Defactoring algorithms: Difference between revisions
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This article discusses how to identify [[enfactoring]] in [[regular temperament]] [[mapping]]s and then [[defactoring|defactor]] it. | This article discusses how to identify [[enfactoring]] in [[regular temperament]] [[mapping]]s and then [[defactoring|defactor]] it. | ||
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Being enfactored tells us that it's wasteful to use this mapping. Specifically, being 2-enfactored tells us that we have 2x as many pitches as we need. Said another way, half of the pitches in our system are bringing nothing to the table, at least not in terms of approximating intervals built out of the 5-limit primes 2, 3, and 5, which is the primary goal of a temperament. | Being enfactored tells us that it's wasteful to use this mapping. Specifically, being 2-enfactored tells us that we have 2x as many pitches as we need. Said another way, half of the pitches in our system are bringing nothing to the table, at least not in terms of approximating intervals built out of the 5-limit primes 2, 3, and 5, which is the primary goal of a temperament. | ||
This is the mapping for [[5-limit]] [[24-ET]]. To be clear, we're not saying there's a major problem with 24 as an [[EDO]]. The point here is only that—if you're after a 5-limit temperament—you may as well use [[12-ET]]<ref>The mathematical term for this idea is [[Wikipedia:Surjective_function|surjective]]: if we understand a mapping as a function, we expect there to be no elements in the range of this function which no domain inputs map to.</ref>. So we would consider 24-ET to stand for 24 Equal Temperoid. | This is the mapping for [[5-limit]] [[24-ET]]. To be clear, we're not saying there's a major problem with 24 as an [[EDO]]. The point here is only that—if you're after a 5-limit temperament—you may as well use [[12-ET]]<ref group="note">The mathematical term for this idea is [[Wikipedia:Surjective_function|surjective]]: if we understand a mapping as a function, we expect there to be no elements in the range of this function which no domain inputs map to.</ref>. So we would consider 24-ET to stand for 24 Equal Temperoid. | ||
Think of it this way: because every element is even, any [[JI]] interval you'd map with with the mapping must come out as an even number of steps of 24-ET, by the definition of the dot product, and every even step of 24-ET is just a step of 12-ET. Examples: {{vector|1 -2 1}}.{{map|24 38 56}} = 24 - 76 + 56 = 4, {{vector|1 1 -1}}.{{map|24 38 56}} = 24 + 38 - 56 = 6. | Think of it this way: because every element is even, any [[JI]] interval you'd map with with the mapping must come out as an even number of steps of 24-ET, by the definition of the dot product, and every even step of 24-ET is just a step of 12-ET. Examples: {{vector|1 -2 1}}.{{map|24 38 56}} = 24 - 76 + 56 = 4, {{vector|1 1 -1}}.{{map|24 38 56}} = 24 + 38 - 56 = 6. | ||
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== Algorithms == | == Algorithms == | ||
In this section, we discuss three methods that defactors the mapping: Smith defactoring, developed by Gene Ward Smith<ref>but the name comes from a different Smith: [[Wikipedia: Henry John Stephen Smith|Henry John Stephen Smith]], for whom the [[Smith normal form]] is named, which this method uses</ref>; Pernet-Stein defactoring, described by Clément Pernet and William Stein; and column Hermite defactoring, developed by Dave and Douglas (the name comes, of course, from Hermite normal form, which it uses). | In this section, we discuss three methods that defactors the mapping: Smith defactoring, developed by Gene Ward Smith<ref group="note">but the name comes from a different Smith: [[Wikipedia: Henry John Stephen Smith|Henry John Stephen Smith]], for whom the [[Smith normal form]] is named, which this method uses</ref>; Pernet-Stein defactoring, described by Clément Pernet and William Stein; and column Hermite defactoring, developed by Dave and Douglas (the name comes, of course, from Hermite normal form, which it uses). | ||
Smith defactoring has not yet been mathematically proven to always defactor mappings, while Pernet-Stein and column Hermite defactoring have been proven. Tests Douglas ran on thousands of random mappings, however, have empirically proven that all three methods work all of the time. Pernet-Stein and column Hermite are more closely related, and so they give the exact same results as each other every time, whereas Smith defactoring sometimes gives different results; however, after taking the HNF of the results, all three do become exactly the same. | Smith defactoring has not yet been mathematically proven to always defactor mappings, while Pernet-Stein and column Hermite defactoring have been proven. Tests Douglas ran on thousands of random mappings, however, have empirically proven that all three methods work all of the time. Pernet-Stein and column Hermite are more closely related, and so they give the exact same results as each other every time, whereas Smith defactoring sometimes gives different results; however, after taking the HNF of the results, all three do become exactly the same. | ||
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Douglas argues that Column Hermite defactoring is the preferable defactoring algorithm particularly when the routine for Hermite normal form also gives a [[Wikipedia: Unimodular matrix|unimodular]] transformation matrix, such as that in [[Wikipedia: Wolfram Language|Wolfram Language]] (formerly Mathematica, see [https://www.wolfram.com/language/ site]). The following reasons are listed: | Douglas argues that Column Hermite defactoring is the preferable defactoring algorithm particularly when the routine for Hermite normal form also gives a [[Wikipedia: Unimodular matrix|unimodular]] transformation matrix, such as that in [[Wikipedia: Wolfram Language|Wolfram Language]] (formerly Mathematica, see [https://www.wolfram.com/language/ site]). The following reasons are listed: | ||
* It is computationally cheap, wasting little resources computing things irrelevant to the result<ref> | * It is computationally cheap, wasting little resources computing things irrelevant to the result<ref group="note"> | ||
Using the following code in Wolfram Language: | Using the following code in Wolfram Language: | ||
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=== Pernet-Stein defactoring === | === Pernet-Stein defactoring === | ||
This algorithm was described in the 2009 paper ''Fast Computation of HNF of Random Integer Matrices'' by Clément Pernet and William Stein.<ref>[https://www.wstein.org/papers/hnf/pernet-stein-fast_computation_of_hnf_of_random_integer_matrices.pdf Clément Pernet and William Stein. ''Fast Computation of HNF of Random Integer Matrices''. Journal of Number Theory.]</ref> | This algorithm was described in the 2009 paper ''Fast Computation of HNF of Random Integer Matrices'' by Clément Pernet and William Stein.<ref group="note">[https://www.wstein.org/papers/hnf/pernet-stein-fast_computation_of_hnf_of_random_integer_matrices.pdf Clément Pernet and William Stein. ''Fast Computation of HNF of Random Integer Matrices''. Journal of Number Theory.]</ref> | ||
For a mapping V of rank ''r'', its defactored form is given by | For a mapping V of rank ''r'', its defactored form is given by | ||
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==== Development note ==== | ==== Development note ==== | ||
At the time Dave and Douglas wrote the first draft of this article and developed column Hermite defactoring, they were unaware of this algorithm. After publicizing column Hermite defactoring, they were referred by [[Graham Breed]] to a similar method in [http://x31eq.com/temper/ Graham's popular online regular temperament tool], implemented as <code>saturate</code><ref>[https://bitbucket.org/x31eq/regular/src/9bc9b5bd8c8e0ced6223b29c3ea487719d120c43/kernel.py#lines-178 Bitbucket | x31eq / regular / kernel.py]</ref> since 2011, and which includes in its commented documentation a link to the aforementioned paper. Unable to reverse-engineer Gene Ward Smith's saturation algorithm, Graham had gone back to the same source Gene had supposedly gotten his inspiration from—the Sage software developed by William Stein, co-author of this paper—and came across this paper. Graham's implementation turned out to be much more similar to the original description by Pernet and Stein than Gene's, differing only by an additional unnecessary use of the HNF at the beginning (while Gene's, by virtue of using the Smith Normal Form, could be said to essentially use some variable number of extraneous uses of HNF). It is not clear how Gene derived his saturation algorithm from Pernet and Stein's work, however, if the fact that Dave and Douglas derived something almost identical to Pernet and Stein's algorithm from Gene's, it suggests that it is not unreasonable for development to lead someone in the opposite direction along the same path. The very close relationship between column Hermite defactoring and Pernet-Stein defactoring will be discussed shortly. | At the time Dave and Douglas wrote the first draft of this article and developed column Hermite defactoring, they were unaware of this algorithm. After publicizing column Hermite defactoring, they were referred by [[Graham Breed]] to a similar method in [http://x31eq.com/temper/ Graham's popular online regular temperament tool], implemented as <code>saturate</code><ref group="note">[https://bitbucket.org/x31eq/regular/src/9bc9b5bd8c8e0ced6223b29c3ea487719d120c43/kernel.py#lines-178 Bitbucket | x31eq / regular / kernel.py]</ref> since 2011, and which includes in its commented documentation a link to the aforementioned paper. Unable to reverse-engineer Gene Ward Smith's saturation algorithm, Graham had gone back to the same source Gene had supposedly gotten his inspiration from—the Sage software developed by William Stein, co-author of this paper—and came across this paper. Graham's implementation turned out to be much more similar to the original description by Pernet and Stein than Gene's, differing only by an additional unnecessary use of the HNF at the beginning (while Gene's, by virtue of using the Smith Normal Form, could be said to essentially use some variable number of extraneous uses of HNF). It is not clear how Gene derived his saturation algorithm from Pernet and Stein's work, however, if the fact that Dave and Douglas derived something almost identical to Pernet and Stein's algorithm from Gene's, it suggests that it is not unreasonable for development to lead someone in the opposite direction along the same path. The very close relationship between column Hermite defactoring and Pernet-Stein defactoring will be discussed shortly. | ||
It should also be noted that toward the very beginning of Dave and Douglas's effort to develop a defactoring algorithm, Thomas McMurray Price described—in a message sent to the Xenharmonic Alliance Discord server—an algorithm almost identical to the Pernet-Stein algorithm, while also still being unaware of the Pernet-Stein paper. At this time, Dave and Douglas could not understand Tom's math well enough to realize that he'd just dropped the solution in their laps. Again, it was not until column Hermite defactoring was published that Tom commented on the findings and brought his ideas back into the conversation that Dave and Douglas realized the close connection between his ideas, Pernet-Stein defactoring, and column Hermite defactoring. | It should also be noted that toward the very beginning of Dave and Douglas's effort to develop a defactoring algorithm, Thomas McMurray Price described—in a message sent to the Xenharmonic Alliance Discord server—an algorithm almost identical to the Pernet-Stein algorithm, while also still being unaware of the Pernet-Stein paper. At this time, Dave and Douglas could not understand Tom's math well enough to realize that he'd just dropped the solution in their laps. Again, it was not until column Hermite defactoring was published that Tom commented on the findings and brought his ideas back into the conversation that Dave and Douglas realized the close connection between his ideas, Pernet-Stein defactoring, and column Hermite defactoring. | ||
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So this implementation begins by finding the unimodular matrix from a column Hermite decomposition. Note that the <code>HermiteDecomposition[]</code> is only available in row-style, so we first transpose the matrix to convert it to column-style. The decomposition in Wolfram returns two items—the unimodular matrix, and the input matrix in Hermite normal form, in that order—and in this case we actually want the unimodular matrix. So we take that with <code>First[]</code>. Then we transpose it again to in effect undo the transposition we did at the beginning. | So this implementation begins by finding the unimodular matrix from a column Hermite decomposition. Note that the <code>HermiteDecomposition[]</code> is only available in row-style, so we first transpose the matrix to convert it to column-style. The decomposition in Wolfram returns two items—the unimodular matrix, and the input matrix in Hermite normal form, in that order—and in this case we actually want the unimodular matrix. So we take that with <code>First[]</code>. Then we transpose it again to in effect undo the transposition we did at the beginning. | ||
The next step is to invert that matrix, which is doable because it is unimodular; a key property of unimodular matrices is that they are always invertible, and because their determinant is | The next step is to invert that matrix, which is doable because it is unimodular; a key property of unimodular matrices is that they are always invertible, and because their determinant is ±1, if they contain all integer entries, their inverse will also contain all integer entries (which it does, and we need it to)<ref group="note">Interesting tidbit regarding [[full-rank]] matrices and unimodular matrices: for square matrices, unimodularity implies full-rank, and while full-rank does not imply unimodularity, it does imply a non-zero determinant.</ref>. | ||
Finally we take only the top ''r'' rows of this. That is found with <code>MatrixRank[m]</code>. | Finally we take only the top ''r'' rows of this. That is found with <code>MatrixRank[m]</code>. | ||
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The basic idea is that the column Hermite decomposition leaves any common row factor the mapping might have had in the HNF part, while preserving in the unimodular part everything that's still meaningful about how the mapping works. So that's why we throw the column HNF away and keep the unimodular part. The rest of the algorithm is basically just "undoing" stuff so we get back to the structure of the matrix that we input. | The basic idea is that the column Hermite decomposition leaves any common row factor the mapping might have had in the HNF part, while preserving in the unimodular part everything that's still meaningful about how the mapping works. So that's why we throw the column HNF away and keep the unimodular part. The rest of the algorithm is basically just "undoing" stuff so we get back to the structure of the matrix that we input. | ||
So inverting is one of those "undo" type operations. To understand why, we have to understand the nature of this decomposition. What the Hermite decomposition does is return a unimodular matrix U and a Hermite normal form matrix H such that if you left-multiply your original matrix A by the unimodular matrix U you get the normal form matrix H, or in other words, UA = H. So, think of it this way. If A is what we input, and we want something sort of like A, but U is what we've taken, and U is multiplied with A in this equality to get H, where H is also kind of like A, then probably what we really want is something like U, but inverted. | So inverting is one of those "undo" type operations. To understand why, we have to understand the nature of this decomposition. What the Hermite decomposition does is return a unimodular matrix U and a Hermite normal form matrix H such that if you left-multiply your original matrix A by the unimodular matrix U you get the normal form matrix H, or in other words, {{nowrap|UA {{=}} H}}. So, think of it this way. If A is what we input, and we want something sort of like A, but U is what we've taken, and U is multiplied with A in this equality to get H, where H is also kind of like A, then probably what we really want is something like U, but inverted. | ||
Finally, we take only the top <math>r</math> rows, which again is an "undo" type operation. Here what we're undoing is that we had to graduate from a rectangle to a square temporarily, storing our important information in the form of this invertible square unimodular matrix temporarily, so we could invert it while keeping it integer, but now we need to get it back into the same type of rectangular shape as we put in. So that's what this part is for.<ref>There is probably some special meaning or information in the rows you throw away here, but we're not sure what it might be.</ref> | Finally, we take only the top <math>r</math> rows, which again is an "undo" type operation. Here what we're undoing is that we had to graduate from a rectangle to a square temporarily, storing our important information in the form of this invertible square unimodular matrix temporarily, so we could invert it while keeping it integer, but now we need to get it back into the same type of rectangular shape as we put in. So that's what this part is for.<ref group="note">There is probably some special meaning or information in the rows you throw away here, but we're not sure what it might be.</ref> | ||
===== The actual defactoring conditions ===== | ===== The actual defactoring conditions ===== | ||
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===== Proof of why column Hermite defactoring (and Pernet-Stein defactoring) work ===== | ===== Proof of why column Hermite defactoring (and Pernet-Stein defactoring) work ===== | ||
The following proof is adapted primarily from Tom Price's thinking: | The following proof is adapted primarily from Tom Price's thinking: | ||
# The input matrix is an | # The input matrix is an ''m''×''n'' matrix A. | ||
# It decomposes into a slightly bigger and square ( | # It decomposes into a slightly bigger and square (''n''×''n'') unimodular matrix U and another ''m''×''n'' matrix which is not exactly A in HNF (because we only have to use unimodular operations so far as to get all the all-zero columns off to one side of A; we don't need to satisfy all of the conventional constraints of HNF), but we'll still call it H. The unimodular matrix is a transformation from A into H, so, {{nowrap|AU {{=}} H}}. | ||
# If we were to actually slice off the all-zero cols we've isolated in H, we'd end up with a slightly smaller and square ( | # If we were to actually slice off the all-zero cols we've isolated in H, we'd end up with a slightly smaller and square (''m''×''m'') matrix. So let's call this little square matrix S (this is our "[[Defactoring algorithms#Finding the greatest factor|greatest factor matrix]]", because its determinant is the greatest factor of A). | ||
# We can left-multiply both sides of our equation by the inverse of S (S{{inv}}) and right-multiply both sides of our equation by the inverse of U (U{{inv}}) to get | # We can left-multiply both sides of our equation by the inverse of S (S{{inv}}) and right-multiply both sides of our equation by the inverse of U (U{{inv}}) to get {{nowrap|S{{inv}}AUU{{inv}} {{=}} S{{inv}}HU{{inv}}}}. The U's cancel out on the left so we end up with {{nowrap|S{{inv}}A {{=}} S{{inv}}HU{{inv}}}}. At first glance we don't seem to have gained any further insight. But there's more we can do from here. | ||
# Because H is just S with a bunch of 0 cols appended, S{{inv}}H is just the identity matrix with a bunch of zero columns appended, in other words it is a truncated identity matrix. We could call that T, and now we have S{{inv}}A = TU{{inv}}. | # Because H is just S with a bunch of 0 cols appended, S{{inv}}H is just the identity matrix with a bunch of zero columns appended, in other words it is a truncated identity matrix. We could call that T, and now we have {{nowrap|S{{inv}}A {{=}} TU{{inv}}}}. | ||
# Multiplying U{{inv}} on the left by a truncated identity matrix is the same as truncating | # Multiplying U{{inv}} on the left by a truncated identity matrix is the same as truncating U{{inv}}. That's how we think of the output of column Hermite defactoring—our supposedly defactored matrix—so let's call that D. We now have {{nowrap|S{{inv}}A {{=}} D}}. (This is how we can see that the Pernet-Stein method of multiplying the input matrix by a transformation matrix that is a truncated and inversed column Hermite normal form of the input is equivalent to our column Hermite method, which takes the other route to the same result: inverting and truncating the unimodular result of the Hermite decomposition.) | ||
# We need to prove now that D has three qualities: | # We need to prove now that D has three qualities: | ||
:: a) It's defactored, | :: a) It's defactored, | ||
:: b) It still represents the same temperament (i.e. it has the same nullspace as A), and | :: b) It still represents the same temperament (i.e. it has the same nullspace as A), and | ||
:: c) It's integer. | :: c) It's an integer. | ||
# Proving (a) is easy. It's defactored because U was unimodular. U's determinant was 1, and neither inverting it nor truncating it would change that. Alternatively, we can prove this by showing how on the other side of the equation, | # Proving (a) is easy. It's defactored because U was unimodular. U's determinant was 1, and neither inverting it nor truncating it would change that. Alternatively, we can prove this by showing how on the other side of the equation, S{{inv}}A is surjective as a function on lattice points (in other words, there's no points in the tempered lattice that JI lattice points don't map to). We begin with the fact that H has the same image as A, because right-multiplication with a unimodular matrix such as U doesn't change the image. Then S has the same image as H, too, and therefore the same image as A, because removing the all-zero columns doesn't change the image either. Now that we've established this, we can assert that S{{inv}}A is surjective by describing a lattice point ''x'' such that {{nowrap|''y'' {{=}} S{{inv}}A''x''}} for any given lattice point ''y''. And because S and A have the same image, we know that {{nowrap|S''y'' {{=}} A''x''}}, and therefore {{nowrap|''y'' {{=}} S{{inv}}A''x''}}. | ||
# Proving (b) is even easier. Multiplying any matrix with an invertible matrix on the left keeps the nullspace the same. | # Proving (b) is even easier. Multiplying any matrix with an invertible matrix on the left keeps the nullspace the same. S{{inv}} is clearly invertible, being itself the inverse of S. A way to understand this is: a non-invertible matrix is the same as a singular matrix, i.e. one whose determinant is 0. So as long as you don't wipe things out by essentially multiplying by 0, the nullspace information is preserved, just scaled. | ||
# Proving (c) is a bit trickier, because | # Proving (c) is a bit trickier, because S{{inv}} is not necessarily an integer matrix. But can show that S{{inv}}A is an integer matrix by showing that it maps lattice points to lattice points. Suppose we have that same equation from the proof of (a), namely that {{nowrap|''y'' {{=}} S{{inv}}A''x''}}, where ''x'' is a lattice point. We want to show that ''y'' is a lattice point. Again, since A and S have the same image (when considered as functions on lattice points), there must be some lattice point ''z'' with {{nowrap|S''z'' {{=}} A''x''}}. But we also know that {{nowrap|S''y'' {{=}} A''x''}}. Since S is invertible, and therefore injective, {{nowrap|''y'' {{=}} ''z''}}, so ''y'' is a lattice point. | ||
===== Relationship with other defactoring methods ===== | ===== Relationship with other defactoring methods ===== | ||
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If column Hermite defactoring is described as "column-Hermite inverse take-r-rows", the analogous way to describe Smith defactoring would be like "row-Hermite column-Hermite row-Hermite column-Hermite row-Hermite column-Hermite … inverse take-r-rows". In other words, the nature of Smith decomposition is to essentially repeatedly Hermite decompose from either angle until Smith normal form has been achieved, which is wasteful and unnecessary in this context, when all that is required is a single column-Hermite decomposition. This helps explain why column Hermite defactoring is more performant in general than Smith defactoring, when code is run against thousands of examples at a time. | If column Hermite defactoring is described as "column-Hermite inverse take-r-rows", the analogous way to describe Smith defactoring would be like "row-Hermite column-Hermite row-Hermite column-Hermite row-Hermite column-Hermite … inverse take-r-rows". In other words, the nature of Smith decomposition is to essentially repeatedly Hermite decompose from either angle until Smith normal form has been achieved, which is wasteful and unnecessary in this context, when all that is required is a single column-Hermite decomposition. This helps explain why column Hermite defactoring is more performant in general than Smith defactoring, when code is run against thousands of examples at a time. | ||
According to Wolfram<ref>[https://reference.wolfram.com/language/ref/SmithDecomposition.html Wolfram Language Documentation | SmithDecomposition]</ref>, "the Smith decomposition can often be found by two iterations of the Hermite decomposition". This notion is echoed in the Sage docs<ref>[https://doc.sagemath.org/html/en/reference/matrices/sage/matrix/matrix_integer_dense.html Sage Reference Manual | Dense matrices over the integer ring]</ref>, which read, "Use Hermite normal form twice to find an invertible matrix whose inverse transforms a matrix with the same row span as self to its saturation," remembered that saturation is the same as defactoring. The reason for the multiple applications of Hermite decomposition to achieve Smith normal form is that you will not necessarily get a diagonal matrix on the first go. But as we know from Smith defactoring, the center matrix of the Smith decomposition—the Smith normal form one—is not the target matrix, so its exact form is not necessary to achieve to accomplish defactoring. | According to Wolfram<ref group="note">[https://reference.wolfram.com/language/ref/SmithDecomposition.html Wolfram Language Documentation | SmithDecomposition]</ref>, "the Smith decomposition can often be found by two iterations of the Hermite decomposition". This notion is echoed in the Sage docs<ref group="note">[https://doc.sagemath.org/html/en/reference/matrices/sage/matrix/matrix_integer_dense.html Sage Reference Manual | Dense matrices over the integer ring]</ref>, which read, "Use Hermite normal form twice to find an invertible matrix whose inverse transforms a matrix with the same row span as self to its saturation," remembered that saturation is the same as defactoring. The reason for the multiple applications of Hermite decomposition to achieve Smith normal form is that you will not necessarily get a diagonal matrix on the first go. But as we know from Smith defactoring, the center matrix of the Smith decomposition—the Smith normal form one—is not the target matrix, so its exact form is not necessary to achieve to accomplish defactoring. | ||
Column Hermite defactoring is very similar to Pernet-Stein defactoring. If you compare the set of methods that they call, they are almost identical; Pernet-Stein just uses matrix multiplication in exchange of column Hermite's concatenation and slice. | Column Hermite defactoring is very similar to Pernet-Stein defactoring. If you compare the set of methods that they call, they are almost identical; Pernet-Stein just uses matrix multiplication in exchange of column Hermite's concatenation and slice. | ||
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===== Relationship with EA ===== | ===== Relationship with EA ===== | ||
Another thought that might help congeal the notion of column Hermite defactoring for you is to use what you know about multimaps (AKA "wedgies"), in particular a) what they are, and b) how to defactor them. The answer to a) is that they are just the list of the largest possible minor determinants (or "largest-minors" for short) of rectangular matrices, or in other words, the closest thing rectangular matrices such as mappings have to a real determinant. And the answer to b) is that you simply divide out the GCD of the entries in this list of largest-minors. So if defactoring a list of largest-minors means dividing common factors out, then it should be little surprise that achieving a real determinant of | Another thought that might help congeal the notion of column Hermite defactoring for you is to use what you know about multimaps (AKA "wedgies"), in particular a) what they are, and b) how to defactor them. The answer to a) is that they are just the list of the largest possible minor determinants (or "largest-minors" for short) of rectangular matrices, or in other words, the closest thing rectangular matrices such as mappings have to a real determinant. And the answer to b) is that you simply divide out the GCD of the entries in this list of largest-minors. So if defactoring a list of largest-minors means dividing common factors out, then it should be little surprise that achieving a real determinant of ±1 is equivalent to defactoring, and thereby that leveraging the unimodularity of the other matrix produced by the Hermite decomposition should be valuable in this capacity. | ||
==== By hand ==== | ==== By hand ==== | ||
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After we know how to do these two things individually, we will learn how to tweak them and assemble them together in order to perform a complete column Hermite defactoring. | After we know how to do these two things individually, we will learn how to tweak them and assemble them together in order to perform a complete column Hermite defactoring. | ||
Fortunately, both of these two processes can be done using a technique you may already be familiar with if you have learned how to calculate the nullspace of a mapping by hand (as demonstrated | Fortunately, both of these two processes can be done using a technique you may already be familiar with if you have learned how to calculate the nullspace of a mapping by hand (as demonstrated [[Dave Keenan & Douglas Blumeyer's guide to RTT/Exploring temperaments#Nullspace|here]]): | ||
# Augmenting your matrix with an identity matrix | # Augmenting your matrix with an identity matrix | ||
# Performing elementary row or column operations until a desired state is achieved<ref>For convenience, here is a summary of the three different techniques and their targets:<br /> | # Performing elementary row or column operations until a desired state is achieved<ref group="note">For convenience, here is a summary of the three different techniques and their targets:<br /> | ||
* Nullspace: augment to the bottom, go until you get columns with all zeros. | * Nullspace: augment to the bottom, go until you get columns with all zeros. | ||
* Hermite: augment to the right, go until echelon form. | * Hermite: augment to the right, go until echelon form. | ||
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# All entries in pivot columns above the pivots ≥ 0 and strictly less than the pivot | # All entries in pivot columns above the pivots ≥ 0 and strictly less than the pivot | ||
One special thing about computing the HNF is that we're not allowed to use all elementary operations; in particular we're not allowed to multiply (or divide) rows. Our main technique, then, will be adding or subtract rows from each other. This, of course, includes adding or subtracting ''multiples'' of rows from each other, because doing so is equivalent to performing those additions or subtractions one at a time (note that adding or subtracting ''multiples'' of rows from each other is significantly different than simply ''multiplying'' a row by itself).<ref>The fact that you're not allowed to multiply or divide is equivalent to the fact that at every step along the way, the augmented matrix remains unimodular.</ref> | One special thing about computing the HNF is that we're not allowed to use all elementary operations; in particular we're not allowed to multiply (or divide) rows. Our main technique, then, will be adding or subtract rows from each other. This, of course, includes adding or subtracting ''multiples'' of rows from each other, because doing so is equivalent to performing those additions or subtractions one at a time (note that adding or subtracting ''multiples'' of rows from each other is significantly different than simply ''multiplying'' a row by itself).<ref group="note">The fact that you're not allowed to multiply or divide is equivalent to the fact that at every step along the way, the augmented matrix remains unimodular.</ref> | ||
So let's begin by subtracting the 1st row from the 2nd row, and let's do it 2 times, because we can see that would get the pivot of the 2nd row pretty close to 0, which is where we're trying to get it, per the 2nd HNF constraint above. | So let's begin by subtracting the 1st row from the 2nd row, and let's do it 2 times, because we can see that would get the pivot of the 2nd row pretty close to 0, which is where we're trying to get it, per the 2nd HNF constraint above. | ||
| Line 339: | Line 341: | ||
<math> | <math> | ||
\left[ \begin{array} {ccc|cc} | \left[ \begin{array} {ccc|cc} | ||
0 & -11 & 4 & 13 & -6 | 0 & -11 & 4 & 13 & -6 \\ | ||
2 & 5 & 4 & -2 & 1 \\ | 2 & 5 & 4 & -2 & 1 \\ | ||
\end{array} \right] | \end{array} \right] | ||
| Line 353: | Line 355: | ||
</math> | </math> | ||
We're actually quite close to done! All we need to do is flip the signs on the 2nd row. But wait, you protest! Isn't that multiplying a row by | We're actually quite close to done! All we need to do is flip the signs on the 2nd row. But wait, you protest! Isn't that multiplying a row by −1, which we specifically forbade? Well, sure, but that just shows we need to clarity what we're concerned about, which is essentially enfactoring. Multiplying by −1 does not change the GCD of the row, where multiplying by −2 or 2 would. Note that because the process for taking the HNF forbids multiplying ''or dividing'', it will never introduce enfactoring where was there was none previously, but it also does not remove enfactoring that is there. | ||
Perhaps another helpful way of thinking about this is that multiplying the row by | Perhaps another helpful way of thinking about this is that multiplying the row by −1 does not alter the potential effects this row could have being added or subtracted from other rows. It merely swaps addition and subtraction. Whereas multiplying the row by any integer with absolute value greater than 1 ''would'' affect the potential effects this row could have being added or subtracted from other rows: it would limit them. | ||
So, let's do that sign flip: | So, let's do that sign flip: | ||
| Line 368: | Line 370: | ||
And we're done! Let's confirm though. | And we're done! Let's confirm though. | ||
# '''All pivots | # '''All pivots > 0?''' Check. The 1st row's pivot is 2 and the 2nd row's pivot is 11. | ||
# '''All entries in pivot columns below the pivots {{=}} 0'''? Check. This only applies to one entry—the bottom right one, below the 1st row's pivot—but it is indeed 0. | # '''All entries in pivot columns below the pivots {{=}} 0'''? Check. This only applies to one entry—the bottom right one, below the 1st row's pivot—but it is indeed 0. | ||
# '''All entries in pivot columns above the pivots ≥ 0 and strictly less than the pivot'''? Check. Again, this only applies to one entry—the center top one, above the 2nd row's pivot of 11—but it is 5, and 5 is indeed non-negative and | # '''All entries in pivot columns above the pivots ≥ 0 and strictly less than the pivot'''? Check. Again, this only applies to one entry—the center top one, above the 2nd row's pivot of 11—but it is 5, and 5 is indeed non-negative and < 11. | ||
And so, we have performed the Hermite decomposition. The matrix to the left of the augmentation line—the one in place of our original matrix—is that original matrix in HNF: | And so, we have performed the Hermite decomposition. The matrix to the left of the augmentation line—the one in place of our original matrix—is that original matrix in HNF: | ||
| Line 420: | Line 422: | ||
<math> | <math> | ||
\left[ \begin{array} { | \left[ \begin{array} {ccc|ccc} | ||
3 & -2 & 4 | 3 & -2 & 4 & 1 & 0 & 0 \\ | ||
1 & 0 & | 1 & 0 & 2 & 0 & 1 & 0 \\ | ||
0 & 1 & 0 & 0 & 0 & 1 \\ | |||
\end{array} \right] | |||
1 | |||
0 & 1 & 0 | |||
0 & 0 & 1 \\ | |||
</math> | </math> | ||
| Line 436: | Line 434: | ||
<math> | <math> | ||
\left[ \begin{array} { | \left[ \begin{array} {ccc|ccc} | ||
1 & 0 & 2 | 1 & 0 & 2 & 0 & 1 & 0 \\ | ||
0 & 1 & 0 \\ | 0 & 1 & 0 & 0 & 0 & 1 \\ | ||
3 & -2 & 4 & 1 & 0 & 0 \\ | |||
\end{array} \right] | |||
0 & 1 | |||
1 & 0 & 0 \\ | |||
</math> | </math> | ||
Okay, now let's target the bottom-right entry. How can we make that 3 into a 0? Let's subtract the 1st row from the 3rd row 3 times: | Okay, now let's target the bottom-right entry. How can we make that 3 into a 0? Let's subtract the 1st row from the 3rd row 3 times: | ||
<math>\left[ \begin{array} { | <math> | ||
\color{blue}1 & \color{blue}0 & \color{blue}2 | \left[ \begin{array} {ccc|ccc} | ||
\color{blue}1 & \color{blue}0 & \color{blue}2 & \color{blue}0 & \color{blue}1 & \color{blue}0 \\ | |||
\color{ | 0 & 1 & 0 & 0 & 0 & 1 \\ | ||
\color{red}0 & \color{red}-2 & \color{red}-2 & \color{red}1 & \color{red}-3 & \color{red}0 \\ | |||
\color{ | \end{array} \right] | ||
\color{red}1 & \color{red}-3 & \color{red}0 \\ | |||
</math> | </math> | ||
Okay, let's next target the bottom-center entry. How can we make that | Okay, let's next target the bottom-center entry. How can we make that −2 into a 0? Let's add the 2nd row to the 3rd row 2 times: | ||
<math> | <math> | ||
\left[ \begin{array} { | \left[ \begin{array} {ccc|ccc} | ||
1 & 0 & 2 \\ | 1 & 0 & 2 & 0 & 1 & 0 \\ | ||
\color{blue}0 & \color{blue}1 & \color{blue}0 | \color{blue}0 & \color{blue}1 & \color{blue}0 & \color{blue}0 & \color{blue}0 & \color{blue}1 \\ | ||
\color{ | \color{red}0 & \color{red}0 & \color{red}-2 & \color{red}1 & \color{red}-3 & \color{red}2 \\ | ||
\end{array} \right] | |||
\color{ | |||
\color{red}1 & \color{red}-3 & \color{red}2 \\ | |||
</math> | </math> | ||
| Line 477: | Line 464: | ||
<math> | <math> | ||
\left[ \begin{array} { | \left[ \begin{array} {ccc|ccc} | ||
\color{red}1 & \color{red}0 & \color{red}0 | \color{red}1 & \color{red}0 & \color{red}0 & \color{red}1 & \color{red}-2 & \color{red}2 \\ | ||
0 & 1 & 0 & 0 & 0 & 1 \\ | |||
\color{ | \color{blue}0 & \color{blue}0 & \color{blue}-2 & \color{blue}1 & \color{blue}-3 & \color{blue}2 \\ | ||
\end{array} \right] | |||
\color{ | |||
\color{blue}1 & \color{blue}-3 & \color{blue}2 \\ | |||
</math> | </math> | ||
Finally, we just need to divide the 3rd row by | Finally, we just need to divide the 3rd row by −2. Yes, unlike with the Hermite decomposition, all elementary row operations are permitted, including multiplying or dividing rows. And in this case there's no restrictions against non-integers (which we didn't even explicitly mention when doing the HNF, but yes, HNF requires integers). So here's where we end up: | ||
<math> | <math> | ||
\left[ \begin{array} { | \left[ \begin{array} {ccc|ccc} | ||
1 & 0 & 0 \\ | 1 & 0 & 0 & 1 & -2 & 2 \\ | ||
0 & 1 & 0 \\ | 0 & 1 & 0 & 0 & 0 & 1 \\ | ||
\color{red}0 & \color{red}0 & \color{red}1 | \color{red}0 & \color{red}0 & \color{red}1 & \color{red}-\frac12 & \color{red}\frac32 & \color{red}-1 \\ | ||
\end{array} \right] | |||
\color{red}-\frac12 & \color{red}\frac32 & \color{red}-1 \\ | |||
</math> | </math> | ||
As with the Hermite decomposition, we have a convenient way to check our work at the end, which involves matrix multiplication. With Hermite, we verified that left-multiplying our original matrix by the unimodular matrix resulted in the HNF. With inversion, we verify that left-multiplying<ref>or right-multiplying, in this case; it doesn't matter</ref> our original matrix by the inverse results in the identity matrix. And indeed: | As with the Hermite decomposition, we have a convenient way to check our work at the end, which involves matrix multiplication. With Hermite, we verified that left-multiplying our original matrix by the unimodular matrix resulted in the HNF. With inversion, we verify that left-multiplying<ref group="note">or right-multiplying, in this case; it doesn't matter</ref> our original matrix by the inverse results in the identity matrix. And indeed: | ||
<math> | <math> | ||
| Line 530: | Line 509: | ||
</math> | </math> | ||
This matrix is chosen specifically to demonstrate the importance of the unimodularity of the other matrix produced by the Hermite decomposition. A unimodular matrix is defined by having a determinant of ±1. And what does this have to do with inverses? Well, take a look at the determinant of our original matrix here, {{rket|{{map|3 -2 4}} {{map|1 0 2}} {{map|0 1 0}}}}. It's 2. The determinant of an invertible matrix will tell you what the LCM of all the denominators in the inverse will be.<ref>If you're familiar with the formula for the Moore-Penrose inverse of rectangular matrices, you may recognize this fact as akin to how you multiply the outside of the pseudoinverse by the reciprocal of the determinant of the matrix.</ref><ref>This may also shed some light on the fact that the only square matrices that are not invertible are those with determinants equal to 0.</ref> And so, the fact that the other matrix produced by the Hermite decomposition is unimodular means that not only is it invertible, if it has only integer terms (which it will, being involved in HNF), then its inverse will also have only integer terms. And this is important because the inverse of a Hermite unimodular matrix is just one step away from the defactored form of an input matrix. | This matrix is chosen specifically to demonstrate the importance of the unimodularity of the other matrix produced by the Hermite decomposition. A unimodular matrix is defined by having a determinant of ±1. And what does this have to do with inverses? Well, take a look at the determinant of our original matrix here, {{rket|{{map|3 -2 4}} {{map|1 0 2}} {{map|0 1 0}}}}. It's 2. The determinant of an invertible matrix will tell you what the LCM of all the denominators in the inverse will be.<ref group="note">If you're familiar with the formula for the Moore-Penrose inverse of rectangular matrices, you may recognize this fact as akin to how you multiply the outside of the pseudoinverse by the reciprocal of the determinant of the matrix.</ref><ref group="note">This may also shed some light on the fact that the only square matrices that are not invertible are those with determinants equal to 0.</ref> And so, the fact that the other matrix produced by the Hermite decomposition is unimodular means that not only is it invertible, if it has only integer terms (which it will, being involved in HNF), then its inverse will also have only integer terms. And this is important because the inverse of a Hermite unimodular matrix is just one step away from the defactored form of an input matrix. | ||
}} | }} | ||
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\end{matrix} \right] \end{array} </math> | \end{matrix} \right] \end{array} </math> | ||
And we've completed the first step!<ref>Note that while the HNF is unique, the unimodular matrix is not. Because the 3rd row of the left matrix—the one in HNF—is all 0's, any number of that row can be added to either of the other two rows without altering the HNF at all, but with affecting the unimodular matrix on the right. Please feel free to experiment yourself, but I expect you will find that the inverse of any matrix you come up with this way, transposed, trimmed, and HNF'd, will give you the same canonical form—no need to worry about the exact path you happen to take to the HNF in the first step.</ref> The original matrix is now in HNF. So the next step is to take the other matrix we've been working on—the unimodular one from the Hermite decomposition—and invert it. Again, since we're in a transposed state, we're going to do the by-hand inversion technique, but to the bottom using elementary column operations rather than to the right using elementary row operations. | And we've completed the first step!<ref group="note">Note that while the HNF is unique, the unimodular matrix is not. Because the 3rd row of the left matrix—the one in HNF—is all 0's, any number of that row can be added to either of the other two rows without altering the HNF at all, but with affecting the unimodular matrix on the right. Please feel free to experiment yourself, but I expect you will find that the inverse of any matrix you come up with this way, transposed, trimmed, and HNF'd, will give you the same canonical form—no need to worry about the exact path you happen to take to the HNF in the first step.</ref> The original matrix is now in HNF. So the next step is to take the other matrix we've been working on—the unimodular one from the Hermite decomposition—and invert it. Again, since we're in a transposed state, we're going to do the by-hand inversion technique, but to the bottom using elementary column operations rather than to the right using elementary row operations. | ||
For our first step, let's add the 1st column to the 2nd column. That will get us a 0 in the top-center entry. Remember, we're trying to get the top-right matrix to look like an identity matrix. | For our first step, let's add the 1st column to the 2nd column. That will get us a 0 in the top-center entry. Remember, we're trying to get the top-right matrix to look like an identity matrix. | ||
| Line 710: | Line 689: | ||
0 & 11 \\ | 0 & 11 \\ | ||
0 & 0 \\ | 0 & 0 \\ | ||
\end{ | \end{matrix} \right. \\ \begin{array} {l} \\ \\ \\ \end{array} \end{array} & \rule[2.5mm]{0.375mm}{18mm} & \left. \begin{matrix} | ||
1 & \color{blue}0 & \color{red}0 \\ | 1 & \color{blue}0 & \color{red}0 \\ | ||
0 & \color{blue}-1 & \color{red}0 \\ | 0 & \color{blue}-1 & \color{red}0 \\ | ||
| Line 805: | Line 784: | ||
\end{matrix} \right]</math> | \end{matrix} \right]</math> | ||
The pivots are 1 and 11, so that 11 tells us that we had a common factor of 11<ref>In the doubly-enfactored case of {{rket|{{map|17 16 -4}} {{map|4 -4 1}}}}, i.e. with a common factor of 33 = 3 | The pivots are 1 and 11, so that 11 tells us that we had a common factor of 11<ref group="note">In the doubly-enfactored case of {{rket|{{map|17 16 -4}} {{map|4 -4 1}}}}, i.e. with a common factor of {{nowrap|33 {{=}} 3 × 11}}, the two pivots of the HNF are 3 and 11, putting each of them on display separately.</ref><ref group="note">It's interesting to observe that while the 11-enfactoring can be observed in the original matrix as a linear combination of 2 of the 1st row with −3 of the 2nd row, i.e. 2{{map|6 5 -4}} + (−3){{map|4 -4 1}} {{=}} {{map|0 22 -11}}, the linear combination of ''columns'', i.e. slicing the original {{rket|{{map|6 5 -4}} {{map|4 -4 1}}}} mapping the other direction like {{rbra|{{vector|6 4}} {{vector|5 -4}} {{vector|-4 1}}}}, that leads to the revelation of this 11 is completely different: (−1){{vector|6 4}} + 2{{vector|5 -4}} + 1{{vector|-4 1}} {{=}} {{vector|0 11}}.</ref>. You could say that the HNF is useful for identifying common factors, but not for removing them. But if you leave them behind in the column-style HNF, the information that is retained in the unimodular matrix which is the other product of the Hermite decomposition, is enough to preserve everything important about the temperament, to get you back to where you started via an inverse and a trimming of extraneous rows. | ||
}} | }} | ||
== Development == | == Development == | ||
At the time [[Dave Keenan]] and [[Douglas Blumeyer]] began their investigation into exterior algebra (EA), most of the math involved in RTT could be handled using only linear algebra (LA), a relatively basic and commonplace subject that many people get a chance to learn in high school or university along with subjects like calculus or trigonometry. But there was one crucial task which LA had not proven able to handle yet: providing a "fingerprint" | At the time [[Dave Keenan]] and [[Douglas Blumeyer]] began their investigation into exterior algebra (EA), most of the math involved in RTT could be handled using only linear algebra (LA), a relatively basic and commonplace subject that many people get a chance to learn in high school or university along with subjects like calculus or trigonometry. But there was one crucial task which LA had not proven able to handle yet: providing a "fingerprint"—a unique mathematical representation—for each distinct temperament, to allow it to be recognized as the same temperament even though it might be derived in different ways, or in other words, a canonical form for them. For many years, EA had provided this service for RTT, using a structure called a "[[wedgie]]". | ||
Dave and Douglas began their investigations with the hypothesis that canonicalization via wedgies was the primary reason it was important for RTT beginners to learn EA, and that if a canonical form could be developed using only LA, then EA could be reframed as an advanced topic. Gene himself, upon introducing the wedgie (which he initially called a "wedge invariant"), dismissed it as a bad idea to use for identifying temperaments: | Dave and Douglas began their investigations with the hypothesis that canonicalization via wedgies was the primary reason it was important for RTT beginners to learn EA, and that if a canonical form could be developed using only LA, then EA could be reframed as an advanced topic. Gene himself, upon introducing the wedgie (which he initially called a "wedge invariant"), dismissed it as a bad idea to use for identifying temperaments: | ||
<blockquote>''Since this is an invariant of the temperament, it would be a good thing to use to refer to it, but for the fact that it is opaque and does not immediately tell us how to define the temperament.''<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_1545.html#1545 Yahoo! Tuning Group | ''Standardizing our wedge product'']</ref></blockquote> | <blockquote>''Since this is an invariant of the temperament, it would be a good thing to use to refer to it, but for the fact that it is opaque and does not immediately tell us how to define the temperament.''<ref group="note">[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_1545.html#1545 Yahoo! Tuning Group | ''Standardizing our wedge product'']</ref></blockquote> | ||
Regarding any other advantages EA brought to the RTT table for beginners: they did not find any. The only minor advantage identified was how the largest-minors of the mapping which wedgies are a list of could also be read as a list of denominators of unit fractions of the tempered lattice which are capable of being generated by the associated combination of primes whose columns in the mapping were used in the calculation of the corresponding largest-minor (this idea is discussed in more detail [[ | Regarding any other advantages EA brought to the RTT table for beginners: they did not find any. The only minor advantage identified was how the largest-minors of the mapping which wedgies are a list of could also be read as a list of denominators of unit fractions of the tempered lattice which are capable of being generated by the associated combination of primes whose columns in the mapping were used in the calculation of the corresponding largest-minor (this idea is discussed in more detail [[Dave Keenan & Douglas Blumeyer's guide to EA for RTT#Multicomma entries: tempered lattice fractions generated by prime combinations|here]]). Furthermore, several disadvantages of EA were identified, the main one being that it is more challenging to learn and use, involving higher level mathematical concepts than LA. | ||
Regarding the development of a canonical form for temperaments using only linear algebra, Dave and Douglas did manage to develop such a form, which is documented here: [[defactored Hermite form]]. It was Gene himself who first described this form (as the result of his "saturation" algorithm), so he either did not realize the full implications of his discovery, or it was simply not popularized and plugged in with the rest of the hive knowledge. | Regarding the development of a canonical form for temperaments using only linear algebra, Dave and Douglas did manage to develop such a form, which is documented here: [[defactored Hermite form]]. It was Gene himself who first described this form (as the result of his "saturation" algorithm), so he either did not realize the full implications of his discovery, or it was simply not popularized and plugged in with the rest of the hive knowledge. | ||
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==== Nullspace'n'back defactoring ==== | ==== Nullspace'n'back defactoring ==== | ||
It's fairly self-explanatory: '''Nullspace'n'back defactoring''' is a defactoring algorithm that works by finding a basis for the nullspace of a mapping, and then finding a basis for the nullspace of that result, which brings one back to a mapping. Depending on the implementation of your math library, nullspace'n'back defactoring ''may'' work<ref>It is proven to work in this other paper by Pernet, where the terms "right kernel basis" and "left kernel basis" are used instead: https://hal.archives-ouvertes.fr/hal-01829139, however, no one involved in the xenharmonic project as of yet has invested the time and effort necessary to understand and confirm this proof. Pernet uses (and indeed has contributed in major part to) the Sage math library, where nullspace'n'back ''does'' work, but it seems likely to work only because their code explicitly defactors matrices whenever taking the nullspace, per our appraisal of their source code! Wolfram Language, on the other hand, which our RTT library uses, does not do this. Sintel has recommended that it does not make sense to return enfactored nullspace bases and that this therefore constitutes a bug in Wolfram Language; they have been subsequently contacted about the issue. Pernet himself has yet to weigh in but has been contacted about this by email.</ref>. For example, {{rket|{{map|7 11 16}} {{map|22 35 51}}}} has a nullspace basis of [{{vector|1 -5 3}}], and when we take the nullspace of that with Wolfram Language, we get {{rket|{{map|3 0 -1}} {{map|0 3 5}}}} which is a classic case of hidden enfactoring as described above, so it does not work in this case. | It's fairly self-explanatory: '''Nullspace'n'back defactoring''' is a defactoring algorithm that works by finding a basis for the nullspace of a mapping, and then finding a basis for the nullspace of that result, which brings one back to a mapping. Depending on the implementation of your math library, nullspace'n'back defactoring ''may'' work<ref group="note">It is proven to work in this other paper by Pernet, where the terms "right kernel basis" and "left kernel basis" are used instead: https://hal.archives-ouvertes.fr/hal-01829139, however, no one involved in the xenharmonic project as of yet has invested the time and effort necessary to understand and confirm this proof. Pernet uses (and indeed has contributed in major part to) the Sage math library, where nullspace'n'back ''does'' work, but it seems likely to work only because their code explicitly defactors matrices whenever taking the nullspace, per our appraisal of their source code! Wolfram Language, on the other hand, which our RTT library uses, does not do this. Sintel has recommended that it does not make sense to return enfactored nullspace bases and that this therefore constitutes a bug in Wolfram Language; they have been subsequently contacted about the issue. Pernet himself has yet to weigh in but has been contacted about this by email.</ref>. For example, {{rket|{{map|7 11 16}} {{map|22 35 51}}}} has a nullspace basis of [{{vector|1 -5 3}}], and when we take the nullspace of that with Wolfram Language, we get {{rket|{{map|3 0 -1}} {{map|0 3 5}}}} which is a classic case of hidden enfactoring as described above, so it does not work in this case. | ||
==== Sum-and-difference defactoring ==== | ==== Sum-and-difference defactoring ==== | ||
| Line 835: | Line 814: | ||
===== How it works ===== | ===== How it works ===== | ||
Originally Dave was inspired by what [[Paul Erlich]] wrote on [http://tonalsoft.com/enc/t/torsion.aspx the article for torsion on the Tonalsoft site]. The algorithm simply checks every possible combination of sums and differences of rows. You have to check all < | Originally Dave was inspired by what [[Paul Erlich]] wrote on [http://tonalsoft.com/enc/t/torsion.aspx the article for torsion on the Tonalsoft site]. The algorithm simply checks every possible combination of sums and differences of rows. You have to check all {{sfrac|3<sup>''r''</sup> − 1|2}} sums and differences where ''r'' is the rank of the mapping. For example, for a rank-3 mapping with rows {{vector|A B C}}, you would need to check | ||
* A | * A | ||
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This is the furthest progress that was made on SAD defactor before efforts were shifted to column Hermite defactor. This implementation still exhibits problems 2 and 3; only problems 1 and 4 were solved here. Problem 3 started to be solved, but was not quite quashed. | This is the furthest progress that was made on SAD defactor before efforts were shifted to column Hermite defactor. This implementation still exhibits problems 2 and 3; only problems 1 and 4 were solved here. Problem 3 started to be solved, but was not quite quashed. | ||
{{Databox| | {{Databox|Wolfram Language code for SAD defactoring| | ||
<pre> | <pre> | ||
confirmEnfactoredRowReplaced[m_] := Module[{i, enfactoredRowReplaced}, | confirmEnfactoredRowReplaced[m_] := Module[{i, enfactoredRowReplaced}, | ||
| Line 922: | Line 901: | ||
== Footnotes == | == Footnotes == | ||
<references /> | <references group="note" /> | ||
[[Category:Regular temperament theory]] | [[Category:Regular temperament theory]] | ||
| Line 928: | Line 907: | ||
[[Category:Math]] | [[Category:Math]] | ||
[[Category:Pages with proofs]] | [[Category:Pages with proofs]] | ||
[[Category:Algorithms]] | |||