Mixing
approximating a decreasing function with hyperbolic functions
$begingroup$
Let $y = f(x)$, where $x, y in mathbb{R}_1$ and $f in mathcal{C}^1$ with $f'(x) le 0$.
Partition the range using $t$ points ${y_1, ldots, y_t}$ and - on each interval $[y_m, y_{m+1})$ - approximate the function $f$ by $g(x) = frac{b_m}{y}$ for some constant $b_m$, chosen so that $f(x_m) = g(x_m)$ for $x_m = f^{-1}(y_m)$.
This approach is ad hoc. If I wish to minimise the approximation error, $||g(x) - f(x)||$, is there a better way to proceed (e.g. how to choose the ${y_1, ldots, y_t}$)?
(n.b. if $b_{m+1} < b_m$, then $g$ is not a function over the full domain. I am actually interested in the inverse functions, $x = f^{-1}(y)$ and $x = g^{-1}(y)$, where this is not a problem.)
real-analysis functional-analysis approximation-theory
asked Jan 23 at 9:59
Colin RowatColin Rowat
62
$endgroup$
add a comment |
$begingroup$
Let $y = f(x)$, where $x, y in mathbb{R}_1$ and $f in mathcal{C}^1$ with $f'(x) le 0$.
Partition the range using $t$ points ${y_1, ldots, y_t}$ and - on each interval $[y_m, y_{m+1})$ - approximate the function $f$ by $g(x) = frac{b_m}{y}$ for some constant $b_m$, chosen so that $f(x_m) = g(x_m)$ for $x_m = f^{-1}(y_m)$.
This approach is ad hoc. If I wish to minimise the approximation error, $||g(x) - f(x)||$, is there a better way to proceed (e.g. how to choose the ${y_1, ldots, y_t}$)?
(n.b. if $b_{m+1} < b_m$, then $g$ is not a function over the full domain. I am actually interested in the inverse functions, $x = f^{-1}(y)$ and $x = g^{-1}(y)$, where this is not a problem.)
real-analysis functional-analysis approximation-theory
asked Jan 23 at 9:59
Colin RowatColin Rowat
62
$endgroup$
add a comment |
$begingroup$
Let $y = f(x)$, where $x, y in mathbb{R}_1$ and $f in mathcal{C}^1$ with $f'(x) le 0$.
Partition the range using $t$ points ${y_1, ldots, y_t}$ and - on each interval $[y_m, y_{m+1})$ - approximate the function $f$ by $g(x) = frac{b_m}{y}$ for some constant $b_m$, chosen so that $f(x_m) = g(x_m)$ for $x_m = f^{-1}(y_m)$.
This approach is ad hoc. If I wish to minimise the approximation error, $||g(x) - f(x)||$, is there a better way to proceed (e.g. how to choose the ${y_1, ldots, y_t}$)?
(n.b. if $b_{m+1} < b_m$, then $g$ is not a function over the full domain. I am actually interested in the inverse functions, $x = f^{-1}(y)$ and $x = g^{-1}(y)$, where this is not a problem.)
real-analysis functional-analysis approximation-theory
asked Jan 23 at 9:59
Colin RowatColin Rowat
62
$endgroup$
Let $y = f(x)$, where $x, y in mathbb{R}_1$ and $f in mathcal{C}^1$ with $f'(x) le 0$.
Partition the range using $t$ points ${y_1, ldots, y_t}$ and - on each interval $[y_m, y_{m+1})$ - approximate the function $f$ by $g(x) = frac{b_m}{y}$ for some constant $b_m$, chosen so that $f(x_m) = g(x_m)$ for $x_m = f^{-1}(y_m)$.
This approach is ad hoc. If I wish to minimise the approximation error, $||g(x) - f(x)||$, is there a better way to proceed (e.g. how to choose the ${y_1, ldots, y_t}$)?
(n.b. if $b_{m+1} < b_m$, then $g$ is not a function over the full domain. I am actually interested in the inverse functions, $x = f^{-1}(y)$ and $x = g^{-1}(y)$, where this is not a problem.)
real-analysis functional-analysis approximation-theory
real-analysis functional-analysis approximation-theory
asked Jan 23 at 9:59
Colin RowatColin Rowat
62
asked Jan 23 at 9:59
Colin RowatColin Rowat
62
edited Jan 23 at 13:51
Colin Rowat
asked Jan 23 at 9:59
Colin RowatColin Rowat
62
asked Jan 23 at 9:59
Colin RowatColin Rowat
62
asked Jan 23 at 9:59
Colin RowatColin Rowat
62
62
add a comment |
add a comment |
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