Gram--Schmidt ortogonalization for complex functions
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I am trying to perform the Gram--Schmidt orthogonalization on a set of continuos complex functions. Numerical solution would be quite sufficient for me.
I have three continuous functions of complex argument on $[0,pi]$:
$$
f_1(x) = sin(x) \
f_2(x) = sin(2x) \
f_3(x) = sin(3x) \
$$
These are obviously orthogonal and I have checked that my Maple and Octave algorithms confirm that.
However, even a slight detour to the complex plane ruins the final set orthogonality. For example:
$$
f_1(x) = sin(x - 0.01mathrm{i}x) \
f_2(x) = sin(2x) \
$$
Then by the modified Gram-Schmidt:
$$
g_1 = frac{f_1}{sqrt{langle f_1,f_1rangle}} \
g_2 = f_2 - langle g_1,f_2rangle g_1 \
g_2 := frac{g_2}{sqrt{langle g_2,g_2rangle}}
$$
where ${langle r,s rangle} = int_0^pi fg^* mathrm{d}x$.
Yet only the real part of ${langle g_1,g_2 rangle}$ is zero (to the machine precision). The imaginary part is non-vanishing. Since I have tried the same procedure in Maple as well as in Octave obtaining the same results (I mean exactly the same, no implementation bias) I must be missing something more fundamental. Any help and/or hints, please?
complex-analysis orthogonality
asked Jan 25 at 21:19
Victor PiraVictor Pira
878
$endgroup$
add a comment |
$begingroup$
I am trying to perform the Gram--Schmidt orthogonalization on a set of continuos complex functions. Numerical solution would be quite sufficient for me.
I have three continuous functions of complex argument on $[0,pi]$:
$$
f_1(x) = sin(x) \
f_2(x) = sin(2x) \
f_3(x) = sin(3x) \
$$
These are obviously orthogonal and I have checked that my Maple and Octave algorithms confirm that.
However, even a slight detour to the complex plane ruins the final set orthogonality. For example:
$$
f_1(x) = sin(x - 0.01mathrm{i}x) \
f_2(x) = sin(2x) \
$$
Then by the modified Gram-Schmidt:
$$
g_1 = frac{f_1}{sqrt{langle f_1,f_1rangle}} \
g_2 = f_2 - langle g_1,f_2rangle g_1 \
g_2 := frac{g_2}{sqrt{langle g_2,g_2rangle}}
$$
where ${langle r,s rangle} = int_0^pi fg^* mathrm{d}x$.
Yet only the real part of ${langle g_1,g_2 rangle}$ is zero (to the machine precision). The imaginary part is non-vanishing. Since I have tried the same procedure in Maple as well as in Octave obtaining the same results (I mean exactly the same, no implementation bias) I must be missing something more fundamental. Any help and/or hints, please?
complex-analysis orthogonality
asked Jan 25 at 21:19
Victor PiraVictor Pira
878
$endgroup$
1
$begingroup$
Are you sure you are not just having rounding errors?
$endgroup$
– SmileyCraft
Jan 25 at 21:22
2
$begingroup$
Maybe you just need to change $langle g_1,f_2rangle$ to $langle f_2,g_1rangle$. The order does matter, but I do not know off the top of my head which order you need here. Recall $langle x,yrangle=overline{langle y,xrangle}$.
$endgroup$
– SmileyCraft
Jan 25 at 21:24
$begingroup$
@SmileyCraft I cannot fully exclude that but the results in the two mentioned environments match and they do not alter for any computational points refinement. That's suspicious.
$endgroup$
– Victor Pira
Jan 25 at 21:24
$begingroup$
@SmileyCraft YES!!! That was it!
$endgroup$
– Victor Pira
Jan 25 at 21:28
add a comment |
$begingroup$
I am trying to perform the Gram--Schmidt orthogonalization on a set of continuos complex functions. Numerical solution would be quite sufficient for me.
I have three continuous functions of complex argument on $[0,pi]$:
$$
f_1(x) = sin(x) \
f_2(x) = sin(2x) \
f_3(x) = sin(3x) \
$$
These are obviously orthogonal and I have checked that my Maple and Octave algorithms confirm that.
However, even a slight detour to the complex plane ruins the final set orthogonality. For example:
$$
f_1(x) = sin(x - 0.01mathrm{i}x) \
f_2(x) = sin(2x) \
$$
Then by the modified Gram-Schmidt:
$$
g_1 = frac{f_1}{sqrt{langle f_1,f_1rangle}} \
g_2 = f_2 - langle g_1,f_2rangle g_1 \
g_2 := frac{g_2}{sqrt{langle g_2,g_2rangle}}
$$
where ${langle r,s rangle} = int_0^pi fg^* mathrm{d}x$.
Yet only the real part of ${langle g_1,g_2 rangle}$ is zero (to the machine precision). The imaginary part is non-vanishing. Since I have tried the same procedure in Maple as well as in Octave obtaining the same results (I mean exactly the same, no implementation bias) I must be missing something more fundamental. Any help and/or hints, please?
complex-analysis orthogonality
asked Jan 25 at 21:19
Victor PiraVictor Pira
878
$endgroup$
I am trying to perform the Gram--Schmidt orthogonalization on a set of continuos complex functions. Numerical solution would be quite sufficient for me.
I have three continuous functions of complex argument on $[0,pi]$:
$$
f_1(x) = sin(x) \
f_2(x) = sin(2x) \
f_3(x) = sin(3x) \
$$
These are obviously orthogonal and I have checked that my Maple and Octave algorithms confirm that.
However, even a slight detour to the complex plane ruins the final set orthogonality. For example:
$$
f_1(x) = sin(x - 0.01mathrm{i}x) \
f_2(x) = sin(2x) \
$$
Then by the modified Gram-Schmidt:
$$
g_1 = frac{f_1}{sqrt{langle f_1,f_1rangle}} \
g_2 = f_2 - langle g_1,f_2rangle g_1 \
g_2 := frac{g_2}{sqrt{langle g_2,g_2rangle}}
$$
where ${langle r,s rangle} = int_0^pi fg^* mathrm{d}x$.
Yet only the real part of ${langle g_1,g_2 rangle}$ is zero (to the machine precision). The imaginary part is non-vanishing. Since I have tried the same procedure in Maple as well as in Octave obtaining the same results (I mean exactly the same, no implementation bias) I must be missing something more fundamental. Any help and/or hints, please?
complex-analysis orthogonality
complex-analysis orthogonality
asked Jan 25 at 21:19
Victor PiraVictor Pira
878
asked Jan 25 at 21:19
Victor PiraVictor Pira
878
asked Jan 25 at 21:19
Victor PiraVictor Pira
878
asked Jan 25 at 21:19
Victor PiraVictor Pira
878
asked Jan 25 at 21:19
Victor PiraVictor Pira
878
878
1
$begingroup$
Are you sure you are not just having rounding errors?
$endgroup$
– SmileyCraft
Jan 25 at 21:22
2
$begingroup$
Maybe you just need to change $langle g_1,f_2rangle$ to $langle f_2,g_1rangle$. The order does matter, but I do not know off the top of my head which order you need here. Recall $langle x,yrangle=overline{langle y,xrangle}$.
$endgroup$
– SmileyCraft
Jan 25 at 21:24
$begingroup$
@SmileyCraft I cannot fully exclude that but the results in the two mentioned environments match and they do not alter for any computational points refinement. That's suspicious.
$endgroup$
– Victor Pira
Jan 25 at 21:24
$begingroup$
@SmileyCraft YES!!! That was it!
$endgroup$
– Victor Pira
Jan 25 at 21:28
add a comment |
1
$begingroup$
Are you sure you are not just having rounding errors?
$endgroup$
– SmileyCraft
Jan 25 at 21:22
2
$begingroup$
Maybe you just need to change $langle g_1,f_2rangle$ to $langle f_2,g_1rangle$. The order does matter, but I do not know off the top of my head which order you need here. Recall $langle x,yrangle=overline{langle y,xrangle}$.
$endgroup$
– SmileyCraft
Jan 25 at 21:24
$begingroup$
@SmileyCraft I cannot fully exclude that but the results in the two mentioned environments match and they do not alter for any computational points refinement. That's suspicious.
$endgroup$
– Victor Pira
Jan 25 at 21:24
$begingroup$
@SmileyCraft YES!!! That was it!
$endgroup$
– Victor Pira
Jan 25 at 21:28
1
1
$begingroup$
Are you sure you are not just having rounding errors?
$endgroup$
– SmileyCraft
Jan 25 at 21:22
$begingroup$
Are you sure you are not just having rounding errors?
$endgroup$
– SmileyCraft
Jan 25 at 21:22
2
2
$begingroup$
Maybe you just need to change $langle g_1,f_2rangle$ to $langle f_2,g_1rangle$. The order does matter, but I do not know off the top of my head which order you need here. Recall $langle x,yrangle=overline{langle y,xrangle}$.
$endgroup$
– SmileyCraft
Jan 25 at 21:24
$begingroup$
Maybe you just need to change $langle g_1,f_2rangle$ to $langle f_2,g_1rangle$. The order does matter, but I do not know off the top of my head which order you need here. Recall $langle x,yrangle=overline{langle y,xrangle}$.
$endgroup$
– SmileyCraft
Jan 25 at 21:24
$begingroup$
@SmileyCraft I cannot fully exclude that but the results in the two mentioned environments match and they do not alter for any computational points refinement. That's suspicious.
$endgroup$
– Victor Pira
Jan 25 at 21:24
$begingroup$
@SmileyCraft I cannot fully exclude that but the results in the two mentioned environments match and they do not alter for any computational points refinement. That's suspicious.
$endgroup$
– Victor Pira
Jan 25 at 21:24
$begingroup$
@SmileyCraft YES!!! That was it!
$endgroup$
– Victor Pira
Jan 25 at 21:28
$begingroup$
@SmileyCraft YES!!! That was it!
$endgroup$
– Victor Pira
Jan 25 at 21:28
add a comment |
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1
$begingroup$
Are you sure you are not just having rounding errors?
$endgroup$
– SmileyCraft
Jan 25 at 21:22
2
$begingroup$
Maybe you just need to change $langle g_1,f_2rangle$ to $langle f_2,g_1rangle$. The order does matter, but I do not know off the top of my head which order you need here. Recall $langle x,yrangle=overline{langle y,xrangle}$.
$endgroup$
– SmileyCraft
Jan 25 at 21:24
$begingroup$
@SmileyCraft I cannot fully exclude that but the results in the two mentioned environments match and they do not alter for any computational points refinement. That's suspicious.
$endgroup$
– Victor Pira
Jan 25 at 21:24
$begingroup$
@SmileyCraft YES!!! That was it!
$endgroup$
– Victor Pira
Jan 25 at 21:28