From 0e55b3454a4c2528d23bb61c74814958ca890767 Mon Sep 17 00:00:00 2001 From: nessan Date: Sun, 21 Jul 2024 15:12:40 +0100 Subject: [PATCH] Built site for gh-pages --- .nojekyll | 2 +- content/bit_assert/index.html | 6 +- content/faq/index.html | 4 + content/gauss/access.html | 4 + content/gauss/constructors.html | 4 + content/gauss/functor.html | 4 + content/gauss/index.html | 4 + content/gauss/queries.html | 4 + content/gauss/solve.html | 4 + content/index.html | 37 ++--- content/lu/access.html | 4 + content/lu/constructors.html | 4 + content/lu/functor.html | 4 + content/lu/index.html | 4 + content/lu/invert.html | 4 + content/lu/permute.html | 4 + content/lu/queries.html | 4 + content/matrix/access.html | 6 +- content/matrix/add-pop.html | 4 + content/matrix/all.html | 4 + content/matrix/append.html | 4 + content/matrix/capacity.html | 4 + content/matrix/characteristic_polynomial.html | 4 + content/matrix/clear.html | 4 + content/matrix/companion.html | 4 + content/matrix/constructors.html | 14 +- content/matrix/count.html | 6 +- content/matrix/description.html | 4 + content/matrix/dot.html | 4 + content/matrix/echelon.html | 4 + content/matrix/from.html | 10 +- content/matrix/index.html | 8 +- content/matrix/invert.html | 4 + content/matrix/is_special.html | 4 + content/matrix/logical-op-eq.html | 4 + content/matrix/logical-op.html | 4 + content/matrix/pow.html | 4 + content/matrix/print.html | 4 + content/matrix/probability.html | 4 + content/matrix/random.html | 4 + content/matrix/replace.html | 4 + content/matrix/resize.html | 4 + content/matrix/set.html | 4 + content/matrix/set_if.html | 4 + content/matrix/shift.html | 4 + content/matrix/shrink_to_fit.html | 4 + content/matrix/size.html | 4 + content/matrix/specials.html | 4 + content/matrix/stream.html | 4 + content/matrix/sub.html | 4 + content/matrix/swap.html | 4 + content/matrix/to_string.html | 4 + content/matrix/to_vector.html | 4 + content/matrix/transpose.html | 4 + content/matrix/triangle.html | 4 + content/notes/danilevsky.html | 6 +- content/notes/design.html | 4 + content/notes/gf2.html | 10 +- content/notes/index.html | 10 +- content/notes/mod.html | 14 +- content/polynomial/access.html | 14 +- content/polynomial/arithmetic.html | 4 + content/polynomial/constructors.html | 4 + content/polynomial/count.html | 4 + content/polynomial/degree.html | 12 +- content/polynomial/evaluation.html | 13 +- content/polynomial/formatter.html | 4 + content/polynomial/index.html | 74 +++++----- content/polynomial/monic.html | 4 + content/polynomial/power.html | 4 + content/polynomial/random.html | 25 ++-- content/polynomial/reduce.html | 4 + content/polynomial/reference.html | 14 +- content/polynomial/shrink_to_fit.html | 8 +- content/polynomial/size.html | 20 ++- content/polynomial/special.html | 6 +- content/polynomial/split.html | 4 + content/polynomial/squared.html | 11 +- content/polynomial/stream.html | 4 + content/polynomial/sub.html | 4 + content/polynomial/times_x.html | 4 + content/polynomial/to_string.html | 4 + content/vector/access.html | 4 + content/vector/all.html | 4 + content/vector/append.html | 4 + content/vector/blocks.html | 4 + content/vector/capacity.html | 4 + content/vector/clear.html | 4 + content/vector/constructors.html | 26 ++-- content/vector/convolution.html | 4 + content/vector/copy.html | 4 + content/vector/count.html | 4 + content/vector/description.html | 4 + content/vector/diff.html | 4 + content/vector/dot.html | 4 + content/vector/first_set.html | 4 + content/vector/formatter.html | 16 +- content/vector/from.html | 22 +-- content/vector/if_set_call.html | 4 + content/vector/index.html | 6 +- content/vector/indices.html | 4 + content/vector/join.html | 4 + content/vector/logical-op-eq.html | 4 + content/vector/logical-op.html | 4 + content/vector/polynomial_mod.html | 4 + content/vector/push-pop.html | 4 + content/vector/random.html | 6 +- content/vector/reference.html | 14 +- content/vector/replace.html | 4 + content/vector/reserve.html | 4 + content/vector/resize.html | 4 + content/vector/riffled.html | 4 + content/vector/set.html | 4 + content/vector/set_if.html | 4 + content/vector/shift.html | 4 + content/vector/size.html | 4 + content/vector/specials.html | 4 + content/vector/stream.html | 16 +- content/vector/sub.html | 4 + content/vector/swap.html | 4 + content/vector/swap_elements.html | 4 + content/vector/to_string.html | 137 ++++++++---------- content/vector/trimmed.html | 4 + content/vector/unit_floor.html | 4 + index.html | 37 ++--- search.json | 89 ++++++------ .../pseudocode-2.4.1/pseudocode.min.css | 1 - .../pseudocode-2.4.1/pseudocode.min.js | 1 - sitemap.xml | 54 +++---- 129 files changed, 807 insertions(+), 320 deletions(-) delete mode 100644 site_libs/quarto-contrib/pseudocode-2.4.1/pseudocode.min.css delete mode 100644 site_libs/quarto-contrib/pseudocode-2.4.1/pseudocode.min.js diff --git a/.nojekyll b/.nojekyll index bde014f..c7f4f78 100644 --- a/.nojekyll +++ b/.nojekyll @@ -1 +1 @@ -b19c3831 \ No newline at end of file +fa442b9f \ No newline at end of file diff --git a/content/bit_assert/index.html b/content/bit_assert/index.html index ee05223..23cd2c2 100644 --- a/content/bit_assert/index.html +++ b/content/bit_assert/index.html @@ -182,6 +182,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • @@ -677,7 +681,7 @@

    Examples

    1
    -Construct a vector of size 12 and then attempt to set the “last” element. +Construct a vector of size 12 and then attempt to set the “last” element.
    2
    diff --git a/content/faq/index.html b/content/faq/index.html index afedc59..98698d7 100644 --- a/content/faq/index.html +++ b/content/faq/index.html @@ -118,6 +118,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/gauss/access.html b/content/gauss/access.html index c891b3a..3c40eda 100644 --- a/content/gauss/access.html +++ b/content/gauss/access.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/gauss/constructors.html b/content/gauss/constructors.html index 922ebd1..132e14f 100644 --- a/content/gauss/constructors.html +++ b/content/gauss/constructors.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/gauss/functor.html b/content/gauss/functor.html index 992db29..65266cf 100644 --- a/content/gauss/functor.html +++ b/content/gauss/functor.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/gauss/index.html b/content/gauss/index.html index a6aa107..df738b1 100644 --- a/content/gauss/index.html +++ b/content/gauss/index.html @@ -182,6 +182,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/gauss/queries.html b/content/gauss/queries.html index 1c9c7e7..cc148da 100644 --- a/content/gauss/queries.html +++ b/content/gauss/queries.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/gauss/solve.html b/content/gauss/solve.html index 6df1534..b000e76 100644 --- a/content/gauss/solve.html +++ b/content/gauss/solve.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/index.html b/content/index.html index 3d2315a..635b88f 100644 --- a/content/index.html +++ b/content/index.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • @@ -576,7 +580,7 @@

    The bitLibrary

    Introduction

    -

    bit is a header-only C++ library for numerical work in bit-space which mathematicians call GF(2) or \(\FF\), the simplest Galois field with just two elements 0 & 1. All arithmetic operations in bit-space are mod 2 so what starts in bit-space stays in bit-space!

    +

    bit is a header-only C++ library for numerical work in bit-space which mathematicians call GF(2) or \(\FF\), the simplest Galois field with just two elements 0 & 1. All arithmetic operations in bit-space are mod 2 so what starts in bit-space stays in bit-space.

    The library provides vector and matrix classes for performing linear algebra in bit-space. The bit::vector class represents bit_vectors, and the bit::matrix class represents bit-matrices. The library also has a bit::polynomial class to represent bit-polynomials over \(\FF\).

    These classes are efficient and pack the individual bit elements into natural word blocks. You can size/resize the classes at run-time.

    Because arithmetic operations in \(\FF\) are mod 2, addition/subtraction becomes the XOR operation, and multiplication/division becomes the AND operation. The bit library uses those equivalences to efficiently perform most interactions on and between bit-vectors and bit-matrices by simultaneously working on whole blocks of elements.

    @@ -613,11 +617,10 @@

    Example

    { 1 auto M = bit::matrix<>::random(6, 6); 2 auto c = bit::characteristic_polynomial(M); - std::cout << "The bit-matrix M:\n" << M << "\n"; - std::cout << "has characteristic polynomial c(x) = " << c << ".\n"; - std::cout << "The polynomial sum c(M) gives:\n"; -3 std::cout << c(M) << "\n"; -}
    + std::cout << std::format("Bit-matrix M:\n{:p}\n", M); + std::cout << std::format("Characteristic poly c(x) = {}\n", c); +3 std::cout << std::format("c(M) yields:\n{:p}\n", c(M)); +}
    1
    @@ -625,23 +628,23 @@

    Example

    2
    -Computes the characteristic polynomial of \(M\), \(c(x) = c_0 + c_1 x + c_2 x^2 + ... + c_6 x^6\). +Computes its characteristic polynomial \(c(x) = c_0 + c_1 x + c_2 x^2 + ... + c_6 x^6\).
    3
    -Verifies that \(M\) satisfies its own characteristic equation \(c(M) = 0\), as expected from the Cayley Hamilton theorem. +Verifies that \(M\) satisfies its own characteristic equation \(c(M) = 0\), as expected from the Cayley Hamilton theorem.

    Sample Output (varies from run to run):

    -
    The bit-matrix M:
-│0 1 1 0 0 0│
-│0 0 1 0 1 0│
-│1 1 0 0 0 1│
-│0 0 0 0 0 1│
-│0 1 0 0 1 1│
-│1 1 0 1 0 1│
-has characteristic polynomial c(x) = x^1 + x^4 + x^6.
-The polynomial sum c(M) gives:
+
    Bit-matrix M:
+│1 1 0 1 1 0│
+│0 0 0 0 0 1│
+│0 1 1 1 0 0│
+│0 0 0 1 0 1│
+│1 1 1 0 1 0│
+│1 0 0 1 1 1│
+Characteristic poly c(x) = x^1 + x^2 + x^3 + x^5 + x^6
+c(M) yields:
 │0 0 0 0 0 0│
 │0 0 0 0 0 0│
 │0 0 0 0 0 0│
diff --git a/content/lu/access.html b/content/lu/access.html
index 5bfee84..5c11346 100644
--- a/content/lu/access.html
+++ b/content/lu/access.html
@@ -181,6 +181,10 @@
   
  • 
      
 Matrices
+  
    •   
+  
  • 
+     
+Polynomials
   
    •   
   
  • 
      
diff --git a/content/lu/constructors.html b/content/lu/constructors.html
index 82092ae..e28f900 100644
--- a/content/lu/constructors.html
+++ b/content/lu/constructors.html
@@ -181,6 +181,10 @@
   
  • 
      
 Matrices
+  
    •   
+  
  • 
+     
+Polynomials
   
    •   
   
  • 
      
diff --git a/content/lu/functor.html b/content/lu/functor.html
index 022c286..7ce6a83 100644
--- a/content/lu/functor.html
+++ b/content/lu/functor.html
@@ -181,6 +181,10 @@
   
  • 
      
 Matrices
+  
    •   
+  
  • 
+     
+Polynomials
   
    •   
   
  • 
      
diff --git a/content/lu/index.html b/content/lu/index.html
index 7facfa2..36a094d 100644
--- a/content/lu/index.html
+++ b/content/lu/index.html
@@ -182,6 +182,10 @@
   
  • 
      
 Matrices
+  
    •   
+  
  • 
+     
+Polynomials
   
    •   
   
  • 
      
diff --git a/content/lu/invert.html b/content/lu/invert.html
index 7f5393a..56ce4b5 100644
--- a/content/lu/invert.html
+++ b/content/lu/invert.html
@@ -181,6 +181,10 @@
   
  • 
      
 Matrices
+  
    •   
+  
  • 
+     
+Polynomials
   
    •   
   
  • 
      
diff --git a/content/lu/permute.html b/content/lu/permute.html
index 31939dc..7ead2b5 100644
--- a/content/lu/permute.html
+++ b/content/lu/permute.html
@@ -181,6 +181,10 @@
   
  • 
      
 Matrices
+  
    •   
+  
  • 
+     
+Polynomials
   
    •   
   
  • 
      
diff --git a/content/lu/queries.html b/content/lu/queries.html
index f0b5346..1dcf05b 100644
--- a/content/lu/queries.html
+++ b/content/lu/queries.html
@@ -181,6 +181,10 @@
   
  • 
      
 Matrices
+  
    •   
+  
  • 
+     
+Polynomials
   
    •   
   
  • 
      
diff --git a/content/matrix/access.html b/content/matrix/access.html
index d307d38..34d7966 100644
--- a/content/matrix/access.html
+++ b/content/matrix/access.html
@@ -181,6 +181,10 @@
   
  • 
      
 Matrices
+  
    •   
+  
  • 
+     
+Polynomials
   
    •   
   
  • 
      
@@ -609,7 +613,7 @@ 
    bit::matrix — Element Access<
    • 5
    -Synonyms for the row(i) methods to allow for alternate “C” style indexing a la matrix[i][j]. +Synonyms for the row(i) methods to allow for alternate C style indexing a la matrix[i][j].
    6
    diff --git a/content/matrix/add-pop.html b/content/matrix/add-pop.html index a295ce1..8943b10 100644 --- a/content/matrix/add-pop.html +++ b/content/matrix/add-pop.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/matrix/all.html b/content/matrix/all.html index ad15e79..88032f5 100644 --- a/content/matrix/all.html +++ b/content/matrix/all.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/matrix/append.html b/content/matrix/append.html index 9aff092..e1be88f 100644 --- a/content/matrix/append.html +++ b/content/matrix/append.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/matrix/capacity.html b/content/matrix/capacity.html index fbac0e4..5d31d51 100644 --- a/content/matrix/capacity.html +++ b/content/matrix/capacity.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/matrix/characteristic_polynomial.html b/content/matrix/characteristic_polynomial.html index 513a5f1..09b534f 100644 --- a/content/matrix/characteristic_polynomial.html +++ b/content/matrix/characteristic_polynomial.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/matrix/clear.html b/content/matrix/clear.html index 6bdbcbd..775e7d9 100644 --- a/content/matrix/clear.html +++ b/content/matrix/clear.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/matrix/companion.html b/content/matrix/companion.html index 1e566ef..43d49f7 100644 --- a/content/matrix/companion.html +++ b/content/matrix/companion.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/matrix/constructors.html b/content/matrix/constructors.html index 67c0cdb..37517cb 100644 --- a/content/matrix/constructors.html +++ b/content/matrix/constructors.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • @@ -834,11 +838,11 @@

    Method Arguments

    • 2
    -Construction from the same binary strings, each with a binary prefix "0b". +Construction from the same binary strings, each with a binary prefix 0b.
    3
    -Construction from the same digits, but each is now interpreted as a hex character thanks to the "0x" prefix. Here, semi-colons separate rows. +Construction from the same digits, but each is now interpreted as a hex character thanks to the 0x prefix. Here, semi-colons separate rows.
    4
    @@ -846,15 +850,15 @@

    Method Arguments

    5
    -Construction where the final characters have a suffix "_8" so are parsed as base-8 numbers. +Construction where the final characters have a suffix _8 so are parsed as base-8 numbers.
    6
    -Construction where the final characters have a suffix "_4" so are parsed as base-4 numbers. +Construction where the final characters have a suffix _4 so are parsed as base-4 numbers.
    7
    -Construction where the final characters have a suffix "_2" so are parsed as base-2 numbers. +Construction where the final characters have a suffix _2 so are parsed as base-2 numbers.

    Output

    diff --git a/content/matrix/count.html b/content/matrix/count.html index 6b31bb4..3494623 100644 --- a/content/matrix/count.html +++ b/content/matrix/count.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • @@ -581,7 +585,7 @@

    bit::matrix — Bit Counts

    3
    -Return count_diagonal() % 2–the “sum” of the diagonal elements. +Return count_diagonal() % 2–the “sum” of the diagonal elements.

    Example

    diff --git a/content/matrix/description.html b/content/matrix/description.html index 265341a..173849d 100644 --- a/content/matrix/description.html +++ b/content/matrix/description.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/matrix/dot.html b/content/matrix/dot.html index fafe9f6..2143d09 100644 --- a/content/matrix/dot.html +++ b/content/matrix/dot.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/matrix/echelon.html b/content/matrix/echelon.html index 1061211..26e1612 100644 --- a/content/matrix/echelon.html +++ b/content/matrix/echelon.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/matrix/from.html b/content/matrix/from.html index 2c3fa8e..3d93a7b 100644 --- a/content/matrix/from.html +++ b/content/matrix/from.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • @@ -668,15 +672,15 @@

    bit::matrix — Construction fr
    4
    -Construction where the row characters have a suffix "_8", so parsed as base-8 numbers. +Construction where the row characters have a suffix _8, so parsed as base-8 numbers.
    5
    -Construction where the row characters have a suffix "_4", so parsed as base-4 numbers. +Construction where the row characters have a suffix _4, so parsed as base-4 numbers.
    6
    -Construction where the row characters have a suffix "_2", so parsed as base-2 numbers. +Construction where the row characters have a suffix _2, so parsed as base-2 numbers.

    Output

    diff --git a/content/matrix/index.html b/content/matrix/index.html index 962a29a..aa4843c 100644 --- a/content/matrix/index.html +++ b/content/matrix/index.html @@ -182,6 +182,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • @@ -1128,11 +1132,11 @@

    Other methods

    matrix::probability_invertible -Returns the probability that a “fair” square bit-matrix is invertible. +Returns the probability that a “fair” square bit-matrix is invertible. matrix::probability_singular -Returns the probability that a “fair” square bit-matrix is singular. +Returns the probability that a “fair” square bit-matrix is singular. diff --git a/content/matrix/invert.html b/content/matrix/invert.html index 090af74..ae6a65a 100644 --- a/content/matrix/invert.html +++ b/content/matrix/invert.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/matrix/is_special.html b/content/matrix/is_special.html index b514c83..f9da5b1 100644 --- a/content/matrix/is_special.html +++ b/content/matrix/is_special.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/matrix/logical-op-eq.html b/content/matrix/logical-op-eq.html index ea7f05f..31b02a8 100644 --- a/content/matrix/logical-op-eq.html +++ b/content/matrix/logical-op-eq.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/matrix/logical-op.html b/content/matrix/logical-op.html index 124b228..d1b6895 100644 --- a/content/matrix/logical-op.html +++ b/content/matrix/logical-op.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/matrix/pow.html b/content/matrix/pow.html index a699c93..9b2a7fd 100644 --- a/content/matrix/pow.html +++ b/content/matrix/pow.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/matrix/print.html b/content/matrix/print.html index 1be02f4..922ac3d 100644 --- a/content/matrix/print.html +++ b/content/matrix/print.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/matrix/probability.html b/content/matrix/probability.html index 09d42e9..1e6b6ca 100644 --- a/content/matrix/probability.html +++ b/content/matrix/probability.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/matrix/random.html b/content/matrix/random.html index 0aa7fb5..7b33129 100644 --- a/content/matrix/random.html +++ b/content/matrix/random.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/matrix/replace.html b/content/matrix/replace.html index c9f8737..8963f5d 100644 --- a/content/matrix/replace.html +++ b/content/matrix/replace.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/matrix/resize.html b/content/matrix/resize.html index bd6b9fe..58ba6a9 100644 --- a/content/matrix/resize.html +++ b/content/matrix/resize.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/matrix/set.html b/content/matrix/set.html index 9e1c7dd..fd91363 100644 --- a/content/matrix/set.html +++ b/content/matrix/set.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/matrix/set_if.html b/content/matrix/set_if.html index 01d43fa..7e90804 100644 --- a/content/matrix/set_if.html +++ b/content/matrix/set_if.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/matrix/shift.html b/content/matrix/shift.html index 2a18705..98e3328 100644 --- a/content/matrix/shift.html +++ b/content/matrix/shift.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/matrix/shrink_to_fit.html b/content/matrix/shrink_to_fit.html index 1b97160..42c8702 100644 --- a/content/matrix/shrink_to_fit.html +++ b/content/matrix/shrink_to_fit.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/matrix/size.html b/content/matrix/size.html index 9b17655..e9c1b69 100644 --- a/content/matrix/size.html +++ b/content/matrix/size.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/matrix/specials.html b/content/matrix/specials.html index 1c84158..b5031a9 100644 --- a/content/matrix/specials.html +++ b/content/matrix/specials.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/matrix/stream.html b/content/matrix/stream.html index 775322d..862acce 100644 --- a/content/matrix/stream.html +++ b/content/matrix/stream.html @@ -181,6 +181,10 @@
  • Matrices +
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  • diff --git a/content/matrix/sub.html b/content/matrix/sub.html index da5d6f2..058ebff 100644 --- a/content/matrix/sub.html +++ b/content/matrix/sub.html @@ -181,6 +181,10 @@
  • Matrices +
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  • diff --git a/content/matrix/swap.html b/content/matrix/swap.html index f1438f5..06d885f 100644 --- a/content/matrix/swap.html +++ b/content/matrix/swap.html @@ -181,6 +181,10 @@
  • Matrices +
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  • diff --git a/content/matrix/to_string.html b/content/matrix/to_string.html index 3344213..8f32474 100644 --- a/content/matrix/to_string.html +++ b/content/matrix/to_string.html @@ -181,6 +181,10 @@
  • Matrices +
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  • diff --git a/content/matrix/to_vector.html b/content/matrix/to_vector.html index 4adb2e3..113fe4b 100644 --- a/content/matrix/to_vector.html +++ b/content/matrix/to_vector.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/matrix/transpose.html b/content/matrix/transpose.html index 2e6627f..90e6462 100644 --- a/content/matrix/transpose.html +++ b/content/matrix/transpose.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/matrix/triangle.html b/content/matrix/triangle.html index 5759eed..ef2848e 100644 --- a/content/matrix/triangle.html +++ b/content/matrix/triangle.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/notes/danilevsky.html b/content/notes/danilevsky.html index fba8436..6af72f7 100644 --- a/content/notes/danilevsky.html +++ b/content/notes/danilevsky.html @@ -151,6 +151,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • @@ -647,7 +651,7 @@

    Companion Matrices

    You can make the same determinant expansion argument if the polynomial coefficients were in the final column instead of the top row. That is the version seen in some expositions.

    -

    \(C\) is a “companion” for the polynomial \(c(\lambda)\) and is known as a companion matrix.

    +

    \(C\) is a “companion” for the polynomial \(c(\lambda)\) and is known as a companion matrix.

    Frobenius Form

    First, note that you can create many others once you have a matrix (like the \(C\) above) with the desired characteristic polynomial. The reason is that similar matrices all have the same characteristic polynomial. So if \(M\) is any invertible \(n \times n\) matrix, then \(M^{-1} \cdot C \cdot M\) will have the same characteristic polynomial as \(C\).

    diff --git a/content/notes/design.html b/content/notes/design.html index 43878b7..7258583 100644 --- a/content/notes/design.html +++ b/content/notes/design.html @@ -183,6 +183,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/notes/gf2.html b/content/notes/gf2.html index 5de6ea4..2145856 100644 --- a/content/notes/gf2.html +++ b/content/notes/gf2.html @@ -151,6 +151,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • @@ -547,10 +551,10 @@

    Introduction

    • Some things are different!

    -

    Over \(\R\), the only vector of size zero in the zero vector.
    +

    Over \(\R\), the only self-orthogonal vector is the zero vector.
    If \(\mathbf{x}\) is a vector over \(\R\) then \[ \mathbf{x} \cdot \mathbf{x} = 0 \iff \mathbf{x} = \mathbf{0}, -\] Put another way, the only self-orthogonal vector over \(\R\) is the zero vector.

    +\] Put another way, the only vector of size 0 over \(\R\) is the zero vector.

    That is not true for vectors over \(\FF\).

    For example, if \(\mathbf{v} = \{1, 1\}\) is thought of as a vector over \(\FF\) then \[ \mathbf{v} \cdot \mathbf{v} = \{1, 1\} \cdot \{1, 1\} = 1 + 1 = 2 \rightarrow 0 \text{ mod 2}. @@ -623,7 +627,7 @@

    Some things are si

    Gaussian Elimination in \(\FF\)

    Suppose that \(A\) is an \(n \times n\) matrix over \(\FF\) and \(b\) is a compatibly sized bit-vector where we are interested in finding an \(x\) satisfying \(A \cdot x = b\). Then the pseudocode for Gaussian elimination looks like:

    -
    +
    \begin{algorithm} \caption{Gaussian Elimination in $F_2$} \begin{algorithmic} \Procedure{Solve}{$A, b, n$} \For {$j = 0$ \To $n - 1$} \State $s = j$ \While {$A(s,j) = 0$} \State $s = s + 1$ \EndWhile \If {$s > n$} \Continue \EndIf \If {$ s \ne j$} \State swap rows $s$ and $j$ in the matrix $A$ \State swap elements $s$ and $j$ in the vector $b$ \EndIf \For {$i = j+1$ \To $n$} \If {$A(i,j) == 1$} \State replace row $i$ in $A$ with the sum of rows $i$ and $j$ \State replace element $i$ in $b$ with the sum of elements $i$ and $j$ \EndIf \EndFor \EndFor \EndProcedure \end{algorithmic} \end{algorithm}
    diff --git a/content/notes/index.html b/content/notes/index.html index 3b993a2..de1416b 100644 --- a/content/notes/index.html +++ b/content/notes/index.html @@ -148,6 +148,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • @@ -535,7 +539,7 @@

    Technical Notes

    Introduction

    -

    There isn’t a lot of material on the web devoted to linear algebra over GF(2).
    +

    There isn’t a lot of material on the web devoted to computational work in GF(2).
    We add some here:

    @@ -552,7 +556,7 @@

    Introduction

    - + @@ -563,7 +567,7 @@

    Introduction

    - + diff --git a/content/notes/mod.html b/content/notes/mod.html index 907bf2e..28e7e25 100644 --- a/content/notes/mod.html +++ b/content/notes/mod.html @@ -184,6 +184,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • @@ -530,7 +534,7 @@

    On this page
    @@ -636,10 +640,10 @@

    An Iterative Techni \end{cases} \]

    -
    -

    Multiply and Square

    +
    +

    Square and Multiply

    This iterative scheme is too slow for large \(N\).

    -

    We can speed it up considerably by using a “multiply and square” approach, which uses the decomposition: \[ +

    We can speed it up considerably by using a “square and multiply” approach, which uses the decomposition: \[ x^N = \begin{cases} x \, \left( x^{\frac{N-1}{2}} \right)^2 & N \text{ odd,} \\ @@ -651,7 +655,7 @@

    Multiply and SquareThe second case above shows that we will also need to do some squaring steps where \(s(x)\) starts as just \(s(x) = x\) and then gets squared as often as needed and, of course, reduced modulo \(P(x)\): \[ s(x) \mid P(x) = x \mid P(x) \rightarrow x^2 \mid P(x) \rightarrow x^4 \mid P(x) \rightarrow \cdots \] Now \(x \mid P(x) = x\) so we can initialize \(\mathbf{s} = \mathbf{u}_1\) the unit vector with a single set bit at index number 1.

    -
    +
    \begin{algorithm} \caption{Computing $x^N$ mod $P(x)$} \begin{algorithmic} \Procedure{Mod}{$N, \mathbf{p}$} \State $n = \text{len}(\mathbf{p}) + 1 \State $M = N - n$ \State $\mathbf{r} = \mathbf{p}$ \State set $s$ to the appropriate unit bit-vector \While {$M \ne 0$} \If $M$ is odd \State $\mathbf{r} \equiv r(x)*s(x) \mid P(x) \EndIf \State $M \ll 1$ \If $M = 0$ \Break \EndIf \State set $\mathbf{s} \equiv s(x)^2 \mid P(x)$ \EndWhile \EndProcedure \end{algorithmic} \end{algorithm}
    diff --git a/content/polynomial/access.html b/content/polynomial/access.html index 88ad618..dfc7650 100644 --- a/content/polynomial/access.html +++ b/content/polynomial/access.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • @@ -583,23 +587,23 @@

    bit::polynomial — Coefficient
    1
    -Accesses the value for coefficient i. +Read-only access to coefficient i.
    2
    -Returns a polynomial::reference object—allows modification of the coefficient at index i. +Returns a polynomial::reference object — allows modification of coefficient i.
    3
    -Another way to access the value for coefficient i. +Another way to get read-only access to coefficient i..
    4
    -Another way to set the value for coefficient i to val. +Set the value of coefficient i to val.
    5
    -This is a synonym for set(i, false). +Set the value of coefficient i to false.
    6
    diff --git a/content/polynomial/arithmetic.html b/content/polynomial/arithmetic.html index 2cea701..c669d82 100644 --- a/content/polynomial/arithmetic.html +++ b/content/polynomial/arithmetic.html @@ -181,6 +181,10 @@
  • Matrices +
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  • diff --git a/content/polynomial/constructors.html b/content/polynomial/constructors.html index d944c00..6198341 100644 --- a/content/polynomial/constructors.html +++ b/content/polynomial/constructors.html @@ -181,6 +181,10 @@
  • Matrices +
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  • diff --git a/content/polynomial/count.html b/content/polynomial/count.html index 69dbdeb..fd80175 100644 --- a/content/polynomial/count.html +++ b/content/polynomial/count.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
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  • diff --git a/content/polynomial/degree.html b/content/polynomial/degree.html index 3c77f58..7f752ec 100644 --- a/content/polynomial/degree.html +++ b/content/polynomial/degree.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • @@ -581,11 +585,11 @@

    bit::polynomial — Polynomial

    • 2
    -Returns true if the polynomial is monic—i.e. has no trailing zero coefficients. +Returns true if the polynomial is monic — i.e. has no trailing zero coefficients.
    3
    -Eliminates any trailing zero coefficients and returns a reference to this, now monic, polynomial. +Eliminates any trailing zero coefficients and returns a reference to this polynomial.
    @@ -599,7 +603,7 @@

    bit::polynomial — Polynomial

    The zero polynomial has no degree though by convention it is often set to \(-\infty\). Our polynomial class has a special “not a degree” constant for this purpose polynomial::ndeg.

    -

    Polynomial methods usually need to treat the zero-polynomial as a special, generally trivial, edge case. The make_monic() method does nothing to a zero polynomial.

    +

    The make_monic() method does nothing to a zero polynomial.

    • @@ -619,7 +623,7 @@

    Degree versus Size

    -

    Calling make_monic on a non-zero polynomial simply ensures that size() == degree() + 1. No memory is released by this operation—see the polynomial::shrink_to_fit method.

    +

    Calling make_monic on a non-zero polynomial simply ensures that size() == degree() + 1. No memory is released by this operation — see the polynomial::shrink_to_fit method.

    diff --git a/content/polynomial/evaluation.html b/content/polynomial/evaluation.html index d0e62ec..9ece604 100644 --- a/content/polynomial/evaluation.html +++ b/content/polynomial/evaluation.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • @@ -572,7 +576,8 @@

    bit::polynomial — Polynomial

    We have methods to evaluate a polynomial for scalar and bit-matrix arguments.

    1constexpr bool operator()(bool x) const;
-2constexpr matrix_type operator()(const matrix_type& M) const;
    + +2constexpr matrix_type operator()(const matrix_type& M) const;
    1
    @@ -580,14 +585,14 @@

    bit::polynomial — Polynomial

    2
    -This evaluates the polynomial for a square bit-matrix argument M. +This evaluates the polynomial for a square bit-matrix argument M.
    +
    +

    Scalar Arguments

    Let \[ p(x) = p_0 + p_1 x + p_2 x^2 + \cdots + p_{n-1} x^{n-1}. \]

    -
    -

    Scalar Arguments

    In GF(2), arithmetic is mod 2, which means that for any scalar argument \(x\) \[ p(x) = p_0 + p_1 x + p_2 x + \cdots + p_{n-1} x. \] If \(x = 0\), this is just p[0], while if \(x = 1\), it is the count of ones (mod 2) in the polynomial coefficients.

    diff --git a/content/polynomial/formatter.html b/content/polynomial/formatter.html index 0374a21..bdc8419 100644 --- a/content/polynomial/formatter.html +++ b/content/polynomial/formatter.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/polynomial/index.html b/content/polynomial/index.html index 94a5e22..fa67a0b 100644 --- a/content/polynomial/index.html +++ b/content/polynomial/index.html @@ -182,6 +182,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • @@ -591,22 +595,21 @@

    Introduction

    If \(p(x)\) is the bit-polynomial: \[ p(x) = p_0 + p_1 x + p_2 x^2 + \cdots + p_{n-1} x^{n-1}, \] then the argument \(x\) and the polynomial coefficients \(p_0, p_1, \ldots\) are all elements of \(\FF\).

    -

    The bit::polynomial class is a wrapper around a bit::vector used to hold the polynomial coefficients. Instance methods forward much of their work to that data member.

    -

    Some fundamental bit-polynomial methods do need a separate implementation. For example, bit-vector addition only makes sense for two equal-sized bit-vectors, but of course, we have to be able to add two polynomials of different degrees.

    +

    The bit::polynomial class holds the polynomial coefficients in a bit::vector. Instance methods forward much of their work to that data member. However, some bit-polynomial methods need a separate implementation. For example, bit-vector addition only makes sense for two equal-sized bit-vectors, but of course, we have to be able to add two polynomials of different degrees.

    -Efficiency +Polynomial size and degree
    -

    A bit-polynomial has a size \(n\) which is just its number of coefficients. It also has a degree \(d\) which is the index of the highest non-trivial power in the polynomial. Monic polynomials are nonzero and have \(d = n+1\) which means there are no trailing zero coefficients.

    +

    The size of a bit-polynomial is the number of its coefficients. The degree of a bit-polynomial is the index of the highest non-trivial power in the polynomial. Monic polynomials are nonzero and have no trailing zero coefficients.

    Operations on and between polynomials generally can ignore trailing zero coefficients. This can be an important efficiency consideration in some cases.

    Algorithms and methods in the bit::polynomial class allow for this and internally work efficiently even if the polynomials are not monic. They do that by reimplementing some of the core bit::vector functionality to only consider underlying storage blocks up to and including the one containing the highest non-trivial power.

    -

    Polynomial methods usually need to treat the zero-polynomial as a special, generally trivial, edge case.

    +

    We also note that polynomial methods usually need to treat the zero-polynomial as a special, generally trivial, edge case.

    • @@ -618,7 +621,10 @@

    Declaration

    Allocator = std::allocator<Block>> class polynomial; } -

    The bit::polynomial class has an underlying bit-vector data member holding the polynomial coefficients that use the two template parameters (see the documentation for bit::vector).

    +

    The bit::polynomial class holds the polynomial coefficients in a bit::vector data member that uses the two template parameters.

    +

    The two template parameters add some visual clutter, but they both have reasonable defaults and disappear entirely in most uses. For example, your code might have a simple line like:

    +
    bit::polynomial p{32};
    +

    This code creates a polynomial with 32 coefficients set to 0 by default.

    Template Parameters

    @@ -643,8 +649,8 @@

    Template Parameters

    The default Block is 64-bits, the native size for many modern CPUs.

    Of course, if you need to use many smaller degree bit-polynomials and have concerns about conserving space, you might use a different Block. Perhaps if the bit-polynomials all have degrees that fit in 8 bits, you might have code along the lines:

    -
    using poly_type= bit::polynomial<uint8_t>;
-poly_type p = ...
    +
    using poly_type= bit::polynomial<uint8_t>;
+poly_type p = ...
    @@ -677,11 +683,11 @@

    Class Constants

    • - + - + @@ -702,8 +708,8 @@

    Construction

    Working in GF(2)Notes about the vagaries of doing linear algebra over \(\FF\).Notes about the vagaries of working in \(\FF\).
    Library Design Notes An algorithm to compute the characteristic polynomial of a square matrix — the algorithm isn’t well known, so we review it here.
    Computing \(x^N\) mod \(P(x)\)Polynomial Reduction An algorithm to compute \(x^N\) mod \(P(x)\) where \(P(x)\) is a polynomial over \(\FF\) and \(N\) is potentially a huge integer.
    vector_typeAlias for bit::vector—the type used to store the polynomial coefficients.Alias for bit::vector — the type used to store the polynomial coefficients.
    matrix_typeAlias for bit::matrix—polynomials can be evaluated for scalar and square bit-matrix arguments of this type.Alias for bit::matrix — polynomials can be evaluated for scalar and square bit-matrix arguments of this type.
    ndeg
    --++ @@ -733,8 +739,8 @@

    Queries

    --++ @@ -769,7 +775,7 @@

    Queries

    - + @@ -808,7 +814,7 @@

    Modifiers

    - + @@ -859,11 +865,11 @@

    Coefficient Access

    - + - +
    polynomial::constantIs this a constant polynomial \(p(x) = 1 \mathrm{ or } 1\)?Is this a constant polynomial \(p(x) = 0 \text{ or } 1\)?
    polynomial::degree
    polynomial::resizeResizes the number of coefficients in the polynomial up or down. Added coefficients are all zeros.Resizes the number of coefficients in the polynomial up or down. Any added coefficients are set to zero.
    polynomial::clear
    polynomial::coefficientsRead-only access all the polynomial coefficients as a bit-vector.Read-only access to the polynomial coefficients as a bit-vector.
    polynomial::set_coefficientsWRite access all the polynomial coefficients as a bit-vector.Write access to the polynomial coefficients as a bit-vector.
    @@ -874,8 +880,8 @@

    Polynomial Evaluatio
    --++ @@ -897,8 +903,8 @@

    Arithmetic

    --++ @@ -936,8 +942,8 @@

    String Conversions

    --++ @@ -959,8 +965,8 @@

    Other Instance Meth

    --++ @@ -971,15 +977,15 @@

    Other Instance Meth

    - + - + - +
    polynomial::subCreate a new polynomial as a distinct sub-polynomial of this one.Create a distinct sub-polynomial of this one.
    polynomial::splitSplit a polynomial \(p(x)\) into two pieces so that \(p(x) = l(x) + x^n h(x)\).Split polynomial \(p(x)\) into \(p(x) = l(x) + x^n h(x)\).
    polynomial::reduceReduces \(x^e\) by this polynomial where the exponent \(e\) can be very large.Reduces \(x^e\) by this polynomial (\(e\) can be very large).
    @@ -991,8 +997,8 @@

    Debugging

    --++ @@ -1031,8 +1037,8 @@

    Non-member Functions<

    --++ @@ -1063,7 +1069,7 @@

    Non-member Functions<

    - +
    polynomial::formatterConnect the bit::polynomial class to std::format and friends.Connect the bit::polynomial class to std::format.
    diff --git a/content/polynomial/monic.html b/content/polynomial/monic.html index 44c9133..513da04 100644 --- a/content/polynomial/monic.html +++ b/content/polynomial/monic.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/polynomial/power.html b/content/polynomial/power.html index 4ee195b..0283fef 100644 --- a/content/polynomial/power.html +++ b/content/polynomial/power.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/polynomial/random.html b/content/polynomial/random.html index 6a46849..6319c01 100644 --- a/content/polynomial/random.html +++ b/content/polynomial/random.html @@ -181,6 +181,10 @@
  • Matrices +
    • +
  • + +Polynomials
  • @@ -568,7 +572,8 @@

    bit::polynomial — Random Poly

    Factory method to construct a fixed degree bit-polynomial whose coefficients come from independent random draws from a Bernoulli distribution

    static bit::polynomial random(std::size_t n, double prob_one = 0.5);
    -

    This method returns a polynomial of degree \(n\) with \(n+1\) coefficients. If \(n > 0\) then \(p_n = 1\) and all the other coefficients in the polynomial are 1 with probability prob_one. If \(n = 0\) then the single coefficient \(p_0\) is 1 with probability prob_one.

    +

    This method returns a polynomial of degree \(n\) with \(n+1\) coefficients.

    +

    If \(n > 0\) then \(p_n = 1\) and all the other coefficients in the polynomial are 1 with probability prob_one. If \(n = 0\) then the single coefficient \(p_0\) is 1 with probability prob_one.

    The default probability value is 0.5, so n coefficient values are determined by tossing a fair coin.

    At the extremes, if this parameter is 1.0, the coefficients will all be 1; if it is 0.0, the elements will all be 0.

    @@ -581,7 +586,7 @@

    bit::polynomial — Random Poly

    • -

    This method throws a std::invalid_argument exception if the prob_one argument is not in the valid range \([0, 1]\).

    +

    This method throws a std::invalid_argument exception if the prob_one argument is not in the valid range \([0, 1]\).

    Example

    @@ -589,18 +594,18 @@

    bit::polynomial — Random Poly int main() { auto p0 = bit::polynomial<>::random(0); - std::cout << std::format("p0(x) = {} has {} coefficient with {} one.\n", p0, p0.size(), p0.count1()); + std::cout << std::format("p0(x) = {} has degree {}.\n", p0, p0.degree()); - auto p1 = bit::polynomial<>::random(11); - std::cout << std::format("p1(x) = {} has {} coefficients with {} ones.\n", p1, p1.size(), p1.count1()); + auto p1 = bit::polynomial<>::random(7); + std::cout << std::format("p0(x) = {} has degree {}.\n", p1, p1.degree()); - auto p2 = bit::polynomial<>::random(11, 0.9); - std::cout << std::format("p2(x) = {} has {} coefficients with {} ones.\n", p2, p2.size(), p2.count1()); + auto p2 = bit::polynomial<>::random(7, 0.9); + std::cout << std::format("p0(x) = {} has degree {}.\n", p2, p2.degree()); }

    Output (will vary from run to run)

    -
    p0(x) = 1 has 1 coefficient with 1 one.
-p1(x) = x^3 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 has 12 coefficients with 7 ones.
-p2(x) = 1 + x^1 + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 has 12 coefficients with 12 ones.
    +
    p0(x) = 1 has degree 0.
+p0(x) = 1 + x^3 + x^7 has degree 7.
+p0(x) = 1 + x^1 + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 has degree 7.

    See Also

    polynomial::constructors
    diff --git a/content/polynomial/reduce.html b/content/polynomial/reduce.html index 73de0a9..eb41f77 100644 --- a/content/polynomial/reduce.html +++ b/content/polynomial/reduce.html @@ -181,6 +181,10 @@

  • Matrices +
    • +
  • + +Polynomials
  • diff --git a/content/polynomial/reference.html b/content/polynomial/reference.html index c8b17c8..a914f46 100644 --- a/content/polynomial/reference.html +++ b/content/polynomial/reference.html @@ -7,7 +7,7 @@ -Linear Algebra over GF(2) - bit::polynomial::reference +Linear Algebra over GF(2) - bit::polynomial::reference — Proxy Class