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# Tag: Accelerate

## How to Accelerate my WordPress Website?

## Is there a way to accelerate or pre-cast spells with long casting times?

## Does gaining levels accelerate?

## How to improve and accelerate this iteration 2 D algorithm?

## Accelerate nested sum

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I am using the latest version of WordPress, all of my plugins and theme are updated but it takes too long to load my website pages. Can I know through someone that how could it be resolved?

My Website: https://www.allsamsungringtones.com

Some spells have casting times of 1m (Tsunami, Conjure Elementals, etc). I am a low level caster with no access to Wish. Is there a way to prepare the spell,somehow pre-cast it the day before into an item, to be able to use it in combat in a single turn?

If I’m in a defensive position, I can combine it with Glyph of Warding. But the Glyph can’t move too far, so it won’t work if I’m storming the enemy castle and I need my trusty elementals. Spell scrolls and Rings of Spell Storing also maintain the spell’s casting time, as far as I know. Sorcerers can only quicken 1 action to 1 bonus action.

What other options are there?

In Pathfinder 2e, an adventurer goes up a level by earning 1000 XP.

If it always takes 1000 XP and adventurers are facing greater foes (who I presume are for more XP), then does it follow that going up levels takes less time the higher the level?

I implemented the algorithm descussed here in the 1 D case (when $ g$ is one variable function). Using a relaxation parameter the algorithm became very fast and converges in few iterations (n=5,6,8..) for smooth functions. After this I tried to implement it for 2 D functions, which is really what I need, but It became very slow. I think the problem is in double integration which take a long time. I tried to use Aitken method to accelerate the process but doesn’t work. Also Alex here used Gauss quadrature formula to perform the integration but still not working for me in a good way.

I want to know why exactly the algorithm is too slow in 2 D case, and If some one have any idea to improve the code. May be someone used such algorithm onece.

The basic code is as follows (may be the one from Alex is better)

`f[x_, y_] := x*(1 - x); (* The exact source term to be constructed *) nsoleq = NDSolveValue[{D[u[t, x, y], t] == D[u[t, x, y], x, x] + D[u[t, x, y], y, y] + f[x, y], u[0, x, y] == 0, u[t, x, 0] == 0, u[t, 0, y] == 0, u[t, x, 1] == 0, u[t, 1, y] == 0}, u, {t, 0, 1}, {x, 0, 1}, {y, 0, 1}]; (* nsoleq[1,x,y] is the observation used in construction of \ the source term *) (*The iteration is as follows g[0]=0 for example, g[i]=g[i-1]+a[i]*p[i] p[i] is the gradient descente of the Conjugate gradient method (we \ need to solve to pdes to calculate it) a[i] is a relaxation parameter *) g[0][x_, y_] := 0; n = 20; (* Initialization of the iteration *) Do[nsol[i] = NDSolveValue[{D[u[t, x, y], t] == D[u[t, x, y], x, x] + D[u[t, x, y], y, y] + g[i - 1][x, y], u[0, x, y] == 0, u[t, x, 0] == 0, u[t, 0, y] == 0, u[t, x, 1] == 0, u[t, 1, y] == 0}, u, {t, 0, 1}, {x, 0, 1}, {y, 0, 1}]; (* Solve the direct equation *) nasol[i] = NDSolveValue[{D[v[t, x, y], t] == D[v[t, x, y], x, x] + D[v[t, x, y], y, y], v[0, x, y] == nsol[i][1, x, y] - nsoleq[1, x, y], v[t, x, 0] == 0, v[t, 0, y] == 0, v[t, x, 1] == 0, v[t, 1, y] == 0}, v, {t, 0, 1}, {x, 0, 1}, {y, 0, 1}, Method -> {"MethodOfLines", "SpatialDiscretization" -> {"TensorProductGrid", "MinPoints" -> 5*15 + 1, "MaxPoints" -> 5*15 + 1, "DifferenceOrder" -> Automatic}}]; (* Solve the adjoint equation *) (*p[i]=N[Integrate[nasol[i][t,x,y],{t,0,1}],10]+0.000001*g[i-1][x, y]; *) b = N[Integrate[nasol[i][t, x, y], {t, 0, 1}], 10] + 0.000001*g[i - 1][x, y]; p[i] = Interpolation[ Flatten[Table[{{x, y}, b}, {x, 0, 1, .1}, {y, 0, 1, .1}], 1]]; nsol1[i] = NDSolveValue[{D[u[t, x, y], t] == D[u[t, x, y], x, x] + D[u[t, x, y], y, y] + p[i][x, y], u[0, x, y] == 0, u[t, x, 0] == 0, u[t, 0, y] == 0, u[t, x, 1] == 0, u[t, 1, y] == 0}, u, {t, 0, 1}, {x, 0, 1}, {y, 0, 1}]; (*This is for calculating the parameter of iterarion a[ i] for accelerating the convergence *) a[i] = (NIntegrate[(p[i][x, y])^2, {x, 0, 1}, {y, 0, 1}])/(N[ NIntegrate[(nsol1[i][1, x, y])^2, {x, 0, 1}, {y, 0, 1}], 10]);(*The parameter of iterarion, In this division we have many problems with NIntegrate, especially when we increase the number of iteration *) g[i] = Interpolation[ Table[{{x, y}, g[i - 1][x, y] - a[i]*(p[i][x, y])}, {x, 0, 1, .1}, {y, 0, 1, .1}]~Flatten~1]; (*Iteration *) , {i, 1, n}]; // AbsoluteTiming `

{128.52590416078385`, Null}

` {Plot3D[f[x, y], {x, 0, 1}, {y, 0, 1}, PlotLabel -> "Exact Solution"], Plot3D[g[n][x, y], {x, 0, 1}, {y, 0, 1}, PlotLabel -> "Constructed Solution", PlotRange -> All]} `

I have a expression that is shown below. Basically all tables like mxTable, bTable and bulkTable have been pre-calculated.

The original expression looks like:

Here are the definitions of tables:

`quantity[n1_, n2_, n3_, n4_] := quantity[n3, n4, n1, n2] = quantity[n1, n4, n3, n2] = quantity[n3, n2, n1, n4] = mxtable[[n1, n2, n3, n4]]*\!\( \*UnderoverscriptBox[\(\[Sum]\), \(m1 = 0\), \(n1 - 1\)]\( \*UnderoverscriptBox[\(\[Sum]\), \(m2 = 0\), \(n2 - 1\)]\( \*UnderoverscriptBox[\(\[Sum]\), \(m3 = 0\), \(n3 - 1\)]\( \*UnderoverscriptBox[\(\[Sum]\), \(m4 = 0\), \(n4 - 1\)]\((bTable[\([n1 + 2, m1 + 3]\)] bTable[\([n2 + 2, m2 + 3]\)] bTable[\([n3 + 2, m3 + 3]\)] bTable[\([n4 + 2, m4 + 3]\)] bulk[\([m1 + 1, m2 + 1, m3 + 1, m4 + 1]\)])\)\)\)\)\); mxTable = Table[1.0/ Sqrt[n1 (n1 + 1) n2 (n2 + 1) n3 (n3 + 1) n4 (n4 + 1)], {n1, 1, 50}, {n2, 1, 50}, {n3, 1, 50}, {n4, 1, 50}]; bTable = Table[Binomial[a, b], {a, 0, 250}, {b, 0, 250}]; ratioTable = Table[N[(m24 + 1)/2^(m13 + 3), 300] (-1)^(m13 + m24) (\!\( \*UnderoverscriptBox[\(\(\[Sum]\)\(\ \ \)\), \(k = 0\), \(m24 + 1\)] \(( \*FractionBox[\(1\), \(12\)] N[ \*FractionBox[\(Binomial[m13 + k + 4, k + 4]\), SuperscriptBox[\(2\), \(k\)]], 300] \((k + 4)\) \((k + 3)\) \((k + 2)\) \((k + 1)\))\)\) + \!\( \*UnderoverscriptBox[\(\(\[Sum]\)\(\ \ \)\), \(l = 0\), \(m24 + 3\)] \((N[ \*FractionBox[\(Binomial[m13 + l + 2, l + 2]\), SuperscriptBox[\(2\), \(l\)]], 300] \((l + 2)\) \((l + 1)\) \((m24 + 3)\) \((m24 + 2)\))\)\)), {m13, 0, 100}, {m24, 0, 100}]; bulkTable = Table[ bTable[[m1 + m3 + 1, m1 + 1]] bTable[[m2 + m4 + 1, m2 + 1]] (ratioTable[[m1 + m3 + 1, m2 + m4 + 1]] + ratioTable[[m2 + m4 + 1, m1 + m3 + 1]]), {m1, 0, 50}, {m2, 0, 50}, {m3, 0, 50}, {m4, 0, 50}]; `

Currently I would like to calculate a table of quantity with each argument from 1 to 50. i.e. a 50x50x50x50 table, but it takes a long time, partly because numbers in bTables are very large and also high precision is needed (with 300+ decimal places). Does anyone have a good idea how to accelerate the code, except for writing it in another language like cpp?

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