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23 Row Vectors

Just as in mathematics, a vector in **GAP** is any object which supports appropriate addition and scalar multiplication operations (see Chapter 61). As in mathematics, an especially important class of vectors are those represented by a list of coefficients with respect to some basis. These correspond roughly to the **GAP** concept of *row vectors*.

`‣ IsRowVector` ( obj ) | ( category ) |

A *row vector* is a vector (see `IsVector`

(31.14-14)) that is also a homogeneous list of odd additive nesting depth (see 21.12). Typical examples are lists of integers and rationals, lists of finite field elements of the same characteristic, and lists of polynomials from a common polynomial ring. Note that matrices are *not* regarded as row vectors, because they have even additive nesting depth.

The additive operations of the vector must thus be compatible with that for lists, implying that the list entries are the coefficients of the vector with respect to some basis.

Note that not all row vectors admit a multiplication via `*`

(which is to be understood as a scalar product); for example, class functions are row vectors but the product of two class functions is defined in a different way. For the installation of a scalar product of row vectors, the entries of the vector must be ring elements; note that the default method expects the row vectors to lie in `IsRingElementList`

, and this category may not be implied by `IsRingElement`

(31.14-16) for all entries of the row vector (see the comment in `IsVector`

(31.14-14)).

Note that methods for special types of row vectors really must be installed with the requirement `IsRowVector`

, since `IsVector`

(31.14-14) may lead to a rank of the method below that of the default method for row vectors (see file `lib/vecmat.gi`

).

gap> IsRowVector([1,2,3]); true

Because row vectors are just a special case of lists, all operations and functions for lists are applicable to row vectors as well (see Chapter 21). This especially includes accessing elements of a row vector (see 21.3), changing elements of a mutable row vector (see 21.4), and comparing row vectors (see 21.10).

Note that, unless your algorithms specifically require you to be able to change entries of your vectors, it is generally better and faster to work with immutable row vectors. See Section 12.6 for more details.

The rules for arithmetic operations involving row vectors are in fact special cases of those for the arithmetic of lists, as given in Section 21.11 and the following sections, here we reiterate that definition, in the language of vectors.

Note that the additive behaviour sketched below is defined only for lists in the category `IsGeneralizedRowVector`

(21.12-1), and the multiplicative behaviour is defined only for lists in the category `IsMultiplicativeGeneralizedRowVector`

(21.12-2).

`vec1` + `vec2`

returns the sum of the two row vectors `vec1` and `vec2`. Probably the most usual situation is that `vec1` and `vec2` have the same length and are defined over a common field; in this case the sum is a new row vector over the same field where each entry is the sum of the corresponding entries of the vectors.

In more general situations, the sum of two row vectors need not be a row vector, for example adding an integer vector `vec1` and a vector `vec2` over a finite field yields the list of pointwise sums, which will be a mixture of finite field elements and integers if `vec1` is longer than `vec2`.

`scalar` + `vec`

`vec` + `scalar`

returns the sum of the scalar `scalar` and the row vector `vec`. Probably the most usual situation is that the elements of `vec` lie in a common field with `scalar`; in this case the sum is a new row vector over the same field where each entry is the sum of the scalar and the corresponding entry of the vector.

More general situations are for example the sum of an integer scalar and a vector over a finite field, or the sum of a finite field element and an integer vector.

gap> [ 1, 2, 3 ] + [ 1/2, 1/3, 1/4 ]; [ 3/2, 7/3, 13/4 ] gap> [ 1/2, 3/2, 1/2 ] + 1/2; [ 1, 2, 1 ]

`vec1` - `vec2`

`scalar` - `vec`

`vec` - `scalar`

Subtracting a vector or scalar is defined as adding its additive inverse, so the statements for the addition hold likewise.

gap> [ 1, 2, 3 ] - [ 1/2, 1/3, 1/4 ]; [ 1/2, 5/3, 11/4 ] gap> [ 1/2, 3/2, 1/2 ] - 1/2; [ 0, 1, 0 ]

`scalar` * `vec`

`vec` * `scalar`

returns the product of the scalar `scalar` and the row vector `vec`. Probably the most usual situation is that the elements of `vec` lie in a common field with `scalar`; in this case the product is a new row vector over the same field where each entry is the product of the scalar and the corresponding entry of the vector.

More general situations are for example the product of an integer scalar and a vector over a finite field, or the product of a finite field element and an integer vector.

gap> [ 1/2, 3/2, 1/2 ] * 2; [ 1, 3, 1 ]

`vec1` * `vec2`

returns the standard scalar product of `vec1` and `vec2`, i.e., the sum of the products of the corresponding entries of the vectors. Probably the most usual situation is that `vec1` and `vec2` have the same length and are defined over a common field; in this case the sum is an element of this field.

More general situations are for example the inner product of an integer vector and a vector over a finite field, or the inner product of two row vectors of different lengths.

gap> [ 1, 2, 3 ] * [ 1/2, 1/3, 1/4 ]; 23/12

For the mutability of results of arithmetic operations, see 12.6.

Further operations with vectors as operands are defined by the matrix operations, see 24.3.

`‣ NormedRowVector` ( v ) | ( attribute ) |

returns a scalar multiple

of the row vector `w` = `c` * `v``v` with the property that the first nonzero entry of `w` is an identity element in the sense of `IsOne`

(31.10-5).

gap> NormedRowVector( [ 5, 2, 3 ] ); [ 1, 2/5, 3/5 ]

**GAP** can use compact formats to store row vectors over fields of order at most 256, based on those used by the Meat-Axe [Rin93]. This format also permits extremely efficient vector arithmetic. On the other hand element access and assignment is significantly slower than for plain lists.

The function `ConvertToVectorRep`

(23.3-1) is used to convert a list into a compressed vector, or to rewrite a compressed vector over another field. Note that this function is *much* faster when it is given a field (or field size) as an argument, rather than having to scan the vector and try to decide the field. Supplying the field can also avoid errors and/or loss of performance, when one vector from some collection happens to have all of its entries over a smaller field than the "natural" field of the problem.

`‣ ConvertToVectorRep` ( list[, field] ) | ( function ) |

`‣ ConvertToVectorRep` ( list[, fieldsize] ) | ( function ) |

`‣ ConvertToVectorRepNC` ( list[, field] ) | ( function ) |

`‣ ConvertToVectorRepNC` ( list[, fieldsize] ) | ( function ) |

Called with one argument `list`, `ConvertToVectorRep`

converts `list` to an internal row vector representation if possible.

Called with a list `list` and a finite field `field`, `ConvertToVectorRep`

converts `list` to an internal row vector representation appropriate for a row vector over `field`.

Instead of a `field` also its size `fieldsize` may be given.

It is forbidden to call this function unless `list` is a plain list or a row vector, `field` is a field, and all elements of `list` lie in `field`. Violation of this condition can lead to unpredictable behaviour or a system crash. (Setting the assertion level to at least 2 might catch some violations before a crash, see `SetAssertionLevel`

(7.5-1).)

`list` may already be a compressed vector. In this case, if no `field` or `fieldsize` is given, then nothing happens. If one is given then the vector is rewritten as a compressed vector over the given `field` unless it has the filter `IsLockedRepresentationVector`

, in which case it is not changed.

The return value is the size of the field over which the vector ends up written, if it is written in a compressed representation.

In this example, we first create a row vector and then ask **GAP** to rewrite it, first over `GF(2)`

and then over `GF(4)`

.

gap> v := [Z(2)^0,Z(2),Z(2),0*Z(2)]; [ Z(2)^0, Z(2)^0, Z(2)^0, 0*Z(2) ] gap> RepresentationsOfObject(v); [ "IsPlistRep", "IsInternalRep" ] gap> ConvertToVectorRep(v); 2 gap> v; <a GF2 vector of length 4> gap> ConvertToVectorRep(v,4); 4 gap> v; [ Z(2)^0, Z(2)^0, Z(2)^0, 0*Z(2) ] gap> RepresentationsOfObject(v); [ "IsDataObjectRep", "Is8BitVectorRep" ]

A vector in the special representation over `GF(2)`

is always viewed as `<a GF2 vector of length ...>`

. Over fields of orders 3 to 256, a vector of length 10 or less is viewed as the list of its coefficients, but a longer one is abbreviated.

Arithmetic operations (see 21.11 and the following sections) preserve the compression status of row vectors in the sense that if all arguments are compressed row vectors written over the same field and the result is a row vector then also the result is a compressed row vector written over this field.

`‣ ImmutableVector` ( field, vector[, change] ) | ( operation ) |

returns an immutable vector equal to `vector` which is in the optimal (concerning space and runtime) representation for vectors defined over `field`. This means that vectors obtained by several calls of `ImmutableVector`

for the same `field` are compatible for fast arithmetic without need for field conversion.

The input vector `vector` might change its representation as a side effect of this function, however the result of `ImmutableVector`

is not necessarily *identical* to `vector` if a conversion is not possible.

If `change` is `true`

, then `vector` may be changed to become immutable; otherwise it is copied first.

`‣ NumberFFVector` ( vec, sz ) | ( operation ) |

returns an integer that gives the position of the finite field row vector `vec` in the sorted list of all row vectors over the field with `sz` elements in the same dimension as `vec`. `NumberFFVector`

returns `fail`

if the vector cannot be represented over the field with `sz` elements.

The following operations all perform arithmetic on row vectors. given as homogeneous lists of the same length, containing elements of a commutative ring.

There are two reasons for using `AddRowVector`

(23.4-1) in preference to arithmetic operators. Firstly, the three argument form has no single-step equivalent. Secondly `AddRowVector`

(23.4-1) changes its first argument in-place, rather than allocating a new vector to hold the result, and may thus produce less garbage.

`‣ AddRowVector` ( dst, src[, mul[, from, to]] ) | ( operation ) |

Adds the product of `src` and `mul` to `dst`, changing `dst`. If `from` and `to` are given then only the index range `[ `

is guaranteed to be affected. Other indices `from` .. `to` ]*may* be affected, if it is more convenient to do so. Even when `from` and `to` are given, `dst` and `src` must be row vectors of the *same* length.

If `mul` is not given either then this operation simply adds `src` to `dst`.

`‣ AddCoeffs` ( list1[, poss1], list2[, poss2[, mul]] ) | ( operation ) |

`AddCoeffs`

adds the entries of `list2``{`

`poss2``}`

, multiplied by the scalar `mul`, to `list1``{`

`poss1``}`

. Unbound entries in `list1` are assumed to be zero. The position of the right-most non-zero element is returned.

If the ranges `poss1` and `poss2` are not given, they are assumed to span the whole vectors. If the scalar `mul` is omitted, one is used as a default.

Note that it is the responsibility of the caller to ensure that `list2` has elements at position `poss2` and that the result (in `list1`) will be a dense list.

The function is free to remove trailing (right-most) zeros.

gap> l:=[1,2,3,4];;m:=[5,6,7];;AddCoeffs(l,m); 4 gap> l; [ 6, 8, 10, 4 ]

`‣ MultRowVector` ( list1[, poss1, list2, poss2], mul ) | ( operation ) |

The five argument version of this operation replaces `list1``[`

`poss1``[`

\(i\)`]]`

by

\(i\)`mul`*`list2`[`poss2`[`]]`

for \(i\) between \(1\) and `Length( `

.`poss1` )

The two-argument version simply multiplies each element of `list1`, in-place, by `mul`.

`‣ CoeffsMod` ( list1[, len1], modulus ) | ( operation ) |

returns the coefficient list obtained by reducing the entries in `list1` modulo `modulus`. After reducing it shrinks the list to remove trailing zeroes. If the optional argument `len1` is used, it reduces only first `len1` elements of the list.

gap> l:=[1,2,3,4];;CoeffsMod(l,2); [ 1, 0, 1 ]

The following functions change coefficient lists by shifting or trimming.

`‣ LeftShiftRowVector` ( list, shift ) | ( operation ) |

changes `list` by assigning `list`\([i]\)`:= `

`list`\([i+\textit{shift}]\) and removing the last `shift` entries of the result.

`‣ RightShiftRowVector` ( list, shift, fill ) | ( operation ) |

changes `list` by assigning `list`\([i+\textit{shift}]\)`:= `

`list`\([i]\) and filling each of the `shift` first entries with `fill`.

`‣ ShrinkRowVector` ( list ) | ( operation ) |

removes trailing zeroes from the list `list`.

gap> l:=[1,0,0];;ShrinkRowVector(l);l; [ 1 ]

`‣ RemoveOuterCoeffs` ( list, coef ) | ( operation ) |

removes `coef` at the beginning and at the end of `list` and returns the number of elements removed at the beginning.

gap> l:=[1,1,2,1,2,1,1,2,1];; RemoveOuterCoeffs(l,1); 2 gap> l; [ 2, 1, 2, 1, 1, 2 ]

The following functions perform operations on finite fields vectors considered as code words in a linear code.

`‣ WeightVecFFE` ( vec ) | ( operation ) |

returns the weight of the finite field vector `vec`, i.e. the number of nonzero entries.

`‣ DistanceVecFFE` ( vec1, vec2 ) | ( operation ) |

returns the distance between the two vectors `vec1` and `vec2`, which must have the same length and whose elements must lie in a common field. The distance is the number of places where `vec1` and `vec2` differ.

`‣ DistancesDistributionVecFFEsVecFFE` ( vecs, vec ) | ( operation ) |

returns the distances distribution of the vector `vec` to the vectors in the list `vecs`. All vectors must have the same length, and all elements must lie in a common field. The distances distribution is a list \(d\) of length `Length(`

, such that the value \(d[i]\) is the number of vectors in `vec`)+1`vecs` that have distance \(i+1\) to `vec`.

`‣ DistancesDistributionMatFFEVecFFE` ( mat, F, vec ) | ( operation ) |

returns the distances distribution of the vector `vec` to the vectors in the vector space generated by the rows of the matrix `mat` over the finite field `F`. The length of the rows of `mat` and the length of `vec` must be equal, and all entries must lie in `F`. The rows of `mat` must be linearly independent. The distances distribution is a list \(d\) of length `Length(`

, such that the value \(d[i]\) is the number of vectors in the vector space generated by the rows of `vec`)+1`mat` that have distance \(i+1\) to `vec`.

`‣ AClosestVectorCombinationsMatFFEVecFFE` ( mat, f, vec, l, stop ) | ( operation ) |

`‣ AClosestVectorCombinationsMatFFEVecFFECoords` ( mat, f, vec, l, stop ) | ( operation ) |

These functions run through the `f`-linear combinations of the vectors in the rows of the matrix `mat` that can be written as linear combinations of exactly `l` rows (that is without using zero as a coefficient). The length of the rows of `mat` and the length of `vec` must be equal, and all elements must lie in the field `f`. The rows of `mat` must be linearly independent. `AClosestVectorCombinationsMatFFEVecFFE`

returns a vector from these that is closest to the vector `vec`. If it finds a vector of distance at most `stop`, which must be a nonnegative integer, then it stops immediately and returns this vector.

`AClosestVectorCombinationsMatFFEVecFFECoords`

returns a length 2 list containing the same closest vector and also a vector `v` with exactly `l` non-zero entries, such that `v` times `mat` is the closest vector.

`‣ CosetLeadersMatFFE` ( mat, f ) | ( operation ) |

returns a list of representatives of minimal weight for the cosets of a code. `mat` must be a *check matrix* for the code, the code is defined over the finite field `f`. All rows of `mat` must have the same length, and all elements must lie in the field `f`. The rows of `mat` must be linearly independent.

A list of ring elements can be interpreted as a row vector or the list of coefficients of a polynomial. There are a couple of functions that implement arithmetic operations based on these interpretations. **GAP** contains proper support for polynomials (see 66), the operations described in this section are on a lower level.

The following operations all perform arithmetic on univariate polynomials given by their coefficient lists. These lists can have different lengths but must be dense homogeneous lists containing elements of a commutative ring. Not all input lists may be empty.

In the following descriptions we will always assume that `list1` is the coefficient list of the polynomial `pol1` and so forth. If length parameter `leni` is not given, it is set to the length of `listi` by default.

`‣ ValuePol` ( coeff, x ) | ( operation ) |

Let `coeff` be the coefficients list of a univariate polynomial \(f\), and `x` a ring element. Then `ValuePol`

returns the value \(f(\textit{x})\).

The coefficient of \(\textit{x}^i\) is assumed to be stored at position \(i+1\) in the coefficients list.

gap> ValuePol([1,2,3],4); 57

`‣ ProductCoeffs` ( list1[, len1], list2[, len2] ) | ( operation ) |

Let \(p1\) (and \(p2\)) be polynomials given by the first `len1` (`len2`) entries of the coefficient list `list2` (`list2`). If `len1` and `len2` are omitted, they default to the lengths of `list1` and `list2`. This operation returns the coefficient list of the product of \(p1\) and \(p2\).

gap> l:=[1,2,3,4];;m:=[5,6,7];;ProductCoeffs(l,m); [ 5, 16, 34, 52, 45, 28 ]

`‣ ReduceCoeffs` ( list1[, len1], list2[, len2] ) | ( operation ) |

Let \(p1\) (and \(p2\)) be polynomials given by the first `len1` (`len2`) entries of the coefficient list `list1` (`list2`). If `len1` and `len2` are omitted, they default to the lengths of `list1` and `list2`. `ReduceCoeffs`

changes `list1` to the coefficient list of the remainder when dividing `p1` by `p2`. This operation changes `list1` which therefore must be a mutable list. The operation returns the position of the last non-zero entry of the result but is not guaranteed to remove trailing zeroes.

gap> l:=[1,2,3,4];;m:=[5,6,7];;ReduceCoeffs(l,m); 2 gap> l; [ 64/49, -24/49, 0, 0 ]

`‣ ReduceCoeffsMod` ( list1[, len1], list2[, len2], modulus ) | ( operation ) |

Let \(p1\) (and \(p2\)) be polynomials given by the first `len1` (`len2`) entries of the coefficient list `list1` (`list2`). If `len1` and `len2` are omitted, they default to the lengths of `list1` and `list2`. `ReduceCoeffsMod`

changes `list1` to the coefficient list of the remainder when dividing `p1` by `p2` modulo `modulus`, which must be a positive integer. This operation changes `list1` which therefore must be a mutable list. The operations returns the position of the last non-zero entry of the result but is not guaranteed to remove trailing zeroes.

gap> l:=[1,2,3,4];;m:=[5,6,7];;ReduceCoeffsMod(l,m,3); 1 gap> l; [ 1, 0, 0, 0 ]

`‣ PowerModCoeffs` ( list1[, len1], exp, list2[, len2] ) | ( operation ) |

Let \(p1\) and \(p2\) be polynomials whose coefficients are given by the first `len1` resp. `len2` entries of the lists `list1` and `list2`, respectively. If `len1` and `len2` are omitted, they default to the lengths of `list1` and `list2`. Let `exp` be a positive integer. `PowerModCoeffs`

returns the coefficient list of the remainder when dividing the `exp`-th power of \(p1\) by \(p2\). The coefficients are reduced already while powers are computed, therefore avoiding an explosion in list length.

gap> l:=[1,2,3,4];;m:=[5,6,7];;PowerModCoeffs(l,5,m); [ -839462813696/678223072849, -7807439437824/678223072849 ]

`‣ ShiftedCoeffs` ( list, shift ) | ( operation ) |

produces a new coefficient list `new`

obtained by the rule `new[i+`

and filling initial holes by the appropriate zero.`shift`]:= `list`[i]

gap> l:=[1,2,3];;ShiftedCoeffs(l,2);ShiftedCoeffs(l,-2); [ 0, 0, 1, 2, 3 ] [ 3 ]

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