RE: Identifying Linearly Dependent Equations

*To*: mathgroup at smc.vnet.net*Subject*: [mg81344] RE: [mg81308] Identifying Linearly Dependent Equations*From*: "Tugrul Temel" <t.temel at uvt.nl>*Date*: Thu, 20 Sep 2007 03:45:09 -0400 (EDT)*References*: <200709170734.DAA15516@smc.vnet.net> <fcnkpb$s0k$1@smc.vnet.net> <200709190919.FAA00834@smc.vnet.net>

Dear Jean, Thank you very much for your help. First, I have just noticed the problem with my e-mail address. It has been corrected and now people will be able to identify it more clearly. Second, your help in Math programming has been very useful. The results have improved substantially. Best regards, Tugrul Temel -----Original Message----- From: Jean-Marc Gulliet [mailto:jeanmarc.gulliet at gmail.com] Sent: woensdag 19 september 2007 11:19 To: mathgroup at smc.vnet.net Subject: [mg81344] [mg81308] Identifying Linearly Dependent Equations %uvt_fullname% wrote: > I am facing the following problem: Note that I have changed the title of the thread. A meaningful title will attract more answers (and a valid email address allows you to get faster replies by direct email) and it is easier for people to find a thread that answers their questions. > 1) I have a system of m=12 linear equations, with n=11 variables. > > 2) I want to identify those linearly dependent equations and drop them from > the system to have a solution. > > 3) I applied the following code (somebody from MathGroup provided this last > year), which is the Gram-Schmidt method of orthogonalization: > > > Clear[V]; > > equ = {i - 0.4 p == 0, k - 0.16 z == 0, m - 0.9 p == 0, c - 0.83 r == 0, x - > 0.7 p ==0, p + m - i - c - k - x == 0, y - p + i == 0, > s - r + c == 0, b - m + x == 0, y - r == 0, k - s - b == 0, > z - p - m == 0}; > > {h1, V} = CoefficientArrays[equ, {p, m, i, k, x, c, y, r, s, b, z}]; > > Do[Do[If[V[[n]] != {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, V[[m]] = > (V[[n]].V[[n]]) V[[m]] - (V[[n]].V[[m]]) V[[n]]], {n, 1, m - 1}], {m, 2, > 12}] > > Map[If[# != {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, "keep", "discard"] &, V] > > > This Code always indicates the 6th row to be discarded no matter which order > you input the equations in "equ". > > Isn't there a problem with this code or maybe I am missing something. At a first glance, the code seems fine for symbolic computation (I am rather condiment that what you are facing is numerical instability due to the use of inexact arithmetic). In the example below, we can see that by reversing the sequence of equations, we get nothing to discard. So use exact numbers or rationalize the equations first. In[1]:= equ = {i - 0.4 p == 0, k - 0.16 z == 0, m - 0.9 p == 0, c - 0.83 r == 0, x - 0.7 p == 0, p + m - i - c - k - x == 0, y - p + i == 0, s - r + c == 0, b - m + x == 0, y - r == 0, k - s - b == 0, z - p - m == 0}; dependentQ[eqns_] := Module[{h1, V}, {h1, V} = CoefficientArrays[eqns, {p, m, i, k, x, c, y, r, s, b, z}]; Do[Do[If[V[[n]] != {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, V[[m]] = (V[[n]].V[[n]]) V[[m]] - (V[[n]].V[[m]]) V[[n]]], {n, 1, m - 1}], {m, 2, 12}]; Map[If[# != {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, "keep", "discard"] &, V] ] dependentQ[equ] dependentQ[Reverse@equ] dependentQ[Sort@equ] Out[3]= {"keep", "keep", "keep", "keep", "keep", "keep", "keep", \ "keep", "keep", "keep", "discard", "keep"} Out[4]= {"keep", "keep", "keep", "keep", "keep", "keep", "keep", \ "keep", "keep", "keep", "keep", "keep"} Out[5]= {"keep", "keep", "keep", "keep", "keep", "keep", "keep", \ "keep", "keep", "keep", "discard", "discard"} In[6]:= equ = Rationalize[equ]; dependentQ[equ] dependentQ[Reverse@equ] dependentQ[Sort@equ] Out[7]= {"keep", "keep", "keep", "keep", "keep", "keep", "keep", \ "keep", "keep", "keep", "discard", "keep"} Out[8]= {"keep", "keep", "keep", "keep", "keep", "keep", "discard", \ "keep", "keep", "keep", "keep", "keep"} Out[9]= {"keep", "keep", "keep", "keep", "keep", "keep", "keep", \ "keep", "keep", "discard", "keep", "keep"} -- Jean-Marc

**References**:**Button Help Example***From:*"David Park" <djmpark@comcast.net>

**Identifying Linearly Dependent Equations***From:*Jean-Marc Gulliet <jeanmarc.gulliet@gmail.com>

**Re: Re: Can an arrow be drawn on a 3D plot?**

**Re: Re: plotmarkers in Listplot**

**Identifying Linearly Dependent Equations**

**Re: Button Help Example**