June 2018 Update
This commit is contained in:
Ingy döt Net
parent
ba8067c3b7
commit
22f33d4004
5278 changed files with 84726 additions and 14379 deletions
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@ -1,4 +1,7 @@
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{{omit from|GUISS}}
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;Task:
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Show how to compute the '''reduced row echelon form'''
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(a.k.a. '''row canonical form''') of a matrix.
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@ -37,10 +40,15 @@ Built-in functions or this pseudocode (from Wikipedia) may be used:
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'''end function'''
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For testing purposes, the RREF of this matrix:
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<pre>1 2 -1 -4
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2 3 -1 -11
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-2 0 -3 22</pre>
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<pre>
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1 2 -1 -4
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2 3 -1 -11
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-2 0 -3 22
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</pre>
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is:
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<pre>1 0 0 -8
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0 1 0 1
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0 0 1 -2</pre>
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<pre>
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1 0 0 -8
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0 1 0 1
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0 0 1 -2
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</pre>
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<br><br>
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@ -28,7 +28,7 @@ rref(list l, integer rows, integer columns)
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u = l[i];
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l_spin(l, i, r);
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l.spin(i, r);
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e = u[lead];
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if (e) {
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for (j, f in u) {
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@ -40,7 +40,7 @@ rref(list l, integer rows, integer columns)
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if (i != r) {
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e = v[lead];
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for (j, f in v) {
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v[j] = f - l_q_integer(u, j) * e;
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v[j] = f - u[j] * e;
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}
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}
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}
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@ -52,14 +52,12 @@ rref(list l, integer rows, integer columns)
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}
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void
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display_2(list l, integer rows, integer columns)
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display_2(list l)
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{
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integer i;
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list u;
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i = 0;
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for (i, u in l) {
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l_ucall(u, o_winteger, -1, 4);
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for (, u in l) {
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u.ucall(o_winteger, -1, 4);
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o_byte('\n');
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}
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}
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@ -69,11 +67,11 @@ main(void)
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{
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list l;
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l = l_effect(l_effect(1, 2, -1, -4),
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l_effect(2, 3, -1, -11),
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l_effect(-2, 0, -3, 22));
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l = list(list(1, 2, -1, -4),
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list(2, 3, -1, -11),
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list(-2, 0, -3, 22));
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rref(l, 3, 4);
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display_2(l, 3, 4);
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display_2(l);
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return 0;
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}
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@ -1,10 +1,36 @@
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]mymatrix=: _4]\ 1 2 _1 _4 2 3 _1 _11 _2 0 _3 22
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1 2 _1 _4
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2 3 _1 _11
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_2 0 _3 22
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NB.*pivot v Pivot at row, column
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NB. form: (row,col) pivot M
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pivot=: dyad define
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'r c'=. x
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col=. c{"1 y
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y - (col - r = i.#y) */ (r{y) % r{col
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)
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require 'math/misc/linear'
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gauss_jordan mymatrix
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1 0 0 _8
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0 1 0 1
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0 0 1 _2
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NB.*gauss_jordan v Gauss-Jordan elimination (full pivoting)
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NB. y is: matrix
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NB. x is: optional minimum tolerance, default 1e_15.
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NB. If a column below the current pivot has numbers of magnitude all
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NB. less then x, it is treated as all zeros.
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gauss_jordan=: verb define
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1e_15 gauss_jordan y
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:
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mtx=. y
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'r c'=. $mtx
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rows=. i.r
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i=. j=. 0
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max=. i.>./
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while. (i<r) *. j<c do.
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k=. max col=. | i}. j{"1 mtx
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if. 0 < x-k{col do. NB. if all col < tol, set to 0:
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mtx=. 0 (<(i}.rows);j) } mtx
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else. NB. otherwise sort and pivot:
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if. k do.
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mtx=. (<i,i+k) C. mtx
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end.
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mtx=. (i,j) pivot mtx
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i=. >:i
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end.
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j=. >:j
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end.
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mtx
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)
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@ -1,16 +1,10 @@
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gauss_jordan 2 0 _1 0 0,1 0 0 _1 0,3 0 0 _2 _1,0 1 0 0 _2,:0 1 _1 0 0
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1 0 0 0 _1
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0 1 0 0 _2
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0 0 1 0 _2
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0 0 0 1 _1
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0 0 0 0 0
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gauss_jordan 1 2 3 4 3 1,2 4 6 2 6 2,3 6 18 9 9 _6,4 8 12 10 12 4,:5 10 24 11 15 _4
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1 2 0 0 3 0
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0 0 1 0 0 0
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0 0 0 1 0 0
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0 0 0 0 0 1
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0 0 0 0 0 0
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gauss_jordan 0 1,1 2,:0 5
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1 0
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0 1
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0 0
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require 'math/misc/linear'
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]A=: 1 2 _1 _4 , 2 3 _1 _11 ,: _2 0 _3 22
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1 2 _1 _4
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2 3 _1 _11
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_2 0 _3 22
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gauss_jordan A
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1 0 0 _8
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0 1 0 1
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0 0 1 _2
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@ -1,37 +1,16 @@
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mat=:".;._2]0 :0
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1 0 0 0 0 0 1 0 0 0 0 _1 0 0 0 0 0 0
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1 0 0 0 0 0 0 1 0 0 0 0 _1 0 0 0 0 0
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1 0 0 0 0 0 0 0 1 0 0 0 0 _1 0 0 0 0
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0 1 0 0 0 0 1 0 0 0 0 0 0 0 _1 0 0 0
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0 1 0 0 0 0 0 0 1 0 0 _1 0 0 0 0 0 0
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0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 _1 0
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0 0 1 0 0 0 1 0 0 0 0 0 _1 0 0 0 0 0
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0 0 1 0 0 0 0 0 0 1 0 0 0 0 _1 0 0 0
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0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 _1 0 0
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0 0 0 1 0 0 0 0 0 1 0 0 _1 0 0 0 0 0
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0 0 0 0 1 0 0 1 0 0 0 0 0 _1 0 0 0 0
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0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 _1 0
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0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 _1 0 0
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0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
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0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1
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0 0 0 0 0 1 0 0 0 0 1 0 0 0 _1 0 0 0
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)
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gauss_jordan mat
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1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.435897
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0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.307692
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0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.512821
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0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0.717949
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0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0.487179
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0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
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0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0.205128
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0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0.282051
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0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0.333333
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0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
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0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0.512821
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0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0.641026
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0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0.717949
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0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0.769231
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0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0.512821
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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1
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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0.820513
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gauss_jordan 2 0 _1 0 0,1 0 0 _1 0,3 0 0 _2 _1,0 1 0 0 _2,:0 1 _1 0 0
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1 0 0 0 _1
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0 1 0 0 _2
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0 0 1 0 _2
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0 0 0 1 _1
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0 0 0 0 0
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gauss_jordan 1 2 3 4 3 1,2 4 6 2 6 2,3 6 18 9 9 _6,4 8 12 10 12 4,:5 10 24 11 15 _4
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1 2 0 0 3 0
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0 0 1 0 0 0
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0 0 0 1 0 0
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0 0 0 0 0 1
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0 0 0 0 0 0
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gauss_jordan 0 1,1 2,:0 5
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1 0
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0 1
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0 0
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37
Task/Reduced-row-echelon-form/J/reduced-row-echelon-form-4.j
Normal file
37
Task/Reduced-row-echelon-form/J/reduced-row-echelon-form-4.j
Normal file
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@ -0,0 +1,37 @@
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mat=: 0 ". ];._2 noun define
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1 0 0 0 0 0 1 0 0 0 0 _1 0 0 0 0 0 0
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1 0 0 0 0 0 0 1 0 0 0 0 _1 0 0 0 0 0
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1 0 0 0 0 0 0 0 1 0 0 0 0 _1 0 0 0 0
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0 1 0 0 0 0 1 0 0 0 0 0 0 0 _1 0 0 0
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0 1 0 0 0 0 0 0 1 0 0 _1 0 0 0 0 0 0
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0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 _1 0
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0 0 1 0 0 0 1 0 0 0 0 0 _1 0 0 0 0 0
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0 0 1 0 0 0 0 0 0 1 0 0 0 0 _1 0 0 0
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0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 _1 0 0
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0 0 0 1 0 0 0 0 0 1 0 0 _1 0 0 0 0 0
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0 0 0 0 1 0 0 1 0 0 0 0 0 _1 0 0 0 0
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0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 _1 0
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0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 _1 0 0
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0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
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0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1
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0 0 0 0 0 1 0 0 0 0 1 0 0 0 _1 0 0 0
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)
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gauss_jordan mat
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1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.435897
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0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.307692
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0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.512821
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0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0.717949
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0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0.487179
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0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
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0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0.205128
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0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0.282051
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0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0.333333
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0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
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0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0.512821
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0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0.641026
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0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0.717949
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0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0.769231
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0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0.512821
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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1
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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0.820513
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@ -0,0 +1,80 @@
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// version 1.1.51
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typealias Matrix = Array<DoubleArray>
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/* changes the matrix to RREF 'in place' */
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fun Matrix.toReducedRowEchelonForm() {
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var lead = 0
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val rowCount = this.size
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val colCount = this[0].size
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for (r in 0 until rowCount) {
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if (colCount <= lead) return
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var i = r
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while (this[i][lead] == 0.0) {
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i++
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if (rowCount == i) {
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i = r
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lead++
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if (colCount == lead) return
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}
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}
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val temp = this[i]
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this[i] = this[r]
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this[r] = temp
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if (this[r][lead] != 0.0) {
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val div = this[r][lead]
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for (j in 0 until colCount) this[r][j] /= div
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}
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for (k in 0 until rowCount) {
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if (k != r) {
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val mult = this[k][lead]
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for (j in 0 until colCount) this[k][j] -= this[r][j] * mult
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}
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}
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lead++
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}
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}
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fun Matrix.printf(title: String) {
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println(title)
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val rowCount = this.size
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val colCount = this[0].size
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for (r in 0 until rowCount) {
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for (c in 0 until colCount) {
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if (this[r][c] == -0.0) this[r][c] = 0.0 // get rid of negative zeros
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print("${"% 6.2f".format(this[r][c])} ")
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}
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println()
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}
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println()
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}
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fun main(args: Array<String>) {
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val matrices = listOf(
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arrayOf(
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doubleArrayOf( 1.0, 2.0, -1.0, -4.0),
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doubleArrayOf( 2.0, 3.0, -1.0, -11.0),
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doubleArrayOf(-2.0, 0.0, -3.0, 22.0)
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),
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arrayOf(
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doubleArrayOf(1.0, 2.0, 3.0, 4.0, 3.0, 1.0),
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doubleArrayOf(2.0, 4.0, 6.0, 2.0, 6.0, 2.0),
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doubleArrayOf(3.0, 6.0, 18.0, 9.0, 9.0, -6.0),
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doubleArrayOf(4.0, 8.0, 12.0, 10.0, 12.0, 4.0),
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doubleArrayOf(5.0, 10.0, 24.0, 11.0, 15.0, -4.0)
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)
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)
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for (m in matrices) {
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m.printf("Original matrix:")
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m.toReducedRowEchelonForm()
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m.printf("Reduced row echelon form:")
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}
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}
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@ -1,25 +1,21 @@
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sub rref (@m) {
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@m or return;
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return unless @m;
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my ($lead, $rows, $cols) = 0, +@m, +@m[0];
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for ^$rows -> $r {
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return @m if $lead >= $cols;
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$lead < $cols or return @m;
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my $i = $r;
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until @m[$i][$lead] {
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until @m[$i;$lead] {
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++$i == $rows or next;
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$i = $r;
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++$lead == $cols and return @m;
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}
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@m[$i, $r] = @m[$r, $i];
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my $lv = @m[$r][$lead];
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@m[$i, $r] = @m[$r, $i] if $r != $i;
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my $lv = @m[$r;$lead];
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@m[$r] »/=» $lv;
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for ^$rows -> $n {
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next if $n == $r;
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@m[$n] »-=» @m[$r] »*» @m[$n][$lead];
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@m[$n] »-=» @m[$r] »*» (@m[$n;$lead] // 0);
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}
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++$lead;
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}
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|
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@ -3,44 +3,45 @@ cols=0; w=0; @.=0 /*max cols in a row; max width;
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mat.=; mat.1= ' 1 2 -1 -4 '
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mat.2= ' 2 3 -1 -11 '
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mat.3= ' -2 0 -3 22 '
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do r=1 until mat.r==''; _=mat.r /*build @.row.col from (matrix) mat.X*/
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do c=1 until _=''; parse var _ @.r.c _
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w=max(w, length(@.r.c) +1) /*find the maximum width of an element.*/
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end /*c*/
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cols=max(cols,c) /*remember max number of cols.*/
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do r=1 until mat.r==''; _=mat.r /*build @.row.col from (matrix) mat.X*/
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do c=1 until _=''; parse var _ @.r.c _
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w=max(w, length(@.r.c) + 1) /*find the maximum width of an element.*/
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end /*c*/
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cols=max(cols, c) /*save the maximum number of columns. */
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end /*r*/
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rows=r-1 /*adjust the row count (from DO loop).*/
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rows=r - 1 /*adjust the row count (from DO loop).*/
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call showMat 'original matrix' /*display the original matrix to screen*/
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!=1 /*set the working column pointer to 1.*/
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/* ┌─────────────────────────────────────────◄── Reduced Row Echelon Form on matrix.*/
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/* ┌──────────────────────◄────────────────◄──── Reduced Row Echelon Form on matrix.*/
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do r=1 for rows while cols>! /*begin to perform the heavy lifting. */
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j=r
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do while @.j.!==0; j=j+1
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if j==rows then do; j=r; !=!+1; if cols==! then leave r; end
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j=r /*use a subsitute index for the DO loop*/
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do while @.j.!==0; j=j + 1
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if j==rows then do; j=r; !=! + 1; if cols==! then leave r; end
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end /*while*/
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/* [↓] swap rows J,R (but not if same)*/
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do _=1 for cols while j\==r; parse value @.r._ @.j._ with @.j._ @._._; end /*_*/
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do _=1 for cols while j\==r; parse value @.r._ @.j._ with @.j._ @._._
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end /*_*/
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?=@.r.!
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do d=1 for cols while ?\=1; @.r.d=@.r.d/?; end /*d*/
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/* [↑] divide row J by @.r.p ──unless≡1*/
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do k=1 for rows; ?=@.k.! /*subtract (row K) @.r.s from row K.*/
|
||||
if k==r | ?=0 then iterate /*skip if row K is the same as row R.*/
|
||||
do s=1 for cols; @.k.s=@.k.s - ?*@.r.s; end /*s*/
|
||||
end /*k*/ /* [↑] for the rest of numbers in row.*/
|
||||
!=!+1 /*bump the column pointer. */
|
||||
do d=1 for cols while ?\=1; @.r.d= @.r.d / ?
|
||||
end /*d*/ /* [↑] divide row J by @.r.p ──unless≡1*/
|
||||
do k=1 for rows; ?= @.k.! /*subtract (row K) @.r.s from row K.*/
|
||||
if k==r | ?=0 then iterate /*skip if row K is the same as row R.*/
|
||||
do s=1 for cols; @.k.s= @.k.s - ? * @.r.s
|
||||
end /*s*/
|
||||
end /*k*/ /* [↑] for the rest of numbers in row.*/
|
||||
!=! + 1 /*bump the column pointer. */
|
||||
end /*r*/
|
||||
|
||||
call showMat 'matrix RREF' /*display the reduced row echelon form.*/
|
||||
exit /*stick a fork in it, we're all done. */
|
||||
/*──────────────────────────────────────────────────────────────────────────────────────*/
|
||||
showMat: parse arg title; say; say center(title, 3+(cols+1)*w, '─'); say
|
||||
showMat: parse arg title; say; say center(title, 3 + (cols+1) * w, '─'); say
|
||||
do r=1 for rows; _=
|
||||
do c=1 for cols
|
||||
if @.r.c=='' then do; say "***error*** matrix element isn't defined:"
|
||||
say 'row' row", column" c'.'; exit 13
|
||||
say 'row' r", column" c'.'; exit 13
|
||||
end
|
||||
_=_ right(@.r.c,w)
|
||||
_=_ right(@.r.c, w)
|
||||
end /*c*/
|
||||
say _ /*display a row of the matrix to screen*/
|
||||
end /*r*/
|
||||
return
|
||||
say _ /*display a matrix row to the terminal.*/
|
||||
end /*r*/; return
|
||||
|
|
|
|||
|
|
@ -0,0 +1,60 @@
|
|||
# Project : Reduced row echelon form
|
||||
# Date : 2017/12/27
|
||||
# Author : Gal Zsolt (~ CalmoSoft ~)
|
||||
# Email : <calmosoft@gmail.com>
|
||||
|
||||
matrix = [[1, 2, -1, -4],
|
||||
[2, 3, -1, -11],
|
||||
[ -2, 0, -3, 22]]
|
||||
ref(matrix)
|
||||
for row = 1 to 3
|
||||
for col = 1 to 4
|
||||
if matrix[row][col] = -0
|
||||
see "0 "
|
||||
else
|
||||
see "" + matrix[row][col] + " "
|
||||
ok
|
||||
next
|
||||
see nl
|
||||
next
|
||||
|
||||
func ref(m)
|
||||
nrows = 3
|
||||
ncols = 4
|
||||
lead = 1
|
||||
for r = 1 to nrows
|
||||
if lead >= ncols
|
||||
exit
|
||||
ok
|
||||
i = r
|
||||
while m[i][lead] = 0
|
||||
i = i + 1
|
||||
if i = nrows
|
||||
i = r
|
||||
lead = lead + 1
|
||||
if lead = ncols
|
||||
exit 2
|
||||
ok
|
||||
ok
|
||||
end
|
||||
for j = 1 to ncols
|
||||
temp = m[i][j]
|
||||
m[i][j] = m[r][j]
|
||||
m[r][j] = temp
|
||||
next
|
||||
n = m[r][lead]
|
||||
if n != 0
|
||||
for j = 1 to ncols
|
||||
m[r][j] = m[r][j] / n
|
||||
next
|
||||
ok
|
||||
for i = 1 to nrows
|
||||
if i != r
|
||||
n = m[i][lead]
|
||||
for j = 1 to ncols
|
||||
m[i][j] = m[i][j] - m[r][j] * n
|
||||
next
|
||||
ok
|
||||
next
|
||||
lead = lead + 1
|
||||
next
|
||||
|
|
@ -1,28 +1,27 @@
|
|||
func rref (Array m) {
|
||||
m.is_empty && return;
|
||||
var (lead, rows, cols) = (0, m.len, m[0].len);
|
||||
func rref (M) {
|
||||
var (j, rows, cols) = (0, M.len, M[0].len)
|
||||
|
||||
rows.range.each { |r|
|
||||
lead >= cols && return m;
|
||||
var i = r;
|
||||
for r in (^rows) {
|
||||
j < cols || return M
|
||||
|
||||
while (!m[i][lead]) {
|
||||
++i == rows || next;
|
||||
i = r;
|
||||
++lead == cols && return m;
|
||||
var i = r
|
||||
while (!M[i][j]) {
|
||||
++i == rows || next
|
||||
i = r
|
||||
++j == cols && return M
|
||||
}
|
||||
|
||||
m[i, r] = m[r, i];
|
||||
var lv = m[r][lead];
|
||||
m[r] = (m[r] »/» lv);
|
||||
M[i, r] = M[r, i] if (r != i)
|
||||
M[r] = (M[r] »/» M[r][j])
|
||||
|
||||
rows.range.each { |n|
|
||||
n == r && next;
|
||||
m[n] = (m[n] »-« (m[r] «*« m[n][lead]))
|
||||
for n in (^rows) {
|
||||
next if (n == r)
|
||||
M[n] = (M[n] »-« (M[r] »*» M[n][j]))
|
||||
}
|
||||
++lead;
|
||||
++j
|
||||
}
|
||||
return m
|
||||
|
||||
return M
|
||||
}
|
||||
|
||||
func say_it (message, array) {
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue