I'm trying to write down some notes of my work. The way Maxima would simplify my work is that once I write bunch of equations and I want to change the definition of a variable, I do it and re-evaluate the entire file.
Here is an example of what I'm trying to accomplish:
Question 1:
I have a system of equations and all I want from Maxima is just variables replacement.
eq1: x=a+b+c
eq2: y=d+e+f
eq3: x+y=0
How do I get Maxima to output
eq3: a+b+c+d+e+f = 0
So in the future if I want x to be a+b-c, I just change it and re-evaluate
Question 2:
Similar to before but a bit more complex
eq1: x=a+b+c
eq2: y=d+e+f
eq3: x=y
eq4: a+s+e=0
How do I get Maxima to output
eq3 a+b+c=d+e+f
How do I get Maxima to solve eq1 for a and solve eq2 for e and output
eq4: x-b-c+s+y-d-f = 0
Thank you in advance for your help,
Guido
I think subst and solve can handle the operations you want here.
(%i1) eq1: x=a+b+c;
(%o1) x = c + b + a
(%i2) eq2: y=d+e+f;
(%o2) y = f + e + d
(%i3) eq3: x+y=0;
(%o3) y + x = 0
(%i4) subst ([eq1, eq2], eq3);
(%o4) f + e + d + c + b + a = 0
OK, now here's your second example. Note that solve returns a list of equations.
(%i5) eq3: x=y;
(%o5) x = y
(%i6) eq4: a+s+e=0;
(%o6) s + e + a = 0
(%i7) subst ([eq1, eq2], eq3);
(%o7) c + b + a = f + e + d
(%i8) solve (eq1, a);
(%o8) [a = x - c - b]
(%i9) solve (eq2, e);
(%o9) [e = y - f - d]
(%i10) append (%o8, %o9);
(%o10) [a = x - c - b, e = y - f - d]
(%i11) subst (%o10, eq4);
(%o11) y + x + s - f - d - c - b = 0
Maxima's solve function is not too powerful; there are many kinds of equations it cannot solve. But it can solve linear equations.
Related
I've created this code and it gives me this error message:
Error using *
Incorrect dimensions for matrix multiplication.
Error in poli3 = sin(pi*a) ...
Below I show one function used in the code. I don't know if the problem comes from the value given by derivadan or what.
x = -1:0.01:1; % Intervalo en el que se evaluará el polinomio de Taylor
y = sin(pi*x); % Función
a = 0;
derivada3 = derivadan(0.01, 3, a);
derivada7 = derivadan(0.01, 7, a);
derivada3_vec = repmat(derivada3, size(x - a));
derivada7_vec = repmat(derivada7, size(x - a));
poli3 = sin(pi*a) + derivada3_vec*(x - a) + (derivada3_vec*(x - a).^2)/factorial(2) + (derivada3_vec*(x - a).^3)/factorial(3);
poli7 = sin(pi*a) + derivada7_vec*(x - a) + (derivada7_vec*(x - a).^2)/factorial(2) + (derivada7_vec*(x - a).^3)/factorial(3) + (derivada7_vec*(x - a).^4)/factorial(4) + (derivada7_vec*(x - a).^5)/factorial(5) + (derivada7_vec*(x - a).^6)/factorial(6) + (derivada7_vec*(x - a).^7)/factorial(7);
figure
plot(x, poli3, 'r', x, poli7, 'b')
legend('Taylor grau 3', 'Taylor grau 7')
title('Grafica Taylor 3 grau vs Grafica Taylor 7 grau')
function Yd = derivadan(h, grado, vecX)
Yd = zeros(size(vecX));
for i = 1:grado
Yd = (vecX(2:end) - vecX(1:end-1)) / h;
vecX = Yd;
end
end
In MATLAB one can go 2 ways when developing taylor series; the hard way and the easy way.
As asked, first the HARD way :
close all;clear all;clc
dx=.01
x = -1+dx:dx:1-dx;
y = sin(pi*x);
a =0;
[va,na]=find(x==a)
n1=3
D3y=zeros(n1,numel(x));
for k=1:1:n1
D3y(k,1:end-k)=dn(dx,k,y);
end
T3=y(na)+sum(1./factorial([1:n1])'.*D3y(:,na).*((x(1:end-n1)-a)'.^[1:n1])',1);
n2=7
D7y=zeros(n2,numel(x));
for k=1:1:n2
D7y(k,1:end-k)=dn(dx,k,y);
end
T7=y(na)+sum([1./factorial([1:n2])]'.*D7y(:,na).*((x(1:end-n2)-a)'.^[1:n2])',1);
figure(1);ax=gca
plot(ax,x(1:numel(T7)),T3(1:numel(T7)),'r')
grid on;hold on
xlabel('x')
plot(ax,x(1:numel(T7)),T7(1:numel(T7)),'b--')
plot(ax,x(1:numel(T7)),y(1:numel(T7)),'g')
axis(ax,[-1 1 -1.2 1.2])
legend('T3', 'T7','sin(pi*x)','Location','northeastoutside')
the support function being
function Yd = dn(h, n, vecX)
Yd = zeros(size(vecX));
for i = 1:n
Yd = (vecX(2:end) - vecX(1:end-1))/h;
vecX = Yd;
end
end
Explanation
1.- The custom function derivadan that I call dn shortens one sample for every unit up in grado where grado is the derivative order.
For instance, the 3rd order derivative is going to be 3 samples shorter than the input function.
This causes product mismatch and when later on attempting plot it's going to cause plot error.
2.- To avoid such mismatchs ALL vectors shortened to the size of the shortest one.
x(1:end-a)
is a samples shorter than x and y and can be used as reference vector in plot.
3.- Call function derivadan (that I call dn) correctly
dn expects as 3rd input (3rd from left) a vector, the function values to differentiate, yet you are calling derivadan with a in 3rd input field. a is scalar and you have set it null. Fixed it.
derivada3 = derivadan(0.01, 3, a);
should be called
derivada3 = derivadan(0.01, 3, y);
same for derivada7
4.- So
error using * ...
error in poly3=sin(pi*a) ...
MATLAB doesn't particularly mean that there's an error right on sin(pi*a) , that could be, but it's not the case here >
MATLAB is saying : THERE'S AN ERROR IN LINE STARTING WITH
poly3=sin(pi*a) ..
MATLAB aborts there.
Same error is found in next line starting with
poly7 ..
Since sin(pi*a)=0 because a=0 yet all other terms in sum for poly3 are repmat outcomes with different sizes all sizes being different and >1 hence attempting product of different sizes.
Operator * requires all terms have same size.
5.- Syntax Error
derivada3_vec = repmat(derivada3, size(x - a))
is built is not correct
this line repeats size(x) times the nth order derivative !
it's a really long sequence.
Now the EASY way
6.- MATLAB already has command taylor
syms x;T3=taylor(sin(pi*x),x,0)
T3 = (pi^5*x^5)/120 - (pi^3*x^3)/6 + pi*x
syms x;T3=taylor(sin(pi*x),x,0,'Order',3)
T3 = pi*x
syms x;T3=taylor(sin(pi*x),x,0,'Order',7)
T7 = (pi^5*x^5)/120 - (pi^3*x^3)/6 + pi*x
T9=taylor(sin(pi*x),x,0,'Order',9)
T9 =- (pi^7*x^7)/5040 + (pi^5*x^5)/120 - (pi^3*x^3)/6 + pi*x
It really simplfies taylor series development because it readily generates all that is needed to such endeavour :
syms f(x)
f(x) = sin(pi*x);
a=0
T_default = taylor(f, x,'ExpansionPoint',a);
T8 = taylor(f, x, 'Order', 8,'ExpansionPoint',a);
T10 = taylor(f, x, 'Order', 10,'ExpansionPoint',a);
figure(2)
fplot([T_default T8 T10 f])
axis([-2 3 -1.2 1.2])
hold on
plot(a,f(a),'r*')
grid on;xlabel('x')
title(['Taylor Series Expansion x =' num2str(a)])
a=.5
T_default = taylor(f, x,'ExpansionPoint',a);
T8 = taylor(f, x, 'Order', 8,'ExpansionPoint',a);
T10 = taylor(f, x, 'Order', 10,'ExpansionPoint',a);
figure(3)
fplot([T_default T8 T10 f])
axis([-2 3 -1.2 1.2])
hold on
plot(a,f(a),'r*')
grid on;xlabel('x')
title(['Taylor Series Expansion x =' num2str(a)])
a=1
T_default = taylor(f, x,'ExpansionPoint',a);
T8 = taylor(f, x, 'Order', 8,'ExpansionPoint',a);
T10 = taylor(f, x, 'Order', 10,'ExpansionPoint',a);
figure(4)
fplot([T_default T8 T10 f])
axis([-2 3 -1.2 1.2])
hold on
plot(a,f(a),'r*')
grid on;xlabel('x')
title(['Taylor Series Expansion x =' num2str(a)])
thanks for reading
I checked the line that you were having issues with; it seems that the error is in the derivada3_vec*(x - a) (as well as in the other terms that use derivada3_vec).
Looking at the variable itself: derivada3_vec is an empty vector. Going back further, the derivada3 variable is also an empty vector.
Your issue is in your function derivadan. You're inputting a 1x1 vector (a = [0]), but the function assumes that a is at least 1x2 (or 2x1).
I suspect there are other issues, but this is the cause of your error message.
I am currently trying to create n-nested loops in order to compute a rather tricky sum.
Each term in the sum depends on the last, and therefore the nesting is required. Here is some Julia-written pseudo-code explaining what I'm trying to get:
# u,v are fixed integers
# m is a fixed, n-length array of integers
_
for i_1 in 0:min(m[1], u) |#Trio of Loops we want to repeat n times
for j_1 in 0:min(m[1], v) |
for t_1 in i_1:(i_1 + j_1) _|#1
_
for i_2 in 0:min(m[2], u - i_1) |
for j_2 in 0:min(m[2], v - j_1) |
for t_2 in i_2:(i_2 + j_2) _|#2
#... repeat until nth time...#
_
for i_n in 0:min(m[n],u - sum(#all i_'s up to n-1)) |
for j_n in 0:min(m[n],v - sum(#all j_'s up to n-1)) |
for t_n in i_n:(i_n + j_n) _|#n
#Sum over sum function which depends on i_'s and j_'s and t_'s
X += f(i_1,j_1,t_1)*f(i_2,j_2,t_2)*...*f(i_n,j_n,t_n)
I am completely unsure of how to properly code this for an arbitrary n-number of trios. If the number n is small then obviously this can be explicitly written, but if not, then we are stuck. I assume that there is a recursive way to employ this code, but I am unsure how to go about this.
Any and all help would be appreciated.
Here is a quick attempt at a slightly simplified version of your problem.
function inner(X, f, ms)
is = [Symbol(:i_, n) for n in 1:length(ms)]
js = [Symbol(:j_, n) for n in 1:length(ms)]
fs = [:($f($i, $j)) for (i,j) in zip(is, js)]
:($X += *($(fs...)))
end
inner(:X, :g, [3,4,5]) # :(X += g(i_1, j_1) * g(i_2, j_2) * g(i_3, j_3))
function looper(ex::Expr, ms, u)
isempty(ms) && return ex
n = length(ms)
m_n = ms[n]
i_n = Symbol(:i_,n)
j_n = Symbol(:j_,n)
out = :( for $i_n in 0:min($m_n, $u)
for $j_n in 0:min($m_n, $u)
$ex
end
end)
looper(out, ms[1:end-1], u)
end
looper(inner(:g, [4,5]), [4,5], 3)
# :(for i_1 = 0:min(4, 3)
# for j_1 = 0:min(4, 3)
# for i_2 = 0:min(5, 3)
# for j_2 = 0:min(5, 3)
# X += g(i_1, j_1) * g(i_2, j_2) ...
macro multi(X, f, ms, u)
ms isa Expr && ms.head == :vect || error("expected a literal vector like [1,2,3]")
looper(inner(esc(X), esc(f), ms.args), ms.args, esc(u))
end
#macroexpand #multi X f [4,5] u
# :(for var"#74#i_1" = 0:Main.min(4, u) ...
function multi(X, f, u)
#multi X f [4,5] u
X
end
multi(0.0, atan, 1)
Given that:
Y = 0.299R + 0.587G + 0.114B
What values do we put in for R,G, and B? I’m assuming 0-255. For arguments sake, if R, G, B are each 50, then does it mean Y=0.299(50) + 0.587(500) + 0.11(50)?
The next two are also confusing. How can B - Y even be possible if Y contains Blue then isn’t B - Y just taking away itself?
Cb = 0.564( B − Y )
Cr =0.713(R−Y)
It's just simple (confusing) math ...
Remark: There are few standards of YCbCr following formula applies BT.601, with "full range":
Equation (1): Y = 0.299R + 0.587G + 0.114B
The common definition of YCbCr assumes that R, G, and B are 8 bits unsigned integers in range [0, 255].
There are cases where R, G, B are floating point values in range [0, 1] (normalized values).
There are also HDR cases where range is [0, 1023] for example.
In case R=50, G=50, B=50, you just need to assign the values:
Y = 0.299*50 + 0.587*50 + 0.114*50
Result: Y = 50.
Since Y represents the Luma (line luminescence), and RGB=(50,50,50), is a gray pixel, it does make sense that Y = 50.
The following equations Cb = 0.564(B - Y), Cr = 0.713(R - Y) are incorrect.
Instead of Cb, and Cr they should be named Pb and Pr.
Equation (2): Pb = 0.564(B - Y)
Equation (3): Pr = 0.713(R - Y)
The equations mean that you can compute Y first, and then use the result for computing Pb and Pr.
Remark: don't round the value of Y when you are using it for computing Pb and Pr.
You can also assign Equation (1) in (2) and (3):
Pb = 0.564(B - Y) = 0.564(B - (0.299R + 0.587G + 0.114B)) = 0.4997*B - 0.3311*G - 0.1686*R
Pr = 0.713(R - Y) = 0.713(R - (0.299R + 0.587G + 0.114B)) = 0.4998*R - 0.4185*G - 0.0813*B
There are some small inaccuracies.
Wikipedia is more accurate (but still just a result of mathematical assignments):
Y = 0.299*R + 0.587*G + 0.114*B
Pb = -0.168736*R - 0.331264*G + 0.5*B
Pr = 0.5*R - 0.418688*G - 0.081312*B
In the above formulas the range of Pb, Pr is [-127.5, 127.5].
In the "full range" formula of YCbCr (not YPbPr), an offset of 128 is added to Pb and Pr (so result is always positive).
In case of 8 bits, the final result is limited to range [0, 255] and rounded.
What you're referencing is the conversion of RGB to YPbPr.
Conversion to YCbCr is as follows:
Y = 0.299 * R + 0.587 * G + 0.114 * B
Cb = -0.167 * R - 0.3313 * G - 0.5 * B + 128
Cr = 0.5 * R - 0.4187 * G - 0.0813 * B + 128
Yours is YPbPr (which is better for JPEG Compression, see below):
delta = 0.5
Y = 0.299 * R + 0.587 * G + 0.114 * B (same as above)
Pb: (B - Y) * 0.564 + delta
Pr: (B - Y) * 0.713 + delta
The above answer did a better job of explaining this.
I've been looking into JPEG Compression for implementation in Pytorch and found this thread (https://photo.stackexchange.com/a/8357) useful to explain why we use YPbPr for compression over YCbCr.
Pb and Pr versions are better for image compression because the luminance information (which contains the most detail) is retained in only one (Y) channel, while Pb and Pr would contain the chrominance information. Thus when you're doing down-sampling later down the line, there's less loss of valuable info.
There must be an easier way to do this, optimization method is also welcome. I have an array 'Y' and many parameters that has to be adjusted such that Y nears zero (= 'X') as given in the MWE. Is there a much better procedure to minimize this difference? This is just an example equation, there can be 6 coefficients to optimized.
x = zeros(10,1)
y = rand(10,1)
for a=1:0.1:4
for b=2:0.1:5
for c = 3:0.1:6
z = (a * y .^ 3 + b * y + c) - x
if -1<= range(z) <= 1
a, b, c
break
end
end
end
end
I believe
p = polyfit(y,x,2);
is what you are looking for.
where p will be an array of your [a, b, c] coefficients.
I am trying to create two data sets, one which summarizes data by 2 groups which I have done using the following code:
x = rnorm(1:100)
g1 = sample(LETTERS[1:3], 100, replace = TRUE)
g2 = sample(LETTERS[24:26], 100, replace = TRUE)
aggregate(x, list(g1, g2), mean)
The second needs to summarize the data by the first group and NOT the second group.
If we consider the possible pairs from the previous example:
A - X B - X C - X
A - Y B - Y C - Y
A - Z B - Z C - Z
The second dataset should to summarize the data as the average of the outgroup.
A - not X
A - not Y
A - not Z etc.
Is there a way to manipulate aggregate functions in R to achieve this?
Or I also thought there could be dummy variable that could represent the data in this way, although I am unsure how it would look.
I have found this answer here:
R using aggregate to find a function (mean) for "all other"
I think this indicates that a dummy variable for each pairing is necessary. However if there is anyone who can offer a better or more efficient way that would be appreciated, as there are many pairings in the true data set.
Thanks in advance
First let us generate the data reproducibly (using set.seed):
# same as question but added set.seed for reproducibility
set.seed(123)
x = rnorm(1:100)
g1 = sample(LETTERS[1:3], 100, replace = TRUE)
g2 = sample(LETTERS[24:26], 100, replace = TRUE)
Now we have two solutions both of which use aggregate:
1) ave
# x equals the sums over the groups and n equals the counts
ag = cbind(aggregate(x, list(g1, g2), sum),
n = aggregate(x, list(g1, g2), length)[, 3])
ave.not <- function(x, g) ave(x, g, FUN = sum) - x
transform(ag,
x = NULL, # don't need x any more
n = NULL, # don't need n any more
mean = x/n,
mean.not = ave.not(x, Group.1) / ave.not(n, Group.1)
)
This gives:
Group.1 Group.2 mean mean.not
1 A X 0.3155084 -0.091898832
2 B X -0.1789730 0.332544353
3 C X 0.1976471 0.014282465
4 A Y -0.3644116 0.236706489
5 B Y 0.2452157 0.099240545
6 C Y -0.1630036 0.179833987
7 A Z 0.1579046 -0.009670734
8 B Z 0.4392794 0.033121335
9 C Z 0.1620209 0.033714943
To double check the first value under mean and under mean.not:
> mean(x[g1 == "A" & g2 == "X"])
[1] 0.3155084
> mean(x[g1 == "A" & g2 != "X"])
[1] -0.09189883
2) sapply Here is a second approach which gives the same answer:
ag <- aggregate(list(mean = x), list(g1, g2), mean)
f <- function(i) mean(x[g1 == ag$Group.1[i] & g2 != ag$Group.2[i]]))
ag$mean.not = sapply(1:nrow(ag), f)
ag
REVISED Revised based on comments by poster, added a second approach and also made some minor improvements.