Updates:
Labels: Matrices
Comment #1 on issue 2184 by asmeurer: Wrong rref()
http://code.google.com/p/sympy/issues/detail?id=2184
What should the correct answer be? Maybe this is correct, when you pass
simplifed=True to rref():
In [16]: M.rref(simplified=True)
Out[16]:
⎛⎡
⎛
⎛ 2 ⎞
⎞ ⎤ ⎞
⎜⎢ ⎛ ⎛ qc⋅qs
⎞ ⎞ ⎜ ⎛ qc⋅qs ⎞ ⎜ qc
⎟ ⎟ ⎥ ⎟
⎜⎢ ⎜qc⋅⎜-qc +
──────────⎟ ⎟ ⎜ ⎜-qc + ──────────⎟⋅(-qs + 2⋅qc)⋅⎜qs -
──────────⎟ ⎟ ⎥ ⎟
⎜⎢ ⎛ 2 ⎞ ⎜ ⎝ -qs +
2⋅qc⎠ qs ⎟ ⎜ ⎝ -qs + 2⋅qc⎠ ⎝ -qs +
2⋅qc⎠ qc⋅qs ⎟ ⎥ ⎟
⎜⎢ ⎜ qc ⎟ (-qs + 3⋅qc)⋅⎜────────────────────── +
──────────⎟⋅⎜-qc - ───────────────────────────────────────────────── +
──────────⎟ ⎥ ⎟
⎜⎢ qc⋅⎜qs - ──────────⎟ ⎜ 2 2
-qs + 2⋅qc⎟ ⎜ 2 2 -qs
+ 2⋅qc⎟ ⎥ ⎟
⎜⎢ ⎝ -qs + 2⋅qc⎠ ⎝-4⋅qc⋅qs + qs +
3⋅qc ⎠ ⎝ -4⋅qc⋅qs + qs +
3⋅qc ⎠ qc ⎥ ⎟
⎜⎢1 0 0 ────────────────────── -
─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
- ──────────⎥, [0, 1, 2]⎟
⎜⎢ 2
2
2 -qs +
2⋅qc⎥ ⎟
⎜⎢ -4⋅qc⋅qs + qs +
3⋅qc -4⋅qc⋅qs +
4⋅qc
⎥ ⎟
⎜⎢
⎥
⎟
⎜⎢
⎛
⎛ 2 ⎞ ⎞
⎥ ⎟
⎜⎢
⎜
⎛ qc⋅qs ⎞ ⎜ qc ⎟ ⎟
⎥ ⎟
⎜⎢
⎜
⎜-qc + ──────────⎟⋅(-qs + 2⋅qc)⋅⎜qs - ──────────⎟ ⎟
⎥ ⎟
⎜⎢ ⎛ 2 ⎞ ⎛ qc⋅qs
⎞ ⎜ ⎝ -qs + 2⋅qc⎠ ⎝
-qs + 2⋅qc⎠ qc⋅qs ⎟ ⎥ ⎟
⎜⎢ ⎜ qc ⎟ ⎜-qc + ──────────⎟⋅(-qs +
2⋅qc)⋅(-qs + 3⋅qc)⋅⎜-qc - ─────────────────────────────────────────────────
+ ──────────⎟ ⎥ ⎟
⎜⎢ (-qs + 2⋅qc)⋅⎜qs - ──────────⎟ ⎝ -qs +
2⋅qc⎠ ⎜ 2
2 -qs + 2⋅qc⎟ ⎥ ⎟
⎜⎢ ⎝ -qs +
2⋅qc⎠ ⎝
-4⋅qc⋅qs + qs + 3⋅qc ⎠ ⎥ ⎟
⎜⎢0 1 0 ────────────────────────────── -
───────────────────────────────────────────────────────────────────────────────────────────────────────────────────
⎥ ⎟
⎜⎢ 2
2 ⎛ 2⎞
⎛ 2 2⎞
⎥ ⎟
⎜⎢ -4⋅qc⋅qs + qs +
3⋅qc ⎝-4⋅qc⋅qs + 4⋅qc
⎠⋅⎝-4⋅qc⋅qs + qs + 3⋅qc ⎠
⎥ ⎟
⎜⎢
⎥
⎟
⎜⎢
⎛ ⎛ 2 ⎞
⎞ ⎥ ⎟
⎜⎢ ⎜
⎛ qc⋅qs ⎞ ⎜ qc ⎟
⎟ ⎥ ⎟
⎜⎢ ⎜ ⎜-qc +
──────────⎟⋅(-qs + 2⋅qc)⋅⎜qs - ──────────⎟
⎟ ⎥ ⎟
⎜⎢ ⎜ ⎝
-qs + 2⋅qc⎠ ⎝ -qs + 2⋅qc⎠ qc⋅qs
⎟ ⎥ ⎟
⎜⎢ (-qs + 3⋅qc)⋅⎜-qc -
───────────────────────────────────────────────── +
──────────⎟ ⎥ ⎟
⎜⎢
⎜ 2 2 -qs +
2⋅qc⎟ ⎥ ⎟
⎜⎢
⎝ -4⋅qc⋅qs + qs + 3⋅qc
⎠ ⎥ ⎟
⎜⎢0 0 1
───────────────────────────────────────────────────────────────────────────────────
⎥ ⎟
⎜⎢
2
⎥
⎟
⎜⎢
-4⋅qc⋅qs
+
4⋅qc
⎥ ⎟
⎜⎢
⎥
⎟
⎝⎣0 0
0
0
⎦ ⎠
The correctness of the reduced row echelon form algorithm relies on the
ability to determine if an expression is identically zero. By default,
Matrix.rref() does not try to simplify expressions to zero, but
simplified=True has it do this. So sometimes, it will not reduce symbolic
expressions to zero without this. I would recommend using it for symbolic
matrices (note that for very large/complex matrices, this can slow things
down though).
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