# httk.core.vectors.fracvector module¶

class `httk.core.vectors.fracvector.``FracScalar`(nom, denom)[source]

Represents the fractional number nom/denom. This is a subclass of FracVector with the purpose of making it clear when a scalar fracvector is needed/used.

classmethod `create`(nom, denom=None, simplify=True)[source]

Create a FracScalar.

FracScalar(something)
something may be any object that can be used in the constructor of the Python Fraction class (also works with strings!).
class `httk.core.vectors.fracvector.``FracVector`(noms, denom=1)[source]

FracVector is a general immutable N-dimensional vector (tensor) class for performing linear algebra with fractional numbers.

A FracVector consists of a multidimensional tuple of integer nominators, and a single shared integer denominator.

Since FracVectors are immutable, every operation on a FracVector returns a new FracVector with the result of the operation. A created FracVector never changes. Hence, they are safe to use as keys in dictionaries, to use in sets, etc.

Note: most methods returns FracVector results that are not simplified (i.e., the FracVector returned does not have the smallest possible integer denominator). To return a FracVector with the smallest possible denominator, just call FracVector.simplify() at the last step.

`T`()[source]

Returns the transpose, A^T.

`acos`(prec=None, degrees=False, limit=False)[source]

Return a FracVector where every element is the arccos of the element in the source FracVector.

prec = precision (should be set as a fraction) limit = True requires the denominator to be smaller or equal to precision

`argmax`()[source]

Return the index of the maximum element across all dimensions in the FracVector.

`argmin`()[source]

Return the index of the minimum element across all dimensions in the FracVector.

`asin`(prec=None, degrees=False, limit=False)[source]

Return a FracVector where every element is the arcsin of the element in the source FracVector.

prec = precision (should be set as a fraction) limit = True requires the denominator to be smaller or equal to precision

`ceil`()[source]

Returns the integer that is equal to or just below the value stored in a scalar FracVector.

classmethod `chain_vecs`(vecs)[source]

Optimized chaining of FracVectors.

vecs: a list (or tuple) of fracvectors.

Returns the same thing as
FracVector.create(vecs,chain=True)
i.e., removes outermost dimension and chain the sub-sequences. If input=[[1 2 3],[4,5,6]], then
FracVector.chain(input) -> [1,2,3,4,5,6]

but this method assumes all vectors share the same denominator (it raises an exception if this is not true)

`cos`(prec=None, degrees=False, limit=False)[source]

Return a FracVector where every element is the cosine of the element in the source FracVector.

prec = precision (should be set as a fraction) limit = True requires the denominator to be smaller or equal to precision

classmethod `create`(noms, denom=None, simplify=True, chain=False, min_accuracy=Fraction(1, 10000))[source]

Create a FracVector from various types of sequences.

Simplest use:

```FracVector.create(some_kind_of_sequence)
```

where ‘some_kind_of_sequence’ can be any nested list or tuple of objects that can be used in the constructor of the Python Fraction class (also works with strings!). If any object found while traveling the items has a .to_fractions() method, it will be called and is expected to return a fraction or list or tuple of fractions.

Optional parameters:

• Invocation with denominator: FracVector.create(nominators,denominator) nominators is any sequence, and denominator a common denominator to divide all nominators with
• simplify: boolean, return a FracVector with the smallest possible denominator.
• chain: boolean, remove outermost dimension and chain the sub-sequences. I.e., if input=[[1 2 3],[4,5,6]], then FracVector.create(input) -> [1,2,3,4,5,6]

Relevant: FracVector itself implements .to_fractions(), and hence, the same constructor allows stacking several FracVector objects like this:

```vertical_fracvector = FracVector.create([[fracvector1],[fracvector2]])
horizontal_fracvector = FracVector.create([fracvector1,fracvector2],chain=True)
```
• min_accuracy: set to a boolean to adjust the minimum accuracy assumed in string input. The default is 1/10000, i.e. 0.33 = 0.3300 = 33/100, whereas 0.3333 = 1/3. Set it to None to assume infinite accuracy, i.e., convert exactly whatever string is given (unless a standard deviation is given as a parenthesis after the string.)
classmethod `create_cos`(data, degrees=False, limit=False, find_best_rational=True, prec=Fraction(1, 1000000))[source]

Creating a FracVector as the cosine of the argument data. If data are composed by strings, the standard deviation of the numbers are taken into account, and the best possible fractional approximation to the cosines of the data are returned within the standard deviation.

This is not the same as FracVector.create(data).cos(), which creates the best possible fractional approximations of data and then takes cos on that.

classmethod `create_exp`(data, prec=Fraction(1, 1000000), limit=False)[source]

Creating a FracVector as the exponent of the argument data. If data are composed by strings, the standard deviation of the numbers are taken into account, and the best possible fractional approximation to the cosines of the data are returned within the standard deviation.

This is not the same as FracVector.create(data).exp(), which creates the best possible fractional approximations of data and then takes exp on that.

classmethod `create_sin`(data, degrees=False, limit=False, prec=Fraction(1, 1000000))[source]

Creating a FracVector as the sine of the argument data. If data are composed by strings, the standard deviation of the numbers are taken into account, and the best possible fractional approximation to the cosines of the data are returned within the standard deviation.

This is not the same as FracVector.create(data).sin(), which creates the best possible fractional approximations of data and then takes cos on that.

`cross`(other)[source]

Returns the vector cross product of the 3-element 1D vector with the 3-element 1D vector ‘other’, i.e., A x B.

`det`()[source]

Returns the determinant of the FracVector as a scalar FracVector.

`dim`

This property returns a tuple with the dimensionality of each dimension of the FracVector (the noms are assumed to be a nested list of rectangular shape).

`dot`(other)[source]

Returns the vector dot product of the 1D vector with the 1D vector ‘other’, i.e., A . B or A cdot B. The same as A * B.T().

`exp`(prec=None, limit=False)[source]

Return a FracVector where every element is the exponent of the element in the source FracVector.

prec = precision (should be set as a fraction) limit = True requires the denominator to be smaller or equal to precision

classmethod `eye`(dims)[source]

Create a diagonal one-matrix with the given dimensions

`flatten`()[source]

Returns a FracVector that has been flattened out to a single rowvector

`floor`()[source]

Returns the integer that is equal to or just below the value stored in a scalar FracVector.

classmethod `from_floats`(l, resolution=4294967296)[source]

Create a FracVector from a (nested) list or tuple of floats. You can convert a numpy array with this method if you use A.tolist()

resolution: the resolution used for interpreting the given floating point numbers. Default is 2^32.

classmethod `from_tuple`(t)[source]

Return a FracVector created from the tuple representation: (denom, …noms…), returned by the to_tuple() method.

`ged_prestacked`(other)[source]
`ged_stackedinsert`(pos, other)[source]
`get_append`(other)[source]
`get_extend`(other)[source]
`get_insert`(pos, other)[source]
`get_prepend`(other)[source]
`get_prextend`(other)[source]
`get_stacked`(other)[source]
`inv`()[source]

Returns the matrix inverse, A^-1

`lengthsqr`()[source]

Returns the square of the length of the vector. The same as A * A.T()

`limit_denominator`(max_denom=1000000000)[source]

Returns a FracVector of reduced resolution.

resolution: each element in the returned FracVector is the closest numerical approximation that can is allowed by a fraction with maximally this denominator. Note: since all elements must be put on a common denominator, the result may have a larger denominator than max_denom

`max`()[source]

Return the maximum element across all dimensions in the FracVector. max(fracvector) works for a 1D vector.

`metric_product`(vecA, vecB)[source]
Returns the result of the metric product using the present square FracVector as the metric matrix. The same as
vecA*self*vecB.T().
`min`()[source]

Return the minimum element across all dimensions in the FracVector. max(fracvector) works for a 1D vector.

`mul`(other)[source]

Returns the result of multiplying the vector with ‘other’ using matrix multiplication.

Note that for two 1D FracVectors, A.dot(B) is not the same as A.mul(B), but rather: A.mul(B.T()).

`nargmax`()[source]

Return a list of indices of all maximum elements across all dimensions in the FracVector.

`nargmin`()[source]

Return a list of indices for all minimum elements across all dimensions in the FracVector.

static `nested_map`(op, *ls)

Map an operator over a nested tuple. (i.e., the same as the built-in map(), but works recursively on a nested tuple)

static `nested_map_fractions`(op, *ls)

Map an operator over a nested tuple, but checks every element for a method to_fractions() and uses this to further convert objects into tuples of Fraction.

`nom`

Returns the integer nominator of a scalar FracVector.

`normalize`()[source]

Add/remove an integer +/-N to each element to place it in the range [0,1)

`normalize_half`()[source]

Add/remove an integer +/-N to each element to place it in the range [-1/2,1/2)

This is useful to find the shortest vector C between two points A, B in a space with periodic boundary conditions [0,1):
C = (A-B).normalize_half()
classmethod `pi`(prec=Fraction(1, 1000000), limit=False)[source]

Create a scalar FracVector with a rational approximation of pi to precision prec.

classmethod `random`(dims, minnom=-100, maxnom=100, denom=100)[source]

Create a zero matrix with the given dimensions

`reciprocal`()[source]
classmethod `set_common_denom`(A, B)[source]

Used internally to combine two different FracVectors.

Returns a tuple (A2,B2,denom) where A2 is numerically equal to A, and B2 is numerically equal to B, but A2 and B2 are both set on the same shared denominator ‘denom’ which is the product of the denominator of A and B.

`set_denominator`(set_denom=1000000000)[source]

Returns a FracVector of reduced resolution where every element is the closest numerical approximation using this denominator.

`sign`()[source]

Returns the sign of the scalar FracVector: -1, 0 or 1.

`simplify`()[source]

Returns a reduced FracVector. I.e., each element has the same numerical value but the new FracVector represents them using the smallest possible shared denominator.

`sin`(prec=None, degrees=False, limit=False)[source]

Return a FracVector where every element is the sine of the element in the source FracVector.

prec = precision (should be set as a fraction) limit = True requires the denominator to be smaller or equal to precision

`sqrt`(prec=None, limit=False)[source]

Return a FracVector where every element is the sqrt of the element in the source FracVector.

prec = precision (should be set as a fraction) limit = True requires the denominator to be smaller or equal to precision

classmethod `stack_vecs`(vecs)[source]

Optimized stacking of FracVectors.

vecs = a list (or tuple) of fracvectors.

Returns the same thing as:

```FracVector.create(vecs)
```

but only works if all vectors share the same denominator (raises an exception if this is not true)

`to_float`()[source]

Converts a scalar ExactVector to a single float.

`to_floats`()[source]

Converts the ExactVector to a list of floats.

`to_fraction`()[source]

Converts scalar FracVector to a fraction.

`to_fractions`()[source]

Converts the FracVector to a list of fractions.

`to_int`()[source]

Converts scalar FracVector to an integer (truncating as necessary).

`to_ints`()[source]

Converts the FracVector to a list of integers, rounded off as best possible.

`to_string`(accuracy=8)[source]

Converts the ExactVector to a list of strings.

`to_strings`(accuracy=8)[source]

Converts the ExactVector to a list of strings.

`to_tuple`()[source]

Return a FracVector on tuple representation: (denom, …noms…).

classmethod `use`(old)[source]

Make sure variable is a FracVector, and if not, convert it.

`validate`()[source]
classmethod `zeros`(dims)[source]

Create a zero matrix with the given dimensions

`httk.core.vectors.fracvector.``main`()[source]
`httk.core.vectors.fracvector.``nested_map_fractions_list`(op, *ls)[source]

Map an operator over a nested list, but checks every element for a method to_fractions() and uses this to further convert objects into lists of Fraction.

`httk.core.vectors.fracvector.``nested_map_fractions_tuple`(op, *ls)[source]

Map an operator over a nested tuple, but checks every element for a method to_fractions() and uses this to further convert objects into tuples of Fraction.

`httk.core.vectors.fracvector.``nested_map_list`(op, *ls)[source]

Map an operator over a nested list. (i.e., the same as the built-in map(), but works recursively on a nested list)

`httk.core.vectors.fracvector.``nested_map_tuple`(op, *ls)[source]

Map an operator over a nested tuple. (i.e., the same as the built-in map(), but works recursively on a nested tuple)

`httk.core.vectors.fracvector.``nested_reduce`(op, l, initializer=None)[source]

Same as built-in reduce, but operates on a nested tuple/list/sequence.

`httk.core.vectors.fracvector.``nested_reduce_fractions`(op, l, initializer=None)[source]

Same as built-in reduce, but operates on a nested tuple/list/sequence. Also checks every element for a method to_fractions() and uses this to further convert such elements to lists of fractions.

`httk.core.vectors.fracvector.``nested_reduce_levels`(op, l, level=1, initializer=None)[source]

Same as built-in reduce, but operates on a nested tuple/list/sequence.

`httk.core.vectors.fracvector.``tuple_eye`(dims, onepos=0)[source]

Create a matrix with the given dimensions and 1 on the diagonal

`httk.core.vectors.fracvector.``tuple_index`(dims, uppidx=())[source]

Create a nested tuple where every element is a tuple indicating the position of that tuple

`httk.core.vectors.fracvector.``tuple_random`(dims, minval, maxval)[source]

Create a nested tuple with the given dimensions filled with random numbers between minval and maxval

`httk.core.vectors.fracvector.``tuple_slice`(l, key)[source]

Given a python slice (i.e., what you get to __getitem__ when you write A[3:2]), cut out the relevant nested tuple.

`httk.core.vectors.fracvector.``tuple_zeros`(dims)[source]

Create a netsted tuple with the given dimensions filled with zeroes