Reduced row echelon form: Difference between revisions

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 =={{header|Maxima}}==
+<lang maxima>rref(a):=block([p,q,k],[p,q]:matrix_size(a),a:echelon(a),
-<!-- {{improve|Maxima|Add an implementation of reduced row echelon form.}} -->
+    for i:p thru 2 step -1 do (
+        if not integerp(k:for j thru q do
+            if a[i,j]#0 then return(j)) then return(),
+        for j from i-1 thru 1 step -1 do a:rowop(a,j,i,a[j,k])),
+    a)$
+m:matrix([1,2,-1,-4],[2,3,-1,-11],[-2,0,-3,22]);
-Maxima does not have a built-in function for reduced row echelon form, only echelon form.
+rref(m);
-<lang maxima>(%i28) echelon(matrix([ 1,  2, -1,  -4],
+matrix([1,0,0,-8],[0,1,0,1],[0,0,1,-2])</lang>
-                      [ 2,  3, -1, -11],
-                      [-2,  0, -3,  22]));
-(%o28) matrix([1,0,3/2,-11],[0,1,-4/3,11/3],[0,0,1,-2])</lang>
-However, rref can be programmed as a Maxima batch file<ref>a version of rref by Antoine Chambert-Loir/France, taken from [http://www.ma.utexas.edu/pipermail/maxima/2007/008246.html UT Math: Maxima Mailing List]</ref>:
-<lang maxima>request_rational_matrix(m, pos, fn) :=
-  if every('identity, map(lambda([s], every('ratnump,s)), args(m))) then true else
-    print("Some entries in the matrix are not rational numbers. The result might be wrong.");
-rowswap(m,i,j) := block([n, p, r],
-  require_matrix(m, "first", "rowswap"),
-  require_integer(i, "second", "rowswap"),
-  require_integer(j, "third", "rowswap"),
-  n : length(m),
-  if (i < 1) or (i > n) or (j < 1) or (j > n)
-     then error("Array index out of bounds"),
-  p : copymatrix(m),
-  r : p[i],
-  p[i] : p[j],
-  p[j] : r,
-  p);
-addrow(m,i,j,k) := block([n,p],
-  require_matrix(m, "first", "addrow"),
-  require_integer(i, "second", "addrow"),
-  require_integer(j, "third", "addrow"),
-  require_rational(k, "fourth", "addrow"),
-  n : length(m),
-  if (i < 1) or (i > n) or (j < 1) or (j > n)
-      then error("Array index out of bounds"),
-  p : copymatrix(m),
-  p [i] : p[i] + k * p[j],
-  p);
-rowmul(m,i,k) := block([n,p],
-  require_matrix(m, "first", "addrow"),
-  require_integer(i, "second", "addrow"),
-  require_rational(k, "fourth", "addrow"),
-  n : length(m),
-  if (i < 1) or (i > n) then error("Array index out of bounds"),
-  p : copymatrix(m),
-  p [i] : k * p[i],
-  p);
-rref(m):= block([p,nr,nc,i,j,k,pivot,pivot_row,debug],
-  debug : 0,
-  request_rational_matrix(m," ","rref"),
-  nc: length(first(m)),
-  nr: length(m),
-  if nc = 0 or nr = 0 then
-    error ("The argument to 'rref' must be a matrix with one or more rows and columns"),
-  p:copymatrix(m),
-  ci : 1, cj : 1,
-  while (ci<=nr) and (cj<=nc) do
-  (
-    if (debug = 1) then (
-	    disp(p),
-	    print("curseur en ligne ",ci," et colonne ",cj)),
-    pivot_row : 0, pivot : 0,
-    for k : ci thru nr do (
-       if ( abs(p[k,cj]) > pivot ) then (
-         pivot_row : k,
-         pivot : abs(p[k,cj]))),
-         if (debug = 1) then
-	   print("colonne ",cj," : pivot trouve ligne ", pivot_row,", valeur : ",pivot),
-    if (pivot = 0) then (cj : cj +1)
-    else (
-      p : rowswap(p,ci,pivot_row),
-	if (debug = 1) then      print (".. Echange : ",p),
-      p : rowmul(p,ci,1/p[ci,cj]),
-	if (debug = 1) then     print (".. Normalisation : ",p),
-      for k : 1 thru nr do (
-         if not (k=ci) then (p : addrow (p,k,ci,-p[k,cj]))),
-      ci : ci+1, cj : cj+1)),
-  p
-);</lang>
-If this Maxima batch file is placed in one of the paths listed in the variable <code>file_search_maxima</code> under the name <code>rref.mac</code>, then it can be invoked as follows:
-<lang maxima>load(rref)$
-m:matrix(
-    [1,2,-1,-4],
-    [2,3,-1,-11],
-    [-2,0,-3,22]
-);
-rref(m);</lang>
-yielding the output:
-<math>\left[\begin{matrix}1 & 0 & 0 & -8\\ 0 & 1 & 0 & 1\\ 0 & 0 & 1 & -2\end{matrix}\right]</math>
-<lang maxima>matrix([1,0,0,-8],[0,1,0,1],[0,0,1,-2]);</lang>
 =={{header|OCaml}}==