Mixing
Romberg-Integration relative error
$begingroup$
How can I check if the relative error of two successive diagonal elements is smaller than e.g. $10^{-3}$?
$leftvert frac{T_{1,2}-T_{1,3}}{T_{1,3}}rightvert<0.001$
for a Romberg Tableau of this form
$begin{array}{cccccc}
T_{1,1}\
&backslash\
T_{2,2}&-&T_{1,2}\
&backslash&&backslash\
T_{3,3}&-&T_{2,3}&-&T_{1,3}\
&backslash&&backslash&&backslash\
end{array}$
I am using for $h=frac{b-a}{N_i}$ with $a$, $b$ as integral limits and $N_i=2^i$, $i=1,...$ as Romberg sequence.
I am using the trapezoidal sum to compute
$T_{i,1}=T(h_i)=frac{h_i}{2}left(f(a)+f(b)+sum_{j=1}^{N_i-1}f(a+jcdot h_i)right)$
and all other elements are computed with the following formula:
$T_{(j,j+k)}(f)=T_{(j+1,j+k)}(f)+frac{T_{(j+1,k+1)}(f)-T_{(j,j+k-1)}(f)}{left(frac{h_j}{h_{j+k}}right)^2-1}$
Thank you in advance.
integration numerical-methods numerical-calculus
asked Jan 26 at 15:43
baxbearbaxbear
398
$endgroup$
add a comment |
$begingroup$
How can I check if the relative error of two successive diagonal elements is smaller than e.g. $10^{-3}$?
$leftvert frac{T_{1,2}-T_{1,3}}{T_{1,3}}rightvert<0.001$
for a Romberg Tableau of this form
$begin{array}{cccccc}
T_{1,1}\
&backslash\
T_{2,2}&-&T_{1,2}\
&backslash&&backslash\
T_{3,3}&-&T_{2,3}&-&T_{1,3}\
&backslash&&backslash&&backslash\
end{array}$
I am using for $h=frac{b-a}{N_i}$ with $a$, $b$ as integral limits and $N_i=2^i$, $i=1,...$ as Romberg sequence.
I am using the trapezoidal sum to compute
$T_{i,1}=T(h_i)=frac{h_i}{2}left(f(a)+f(b)+sum_{j=1}^{N_i-1}f(a+jcdot h_i)right)$
and all other elements are computed with the following formula:
$T_{(j,j+k)}(f)=T_{(j+1,j+k)}(f)+frac{T_{(j+1,k+1)}(f)-T_{(j,j+k-1)}(f)}{left(frac{h_j}{h_{j+k}}right)^2-1}$
Thank you in advance.
integration numerical-methods numerical-calculus
asked Jan 26 at 15:43
baxbearbaxbear
398
$endgroup$
$begingroup$
What are the trapezoidal and what the Simpson sums? Are these the diagonals? Is the difference that you want to estimate between trapezoidal sums?
$endgroup$
– LutzL
Jan 26 at 17:15
$begingroup$
I think/hope I added all the missing information.
$endgroup$
– baxbear
Jan 27 at 2:12
add a comment |
$begingroup$
How can I check if the relative error of two successive diagonal elements is smaller than e.g. $10^{-3}$?
$leftvert frac{T_{1,2}-T_{1,3}}{T_{1,3}}rightvert<0.001$
for a Romberg Tableau of this form
$begin{array}{cccccc}
T_{1,1}\
&backslash\
T_{2,2}&-&T_{1,2}\
&backslash&&backslash\
T_{3,3}&-&T_{2,3}&-&T_{1,3}\
&backslash&&backslash&&backslash\
end{array}$
I am using for $h=frac{b-a}{N_i}$ with $a$, $b$ as integral limits and $N_i=2^i$, $i=1,...$ as Romberg sequence.
I am using the trapezoidal sum to compute
$T_{i,1}=T(h_i)=frac{h_i}{2}left(f(a)+f(b)+sum_{j=1}^{N_i-1}f(a+jcdot h_i)right)$
and all other elements are computed with the following formula:
$T_{(j,j+k)}(f)=T_{(j+1,j+k)}(f)+frac{T_{(j+1,k+1)}(f)-T_{(j,j+k-1)}(f)}{left(frac{h_j}{h_{j+k}}right)^2-1}$
Thank you in advance.
integration numerical-methods numerical-calculus
asked Jan 26 at 15:43
baxbearbaxbear
398
$endgroup$
How can I check if the relative error of two successive diagonal elements is smaller than e.g. $10^{-3}$?
$leftvert frac{T_{1,2}-T_{1,3}}{T_{1,3}}rightvert<0.001$
for a Romberg Tableau of this form
$begin{array}{cccccc}
T_{1,1}\
&backslash\
T_{2,2}&-&T_{1,2}\
&backslash&&backslash\
T_{3,3}&-&T_{2,3}&-&T_{1,3}\
&backslash&&backslash&&backslash\
end{array}$
I am using for $h=frac{b-a}{N_i}$ with $a$, $b$ as integral limits and $N_i=2^i$, $i=1,...$ as Romberg sequence.
I am using the trapezoidal sum to compute
$T_{i,1}=T(h_i)=frac{h_i}{2}left(f(a)+f(b)+sum_{j=1}^{N_i-1}f(a+jcdot h_i)right)$
and all other elements are computed with the following formula:
$T_{(j,j+k)}(f)=T_{(j+1,j+k)}(f)+frac{T_{(j+1,k+1)}(f)-T_{(j,j+k-1)}(f)}{left(frac{h_j}{h_{j+k}}right)^2-1}$
Thank you in advance.
integration numerical-methods numerical-calculus
integration numerical-methods numerical-calculus
asked Jan 26 at 15:43
baxbearbaxbear
398
asked Jan 26 at 15:43
baxbearbaxbear
398
edited Jan 27 at 2:12
baxbear
asked Jan 26 at 15:43
baxbearbaxbear
398
asked Jan 26 at 15:43
baxbearbaxbear
398
asked Jan 26 at 15:43
baxbearbaxbear
398
398
$begingroup$
What are the trapezoidal and what the Simpson sums? Are these the diagonals? Is the difference that you want to estimate between trapezoidal sums?
$endgroup$
– LutzL
Jan 26 at 17:15
$begingroup$
I think/hope I added all the missing information.
$endgroup$
– baxbear
Jan 27 at 2:12
add a comment |
$begingroup$
What are the trapezoidal and what the Simpson sums? Are these the diagonals? Is the difference that you want to estimate between trapezoidal sums?
$endgroup$
– LutzL
Jan 26 at 17:15
$begingroup$
I think/hope I added all the missing information.
$endgroup$
– baxbear
Jan 27 at 2:12
$begingroup$
What are the trapezoidal and what the Simpson sums? Are these the diagonals? Is the difference that you want to estimate between trapezoidal sums?
$endgroup$
– LutzL
Jan 26 at 17:15
$begingroup$
What are the trapezoidal and what the Simpson sums? Are these the diagonals? Is the difference that you want to estimate between trapezoidal sums?
$endgroup$
– LutzL
Jan 26 at 17:15
$begingroup$
I think/hope I added all the missing information.
$endgroup$
– baxbear
Jan 27 at 2:12
$begingroup$
I think/hope I added all the missing information.
$endgroup$
– baxbear
Jan 27 at 2:12
add a comment |
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$begingroup$
What are the trapezoidal and what the Simpson sums? Are these the diagonals? Is the difference that you want to estimate between trapezoidal sums?
$endgroup$
– LutzL
Jan 26 at 17:15
$begingroup$
I think/hope I added all the missing information.
$endgroup$
– baxbear
Jan 27 at 2:12