],[0,1,0],[0,0,1],[0,0,0],[0,0,0]])::real^3^5) (list_to_vec([2,3,4,0,0])::real^5)"
value "inconsistent (list_of_list_to_matrix ([[1,0,0],[0,1,0],[3,0,1],[0,7,0],[0,0,9]])::real^3^5) (list_to_vec([2,0,4,0,0])::real^5)"
value "dependent_and_consistent (list_of_list_to_matrix ([[1,0,0],[0,1,0]])::real^3^2) (list_to_vec([3,4])::real^2)"
value "independent_and_consistent (mat 1::real^3^3) (list_to_vec([3,4,5])::real^3)"
subsubsection‹Solving systems of linear equations›
text‹Examples on solving linear systems.›
definition print_result_solve
where "print_result_solve A = (if A = None then None else Some (vec_to_list (fst (the A)), vec_to_list` (snd (the A))))"
value "let A = (list_of_list_to_matrix [[4,5,8],[9,8,7],[4,6,1]]::real^3^3);
b=(list_to_vec [4,5,8]::real^3)
in (print_result_solve (solve A b))"
value "let A = (list_of_list_to_matrix [[0,0,0],[0,0,0],[0,0,1]]::real^3^3);
b=(list_to_vec [4,5,0]::real^3)
in (print_result_solve (solve A b))"
value "let A = (list_of_list_to_matrix [[3,2,5,2,7],[6,4,7,4,5],[3,2,-1,2,-11],[6,4,1,4,-13]]::real^5^4);
b=(list_to_vec [0,0,0,0]::real^4)
in (print_result_solve (solve A b))"
value "let A = (list_of_list_to_matrix [[1,2,1],[-2,-3,-1],[2,4,2]]::real^3^3);
b=(list_to_vec [-2,1,-4]::real^3)
in (print_result_solve (solve A b))"
value "let A = (list_of_list_to_matrix [[1,1,-4,10],[3,-2,-2,6]]::real^4^2);
b=(list_to_vec [24,15]::real^2)
in (print_result_solve (solve A b))"
end