There's no provision in this for avoiding using hidden zeros as pivots, during the linear system solving. I still thnik that, in general, that's a serious issue for this class of problem. Did you mean to have, V1:= LinearSolve(U1[1..3,1..4],Vector(3)); but perhaps got caught by the < issue? This seems like just a bit more "by-hand" work, than calling NullSpace. Also, how can you know that it's "OK" to drop the last row of U1 (as opposed to the 2nd, etc)? What if the 1st and 2nd rows are linear combinations of each other and the 4th is independent of the top 3? Why not just do, LinearSolve(U1,Vector(4)); which is in effect what NullSpace does? acer

Interesting. Using a simplifier (with side relations) for Normalizer on your Matrix U resulted in a spurious all-zero Matrix of eigenvectors when using the Eigenvectors command. So maybe Eigenvectors is not clever enough about how it goes to work in such a situation. But I still think that you might be rightfully concerned with Eigenvectors() using hidden zeros as pivots during the kernel computation. Having computed the eigenvalues as evU, you might then compute the eigenvectors associated with the i'th eigenvalues the good old fashioned way. That is to say, using NullSpace(). You may also find that the Normalizer setting gets the eigenvalues in their simplified form too. with(LinearAlgebra): U := Matrix(1/8*[[0, 0, -8*I, 0], [0, 8*I, 0, 8*I], [I*b, I*(a-1), (2*I)*(a-1+b)+8*I, (2*I)*(a-1-b)], [I*(a-1), -I*b, (2*I)*(a-1-b), -(2*I)*(a-1+b)]]): use_assump := x -> simplify(simplify(x, {a^2+b^2 = 1})): Normalizer := use_assump: evU := Eigenvalues(U); for i from 1 to 4 do NullSpace(U-evU[i]*IdentityMatrix(4)) end do; Normalizer := normal: acer

Interesting. Using a simplifier (with side relations) for Normalizer on your Matrix U resulted in a spurious all-zero Matrix of eigenvectors when using the Eigenvectors command. So maybe Eigenvectors is not clever enough about how it goes to work in such a situation. But I still think that you might be rightfully concerned with Eigenvectors() using hidden zeros as pivots during the kernel computation. Having computed the eigenvalues as evU, you might then compute the eigenvectors associated with the i'th eigenvalues the good old fashioned way. That is to say, using NullSpace(). You may also find that the Normalizer setting gets the eigenvalues in their simplified form too. with(LinearAlgebra): U := Matrix(1/8*[[0, 0, -8*I, 0], [0, 8*I, 0, 8*I], [I*b, I*(a-1), (2*I)*(a-1+b)+8*I, (2*I)*(a-1-b)], [I*(a-1), -I*b, (2*I)*(a-1-b), -(2*I)*(a-1+b)]]): use_assump := x -> simplify(simplify(x, {a^2+b^2 = 1})): Normalizer := use_assump: evU := Eigenvalues(U); for i from 1 to 4 do NullSpace(U-evU[i]*IdentityMatrix(4)) end do; Normalizer := normal: acer

Since nobody has yet stated it in just so many words, I will add that `=` does not by itself bring about assignment in Maple. So, it is not true that, "The symbol = means assignment." Without qualification an assertion like that is completely general. But since it isn't true in Maple in particular, then it also isn't always true. Assignments in Maple are done using a `:=` statement, or using the `assign` operator. As for what `=` can in fact mean in Maple, please see other replies to this subthread whose points I won't repeat here. acer

Since nobody has yet stated it in just so many words, I will add that `=` does not by itself bring about assignment in Maple. So, it is not true that, "The symbol = means assignment." Without qualification an assertion like that is completely general. But since it isn't true in Maple in particular, then it also isn't always true. Assignments in Maple are done using a `:=` statement, or using the `assign` operator. As for what `=` can in fact mean in Maple, please see other replies to this subthread whose points I won't repeat here. acer

The procedure Rsquared below could allow you to compute R^2 for data appearing in the first and second columns of a Matrix and for a given fitting formula in variable t.

Rsquared := proc( formula, T::symbol,  m::Matrix )
local N, ybar, SS_err, SS_tot, SS_reg;
  Digits:=trunc(evalhf(Digits));
  N := op([1,1],m);
  ybar := add( m[i,2], i=1..N )/N;
  SS_tot := add( ( m[i,2] - ybar )^2, i=1..N );
  SS_reg := add( ( eval(formula,T=m[i,1]) - ybar )^2, i=1..N );
  SS_err := add( ( m[i,2] - eval(formula,T=m[i,1]) )^2, i=1..N );
  return 1-SS_err/SS_tot;
end proc:

M := Matrix(6,2, [[1,2],[2,3],[3,4.8],[5,10.2],[6,30.9]] );

form := 0.888 + 0.606*exp(0.649*t);

Rsquared( form, t, M );

You could fix it up to do proper argument checking and validation. Comments could be added to the code. Or you could just examine it and compare to the formulas on that web-page, and try and figure out a bit more of how the Maple programming language works. acer

The procedure Rsquared below could allow you to compute R^2 for data appearing in the first and second columns of a Matrix and for a given fitting formula in variable t.

Rsquared := proc( formula, T::symbol,  m::Matrix )
local N, ybar, SS_err, SS_tot, SS_reg;
  Digits:=trunc(evalhf(Digits));
  N := op([1,1],m);
  ybar := add( m[i,2], i=1..N )/N;
  SS_tot := add( ( m[i,2] - ybar )^2, i=1..N );
  SS_reg := add( ( eval(formula,T=m[i,1]) - ybar )^2, i=1..N );
  SS_err := add( ( m[i,2] - eval(formula,T=m[i,1]) )^2, i=1..N );
  return 1-SS_err/SS_tot;
end proc:

M := Matrix(6,2, [[1,2],[2,3],[3,4.8],[5,10.2],[6,30.9]] );

form := 0.888 + 0.606*exp(0.649*t);

Rsquared( form, t, M );

You could fix it up to do proper argument checking and validation. Comments could be added to the code. Or you could just examine it and compare to the formulas on that web-page, and try and figure out a bit more of how the Maple programming language works. acer

So, you may be interested in the so-called "coefficient of determination", or R^2. See this link for details and explanations. What I showed before is the residual sum of squares, which that page describes as its SS_err term. That pages also gives some formulae for R^2. Perhaps you would like to use the formula which computes R^2 via regression sum of squares. If your X and Y data lie in the first and second columns of Matrix M, then in your code you can access the entries as M[i,1] instead of X[i] and M[i,2] instead of Y[i]. That saves your having to form the X and Y Vectors explicitly. acer

So, you may be interested in the so-called "coefficient of determination", or R^2. See this link for details and explanations. What I showed before is the residual sum of squares, which that page describes as its SS_err term. That pages also gives some formulae for R^2. Perhaps you would like to use the formula which computes R^2 via regression sum of squares. If your X and Y data lie in the first and second columns of Matrix M, then in your code you can access the entries as M[i,1] instead of X[i] and M[i,2] instead of Y[i]. That saves your having to form the X and Y Vectors explicitly. acer

The uploaded file has,

p*n+s = `mod`(0, a)

Did you perhaps intend either of these forms below? 

> restart:
> s := 3: p := 1: a := 4: t := 0:
> for n to 10 do
> if `mod`(p*n+s,a) = 0
> then t := t+1
> else p*n+s
> end if;
> end do;
> t;

                                       3


Or,

> restart:
> s := 3: p := 1: a := 4: t := 0:
> for n to 10 do
> if p*n+s mod a = 0
> then t := t+1
> else p*n+s
> end if;
> end do;
> t;

                                       3


acer

Exactly, Scott. It's not currently possible to get the symbol to display correctly in both places; Subject line and Recent comments sidebar. It's nice to know that Will's looking at some issues with this site. As for the motivation, I was actually trying to represent a key press in a not so unusual way, eg. -. But it's also pretty natural to imagine that the less-than symbol might be wanted, at some point, in the title of a posting on a discussion site for mathematical software. acer

It appears as if the comments (starting with # symbol) and the code in the problem area of your Document are being interpreted by Maple as being all on one line. So anything that appeared after the comment symbol, including the code, was being treated as part of one big long comment. As one big comment, it wouldn't execute. At least, that's what it looked like after conversion to 1D input, and that would exactly match the behaviour being seen. I suggest one of two possible ways to get around this problematic behaviour. The first is to write your code in 1D Math input, preferably in a Worksheet not a Document, where such discrepancies are very much rarer. (You can even change preferences, globally for future sessions, for both those aspects.) The second choice is to use text input for comments when using 2D Math and Documents. The max() routine will give the maximum of an expression sequence. For a list L, the expression sequence of its contents is op(L). The printf() routine can format and print things nicely. It works much like it does in the C language. There is an additional qualifier, %a, which stands for algebraic (I think). For your code and list W, it could be like this printf("Max = %a\n", max(op(W))); acer

It appears as if the comments (starting with # symbol) and the code in the problem area of your Document are being interpreted by Maple as being all on one line. So anything that appeared after the comment symbol, including the code, was being treated as part of one big long comment. As one big comment, it wouldn't execute. At least, that's what it looked like after conversion to 1D input, and that would exactly match the behaviour being seen. I suggest one of two possible ways to get around this problematic behaviour. The first is to write your code in 1D Math input, preferably in a Worksheet not a Document, where such discrepancies are very much rarer. (You can even change preferences, globally for future sessions, for both those aspects.) The second choice is to use text input for comments when using 2D Math and Documents. The max() routine will give the maximum of an expression sequence. For a list L, the expression sequence of its contents is op(L). The printf() routine can format and print things nicely. It works much like it does in the C language. There is an additional qualifier, %a, which stands for algebraic (I think). For your code and list W, it could be like this printf("Max = %a\n", max(op(W))); acer

Please ignore. This was an accidental duplicate post. acer

Please ignore. This was an accidental duplicate post. acer