[math-fun] peculiar 2x2 products
We're used to evaluating an infinite continued fraction as a limit of ratios of elements of a running product of 2x2 matrices, while the actual elements of the product blow up. E.g., for 4 arctan(1) In[1153]:= FoldList[Dot, {{0, 4}, {1, 0}}, Table[{{2*k - 1, k^2}, {1, 0}}, {k, 9}]] Out[1153]= {{{0, 4}, {1, 0}}, {{4, 0}, {1, 1}}, {{12, 16}, {4, 4}}, {{76, 108}, {24, 36}}, {{640, 1216}, {204, 384}}, {{6976, 16000}, {2220, 5100}}, {{92736, 251136}, {29520, 79920}}, {{1456704, 4544064}, {463680, 1446480}}, {{26394624, 93229056}, {8401680, 29675520}}, {{541937664, 2137964544}, {172504080, 680536080}}} In[1154]:= N[Divide @@ # & /@ Rest[%]] Out[1154]= {{4., 0.}, {3., 4.}, {3.16667, 3.}, {3.13725, 3.16667}, {3.14234, 3.13725}, {3.14146, 3.14234}, {3.14161, 3.14146}, {3.14159, 3.14161}, {3.14159, 3.14159}} But here's an infinite matrix product where the elements converge individually: Product[MatrixForm[{{(3*a^2 - 3*a*j + j^2)/(j*(-1 - 3*a + b + j)), a^3/(j*(-1 - 3*a + b + j))}, {1/j, 1}}], {j, b, ∞}] == MatrixForm[{{"?", (a^3*HypergeometricPFQ[{1 + a, 1 + a, 1 + a}, {1 + b, 1 + b}, 1])/b^2}, {"?", HypergeometricPFQ[{a, a, a}, {b, b}, 1]}}] (Product means matrix product, not Mathematica's product.) The ?s converge, too, but I don't know to what. Note that the determinant ≠ 1, but approaches 1 in a very particular way. It's not obvious to me how to scale a matrix to make its running product converge elementwise. I actually have an elementwise convergent 2x2 product for the general 3F2[1], but idiot Mathematica 8.04 takes (literally?) forever to convert a simple sum of Pochhammers to a 3F2. --rwg
On Mon, Jul 23, 2012 at 12:01 AM, Bill Gosper <billgosper@gmail.com> wrote:
We're used to evaluating an infinite continued fraction as a limit of ratios of elements of a running product of 2x2 matrices, while the actual elements of the product blow up. E.g., for 4 arctan(1)
In[1153]:= FoldList[Dot, {{0, 4}, {1, 0}}, Table[{{2*k - 1, k^2}, {1, 0}}, {k, 9}]]
Out[1153]= {{{0, 4}, {1, 0}}, {{4, 0}, {1, 1}}, {{12, 16}, {4, 4}}, {{76, 108}, {24, 36}}, {{640, 1216}, {204, 384}}, {{6976, 16000}, {2220, 5100}}, {{92736, 251136}, {29520, 79920}}, {{1456704, 4544064}, {463680, 1446480}}, {{26394624, 93229056}, {8401680, 29675520}}, {{541937664, 2137964544}, {172504080, 680536080}}}
In[1154]:= N[Divide @@ # & /@ Rest[%]]
Out[1154]= {{4., 0.}, {3., 4.}, {3.16667, 3.}, {3.13725, 3.16667}, {3.14234, 3.13725}, {3.14146, 3.14234}, {3.14161, 3.14146}, {3.14159, 3.14161}, {3.14159, 3.14159}}
But here's an infinite matrix product where the elements converge individually:
Product[MatrixForm[{{(3*a^2 - 3*a*j + j^2)/(j*(-1 - 3*a + b + j)), a^3/(j*(-1 - 3*a + b + j))}, {1/j, 1}}], {j, b, ∞}] == MatrixForm[{{"?", (a^3*HypergeometricPFQ[{1 + a, 1 + a, 1 + a}, {1 + b, 1 + b}, 1])/b^2}, {"?", HypergeometricPFQ[{a, a, a}, {b, b}, 1]}}]
(Product means matrix product, not Mathematica's product.) The ?s converge, too, but I don't know to what. Note that the determinant ≠ 1, but approaches 1 in a very particular way. It's not obvious to me how to scale a matrix to make its running product converge elementwise.
I actually have an elementwise convergent 2x2 product for the general 3F2[1], but idiot Mathematica 8.04 takes (literally?) forever to convert a simple sum of Pochhammers to a 3F2. --rwg
I gave up waiting and simply guessed the general case, and while numerically testing, guessed the "?"s: MProd[MatrixForm[{{((-a)*(b + c - j) + (b - j)*(-c + j))/ ((1 + a + b + c - d - j)*j), -((a*b*c)/((1 + a + b + c - d - j)*j))}, {1/j, 1}}], {j, e, ∞}] == MatrixForm[ {{(a*b*c*HypergeometricPFQ[{1 + a, 1 + b, 1 + c}, {1 + d, 1 + e}, 1])/((-1 + d)*d*e), (a*b*c*HypergeometricPFQ[{1 + a, 1 + b, 1 + c}, {1 + d, 1 + e}, 1])/(d*e)}, {HypergeometricPFQ[{a, b, c}, {d, e}, 1]/(-1 + d), HypergeometricPFQ[{a, b, c}, {d, e}, 1]}}] I.e., the "?"s were just the right column divided by d-1. So we "only" get two useful results instead of four. But from a 2x2! Except the convergence is lousy. I'm working on it. Here are the two Kochflake ring<http://gosper.org/kocharmor-2-8-colored.png> pictures <http://gosper.org/kocharmor-2-8-coloredmore.png> Neil withheld because he was embarrassed about the colors, which were hastily grabbed from a Paint palate for the sole purpose of visualizing the recurrence relations. It looks to me like the boundary dimension is reducing dramatically--possibly to 1?! --rwg I think during an earlier discussion here of the necessity of algebra instruction I mentioned a childhood memory of my great aunt recounting her struggles filling out a New York State tax form after her carpenter husband neglected to charge a customer some prescribed sales tax. She was trying to compute by successive approximations what pre-tax principal he would have to charge so that the added tax totaled what he actually charged. Ironically, I had already read her sister's (my grandmother's) highschool algebra text (which I found in a carton), and could have easily solved her problem. But I said nothing, because I could tell from her tone that she knew I knew. The book was a plain green hardcover containing zero illustrations. I doubt I would have learned anything at all from one of those ridiculous damned picture books foisted on the current generation. It must come as a shock when today's college freshmen first open their texts and find only text.
On Fri, Aug 10, 2012 at 2:39 AM, Bill Gosper <billgosper@gmail.com> wrote:
On Mon, Jul 23, 2012 at 12:01 AM, Bill Gosper <billgosper@gmail.com>wrote:
We're used to evaluating an infinite continued fraction as a limit of ratios of elements of a running product of 2x2 matrices, while the actual elements of the product blow up. E.g., for 4 arctan(1)
In[1153]:= FoldList[Dot, {{0, 4}, {1, 0}}, Table[{{2*k - 1, k^2}, {1, 0}}, {k, 9}]]
Out[1153]= {{{0, 4}, {1, 0}}, {{4, 0}, {1, 1}}, {{12, 16}, {4, 4}}, {{76, 108}, {24, 36}}, {{640, 1216}, {204, 384}}, {{6976, 16000}, {2220, 5100}}, {{92736, 251136}, {29520, 79920}}, {{1456704, 4544064}, {463680, 1446480}}, {{26394624, 93229056}, {8401680, 29675520}}, {{541937664, 2137964544}, {172504080, 680536080}}}
In[1154]:= N[Divide @@ # & /@ Rest[%]]
Out[1154]= {{4., 0.}, {3., 4.}, {3.16667, 3.}, {3.13725, 3.16667}, {3.14234, 3.13725}, {3.14146, 3.14234}, {3.14161, 3.14146}, {3.14159, 3.14161}, {3.14159, 3.14159}}
But here's an infinite matrix product where the elements converge individually:
Product[MatrixForm[{{(3*a^2 - 3*a*j + j^2)/(j*(-1 - 3*a + b + j)), a^3/(j*(-1 - 3*a + b + j))}, {1/j, 1}}], {j, b, ∞}] == MatrixForm[{{"?", (a^3*HypergeometricPFQ[{1 + a, 1 + a, 1 + a}, {1 + b, 1 + b}, 1])/b^2}, {"?", HypergeometricPFQ[{a, a, a}, {b, b}, 1]}}]
(Product means matrix product, not Mathematica's product.) The ?s converge, too, but I don't know to what. Note that the determinant ≠ 1, but approaches 1 in a very particular way. It's not obvious to me how to scale a matrix to make its running product converge elementwise.
I actually have an elementwise convergent 2x2 product for the general 3F2[1], but idiot Mathematica 8.04 takes (literally?) forever to convert a simple sum of Pochhammers to a 3F2. --rwg
I gave up waiting and simply guessed the general case, and while numerically testing, guessed the "?"s:
I.e., the right column can be proven by path invariance, but the left is conjectural.
MProd[MatrixForm[{{((-a)*(b + c - j) + (b - j)*(-c + j))/ ((1 + a + b + c - d - j)*j), -((a*b*c)/((1 + a + b + c - d - j)*j))}, {1/j, 1}}], {j, e, ∞}] == MatrixForm[ {{(a*b*c*HypergeometricPFQ[{1 + a, 1 + b, 1 + c}, {1 + d, 1 + e}, 1])/((-1 + d)*d*e), (a*b*c*HypergeometricPFQ[{1 + a, 1 + b, 1 + c}, {1 + d, 1 + e}, 1])/(d*e)}, {HypergeometricPFQ[{a, b, c}, {d, e}, 1]/(-1 + d), HypergeometricPFQ[{a, b, c}, {d, e}, 1]}}]
I.e., the "?"s were just the right column divided by d-1. So we "only" get two useful results instead of four. But from a 2x2! Except the convergence is lousy. I'm working on it.
Fail. It appears impossible to rapidly compute two pFqs with a 2x2. They seem always to converge like 1/j. Even 2F1[z]! MProd[MatrixForm[{{-(((a + b - j)*z)/(j*(-1 + z))), (a*b*z)/(j - j*z)}, {1/j, 1}}], {j, e, ∞}] == MatrixForm[{{0, (a*b*z*Hypergeometric2F1[1 + a, 1 + b, 1 + e, z])/e}, {0, HypergeometricPFQ[{a, b}, {e}, z]}}] Other attempts to accelerate convergence were messy, producing, e.g. a slow octic/cubic CF.
Here are the two Kochflake ring<http://gosper.org/kocharmor-2-8-colored.png> pictures <http://gosper.org/kocharmor-2-8-coloredmore.png> Neil withheld because he was embarrassed about the colors, which were hastily grabbed from a Paint palate for the sole purpose of visualizing the recurrence relations.
No, he didn't mouthpaint. Duh, I meant palette.
It looks to me like the boundary dimension is reducing
(confirmed by Julian based on In[125]:= N[Max[Abs[Eigenvalues[{{0, 0, 0, -2}, {1, 0, 0, 2}, {0, 1, 0, 3}, {0, 0, 1, 0}}]]], 30] Out[125]= 1.87018472683652509239367973786 )
dramatically--possibly to 1?! --rwg
Also, Julian has discovered how pentaflakes reproduce<http://gosper.org/pentahouse.png>! Tsk. --rwg
I think during an earlier discussion here of the necessity of algebra instruction I mentioned a childhood memory of my great aunt recounting her struggles filling out a New York State tax form after her carpenter husband neglected to charge a customer some prescribed sales tax. She was trying to compute by successive approximations what pre-tax principal he would have to charge so that the added tax totaled what he actually charged. Ironically, I had already read her sister's (my grandmother's) highschool algebra text (which I found in a carton), and could have easily solved her problem. But I said nothing, because I could tell from her tone that she knew I knew.
The book was a plain green hardcover containing zero illustrations. I doubt I would have learned anything at all from one of those ridiculous damned picture books foisted on the current generation.
My eavesdropping teacher-cousin calls them hardcover comic books.
It must come as a shock when today's college freshmen first open their texts and find only text.
On Sun, Aug 12, 2012 at 6:40 PM, Bill Gosper <billgosper@gmail.com> wrote:
On Fri, Aug 10, 2012 at 2:39 AM, Bill Gosper <billgosper@gmail.com> wrote:
On Mon, Jul 23, 2012 at 12:01 AM, Bill Gosper <billgosper@gmail.com>wrote:
We're used to evaluating an infinite continued fraction as a limit of ratios of elements of a running product of 2x2 matrices, while the actual elements of the product blow up. E.g., for 4 arctan(1)
In[1153]:= FoldList[Dot, {{0, 4}, {1, 0}}, Table[{{2*k - 1, k^2}, {1, 0}}, {k, 9}]]
Out[1153]= {{{0, 4}, {1, 0}}, {{4, 0}, {1, 1}}, {{12, 16}, {4, 4}}, {{76, 108}, {24, 36}}, {{640, 1216}, {204, 384}}, {{6976, 16000}, {2220, 5100}}, {{92736, 251136}, {29520, 79920}}, {{1456704, 4544064}, {463680, 1446480}}, {{26394624, 93229056}, {8401680, 29675520}}, {{541937664, 2137964544}, {172504080, 680536080}}}
In[1154]:= N[Divide @@ # & /@ Rest[%]]
Out[1154]= {{4., 0.}, {3., 4.}, {3.16667, 3.}, {3.13725, 3.16667}, {3.14234, 3.13725}, {3.14146, 3.14234}, {3.14161, 3.14146}, {3.14159, 3.14161}, {3.14159, 3.14159}}
But here's an infinite matrix product where the elements converge individually:
Product[MatrixForm[{{(3*a^2 - 3*a*j + j^2)/(j*(-1 - 3*a + b + j)), a^3/(j*(-1 - 3*a + b + j))}, {1/j, 1}}], {j, b, ∞}] == MatrixForm[{{"?", (a^3*HypergeometricPFQ[{1 + a, 1 + a, 1 + a}, {1 + b, 1 + b}, 1])/b^2}, {"?", HypergeometricPFQ[{a, a, a}, {b, b}, 1]}}]
(Product means matrix product, not Mathematica's product.) The ?s converge, too, but I don't know to what. Note that the determinant ≠ 1, but approaches 1 in a very particular way. It's not obvious to me how to scale a matrix to make its running product converge elementwise.
I actually have an elementwise convergent 2x2 product for the general 3F2[1], but idiot Mathematica 8.04 takes (literally?) forever to convert a simple sum of Pochhammers to a 3F2. --rwg
I gave up waiting and simply guessed the general case, and while numerically testing, guessed the "?"s:
I.e., the right column can be proven by path invariance, but the left is conjectural.
MProd[MatrixForm[{{((-a)*(b + c - j) + (b - j)*(-c + j))/ ((1 + a + b + c - d - j)*j), -((a*b*c)/((1 + a + b + c - d - j)*j))}, {1/j, 1}}], {j, e, ∞}] == MatrixForm[ {{(a*b*c*HypergeometricPFQ[{1 + a, 1 + b, 1 + c}, {1 + d, 1 + e}, 1])/((-1 + d)*d*e), (a*b*c*HypergeometricPFQ[{1 + a, 1 + b, 1 + c}, {1 + d, 1 + e}, 1])/(d*e)}, {HypergeometricPFQ[{a, b, c}, {d, e}, 1]/(-1 + d), HypergeometricPFQ[{a, b, c}, {d, e}, 1]}}]
I.e., the "?"s were just the right column divided by d-1. So we "only" get two useful results instead of four. But from a 2x2! Except the convergence is lousy. I'm working on it.
Fail. It appears impossible to rapidly compute two pFqs with a 2x2. They seem always to converge like 1/j. Even 2F1[z]!
MProd[MatrixForm[{{-(((a + b - j)*z)/(j*(-1 + z))), (a*b*z)/(j - j*z)}, {1/j, 1}}], {j, e, ∞}] == MatrixForm[{{0, (a*b*z*Hypergeometric2F1[1 + a, 1 + b, 1 + e, z])/e}, {0, HypergeometricPFQ[{a, b}, {e}, z]}}]
Other attempts to accelerate convergence were messy, producing, e.g. a slow octic/cubic CF.
Here's one that empirically converges like kmax^-2: mprd[{{((-a)*b*c + (((a - k)*(b - k) + c*(a + b - k))* ((a + d - e - k)*(b + d - e - k) + c*(a + b + d - e - k)))/(a + b + c + d - e - 2*k))/ ((1 + a + b + c + d - e - 2*k)*k*(-d + e + k)), (a*b*c*(c*(d - e) + (-d + e)^2 + b*(c + d - e - 2*k) + a*(b + c + d - e - 2*k) - 2*c*k + 3*(-d + e)*k + 3*k^2))/ ((a + b + c + d - e - 2*k)*(1 + a + b + c + d - e - 2*k)* k*(-d + e + k))}, {(1 + ((-a)*(b + c - k) + (b - k)*(-c + k))/ ((a + b + c + d - e - 2*k)*(-d + e + k)))/k, 1 - (a*b*c)/((a + b + c + d - e - 2*k)*k*(-d + e + k))}}, {k, d,∞}] -> {{"?1", (a*b*c*HypergeometricPFQ[{1 + a, 1 + b, 1 + c}, {1 + d, 1 + e}, 1])/(d*e)}, {"?2", HypergeometricPFQ[{a, b, c}, {d, e}, 1]}} ?1 and ?2 will be fairly easy perturbations of the right column, provable I think. You might argue that this matrix is twice as complicated as the jmax^-1 one, so it's like taking twice as many terms. Indeed this would be a wash for positive convergence rates, but twice as many terms for jmax^-1 buys just one more bit! --rwg
Here are the two Kochflake ring<http://gosper.org/kocharmor-2-8-colored.png> pictures <http://gosper.org/kocharmor-2-8-coloredmore.png> Neil withheld because he was embarrassed about the colors, which were hastily grabbed from a Paint palate for the sole purpose of visualizing the recurrence relations.
No, he didn't mouthpaint. Duh, I meant palette.
It looks to me like the boundary dimension is reducing
(confirmed by Julian based on
In[125]:= N[Max[Abs[Eigenvalues[{{0, 0, 0, -2}, {1, 0, 0, 2}, {0, 1, 0, 3}, {0, 0, 1, 0}}]]], 30]
Out[125]= 1.87018472683652509239367973786 )
dramatically--possibly to 1?! --rwg
Also, Julian has discovered how pentaflakes reproduce<http://gosper.org/pentahouse.png>! Tsk. --rwg
I think during an earlier discussion here of the necessity of algebra instruction I mentioned a childhood memory of my great aunt recounting her struggles filling out a New York State tax form after her carpenter husband neglected to charge a customer some prescribed sales tax. She was trying to compute by successive approximations what pre-tax principal he would have to charge so that the added tax totaled what he actually charged. Ironically, I had already read her sister's (my grandmother's) highschool algebra text (which I found in a carton), and could have easily solved her problem. But I said nothing, because I could tell from her tone that she knew I knew.
The book was a plain green hardcover containing zero illustrations. I doubt I would have learned anything at all from one of those ridiculous damned picture books foisted on the current generation.
My eavesdropping teacher-cousin calls them hardcover comic books.
It must come as a shock when today's college freshmen first open their texts and find only text.
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Bill Gosper