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Implement helpers related to lattice bases #39

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Jun 22, 2024
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Implement helpers related to lattice bases #39

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324 changes: 324 additions & 0 deletions src/__tests__/basis.spec.ts
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import {describe, expect, it} from 'vitest';
import {
fractionalGram,
fractionalLenstraLenstraLovasz,
gram,
lenstraLenstraLovasz,
} from '../basis';
import {
applyWeights,
fractionalDot,
fractionalMonzosEqual,
monzoToFraction,
monzosEqual,
toMonzo,
unapplyWeights,
} from '../monzo';
import {dot} from '../number-array';
import {LOG_PRIMES} from '../primes';

const FUZZ = 'FUZZ' in process.env;

describe('Gram process', () => {
it('orthogonalizes a basis', () => {
const basis = [
[1, 2, 3],
[4, -5, 6],
[-7, 8, 9],
];
const {ortho, dual} = gram(basis);
// Leading orientation
expect(monzosEqual(basis[0], ortho[0]));
// Geometric duals
expect(dot(ortho[0], dual[0])).toBeCloseTo(1);
expect(dot(ortho[1], dual[1])).toBeCloseTo(1);
expect(dot(ortho[2], dual[2])).toBeCloseTo(1);
// Orthogonality
expect(dot(ortho[0], ortho[1])).toBeCloseTo(0);
expect(dot(ortho[0], ortho[2])).toBeCloseTo(0);
expect(dot(ortho[1], ortho[2])).toBeCloseTo(0);
// Value
expect(ortho.map(o => o.map(c => c.toFixed(2)))).toEqual([
['1.00', '2.00', '3.00'],
['3.14', '-6.71', '3.43'],
['-7.46', '-1.66', '3.59'],
]);
});

it('handles non-basis', () => {
const basis = [
[1, 2, 3, 4],
[0, 1, 1, 0],
[1, 1, 2, 4],
[-1, 2, 0, 0],
];
const {ortho} = gram(basis);
// Pseudo-orthogonality
expect(dot(ortho[0], ortho[1])).toBeCloseTo(0);
expect(dot(ortho[0], ortho[2])).toBeCloseTo(0);
expect(dot(ortho[0], ortho[3])).toBeCloseTo(0);
expect(dot(ortho[1], ortho[2])).toBeCloseTo(0);
expect(dot(ortho[1], ortho[3])).toBeCloseTo(0);
expect(dot(ortho[2], ortho[3])).toBeCloseTo(0);
});

it.runIf(FUZZ)('Fuzzes for random bases', () => {
for (let k = 0; k < 100000; ++k) {
const basis: number[][] = [];
for (let i = Math.random() * 10; i > 0; --i) {
const row: number[] = [];
for (let j = Math.random() * 10; j > 0; --j) {
row.push(Math.random() * 100 - 50);
}
basis.push(row);
}
const {ortho} = gram(basis);
// Pseudo-orthogonality
for (let i = 0; i < basis.length; ++i) {
for (let j = 0; j < i; ++j) {
expect(dot(ortho[i], ortho[j])).toBeCloseTo(0);
}
}
}
});
});

describe('Gram process for arrays of fractions', () => {
it('orthogonalizes a basis', () => {
const basis = [
[1, 2, 3],
[4, -5, 6],
[-7, 8, 9],
];
const {ortho, dual} = fractionalGram(basis);
// Leading orientation
expect(fractionalMonzosEqual(basis[0], ortho[0]));
// Geometric duals
expect(fractionalDot(ortho[0], dual[0]).toFraction()).toBe('1');
expect(fractionalDot(ortho[1], dual[1]).toFraction()).toBe('1');
expect(fractionalDot(ortho[2], dual[2]).toFraction()).toBe('1');
// Orthogonality
expect(fractionalDot(ortho[0], ortho[1]).n).toBe(0);
expect(fractionalDot(ortho[0], ortho[2]).n).toBe(0);
expect(fractionalDot(ortho[1], ortho[2]).n).toBe(0);
// Value
expect(ortho.map(o => o.map(c => c.toFraction()))).toEqual([
['1', '2', '3'],
['22/7', '-47/7', '24/7'],
['-3483/467', '-774/467', '1677/467'],
]);
});

it('handles non-basis', () => {
const basis = [
[1, 2, 3, 4],
[0, 1, 1, 0],
[1, 1, 2, 4],
[-1, 2, 0, 0],
];
const {ortho} = fractionalGram(basis);
// Pseudo-orthogonality
expect(fractionalDot(ortho[0], ortho[1]).n).toBeCloseTo(0);
expect(fractionalDot(ortho[0], ortho[2]).n).toBeCloseTo(0);
expect(fractionalDot(ortho[0], ortho[3]).n).toBeCloseTo(0);
expect(fractionalDot(ortho[1], ortho[2]).n).toBeCloseTo(0);
expect(fractionalDot(ortho[1], ortho[3]).n).toBeCloseTo(0);
expect(fractionalDot(ortho[2], ortho[3]).n).toBeCloseTo(0);
});
});

describe('LLL basis reduction', () => {
it('can LLL reduce', () => {
const basis = [
[1, 1, 1],
[-1, 0, 2],
[3, 5, 6],
];
const lll = lenstraLenstraLovasz(basis);
// Size-reduction
for (let i = 0; i < 3; ++i) {
for (let j = 0; j < i; ++j) {
expect(
Math.abs(dot(lll.basis[i], lll.gram.dual[j]))
).toBeLessThanOrEqual(0.5);
}
}
// Lovász condition
for (let k = 1; k < 3; ++k) {
const ok = lll.gram.ortho[k];
const ok1 = lll.gram.ortho[k - 1];
const mu = dot(lll.basis[k], lll.gram.dual[k - 1]);
const n1 = dot(ok1, ok1);
expect((n1 * 3) / 4).toBeLessThanOrEqual(dot(ok, ok) + n1 * mu * mu);
}

expect(lll.basis).toEqual([
[0, 1, 0],
[1, 0, 1],
[-1, 0, 2],
]);
});

it('handles non-basis', () => {
const basis = [
[1, 2, 3, 4],
[0, 1, 1, 0],
[1, 1, 2, 4],
[-1, 2, 0, 0],
];
const lll = lenstraLenstraLovasz(basis);
expect(lll.basis).toEqual([
[0, 0, 0, 0],
[0, 1, 1, 0],
[-1, 1, -1, 0],
[0, 0, -1, 4],
]);
});

it('can mess up the basis of miracle with naïve weights', () => {
const basis = ['225/224', '1029/1024'].map(toMonzo);
const lll = lenstraLenstraLovasz(basis);
expect(lll.basis.map(m => monzoToFraction(m).toFraction())).toEqual([
'225/224',
'2401/2400',
]);
});

it('can fix the basis of miracle with Tenney weights', () => {
const basis = ['225/224', '2401/2400']
.map(toMonzo)
.map(m => applyWeights(m, LOG_PRIMES));
const lll = lenstraLenstraLovasz(basis);
const commas = lll.basis
.map(m => unapplyWeights(m, LOG_PRIMES).map(Math.round))
.map(m => monzoToFraction(m).toFraction());
expect(commas).toEqual(['225/224', '1029/1024']);
});

it.runIf(FUZZ)('Fuzzes for random bases', () => {
for (let k = 0; k < 1000; ++k) {
let basis: number[][] = [];
for (let i = Math.random() * 10; i > 0; --i) {
const row: number[] = [];
for (let j = Math.random() * 10; j > 0; --j) {
row.push(Math.random() * 100 - 50);
}
basis.push(row);
}
if (Math.random() < 0.5) {
basis = basis.map(row => row.map(Math.round));
}
const lll = lenstraLenstraLovasz(basis);
if (lll.gram.squaredLengths.every(l => l)) {
// Size-reduction
for (let i = 0; i < basis.length; ++i) {
for (let j = 0; j < i; ++j) {
expect(
Math.abs(dot(lll.basis[i], lll.gram.dual[j]))
).toBeLessThanOrEqual(0.5);
}
}
// Lovász condition
for (let k = 1; k < basis.length; ++k) {
const ok = lll.gram.ortho[k];
const ok1 = lll.gram.ortho[k - 1];
const mu = dot(lll.basis[k], lll.gram.dual[k - 1]);
const n1 = dot(ok1, ok1);
expect((n1 * 3) / 4).toBeLessThanOrEqual(dot(ok, ok) + n1 * mu * mu);
}
}
}
});
});

describe('Precise LLL basis reduction', () => {
it('can LLL reduce', () => {
const basis = [
[1, 1, 1],
[-1, 0, 2],
[3, 5, 6],
];
const lll = fractionalLenstraLenstraLovasz(basis);
// Size-reduction
for (let i = 0; i < 3; ++i) {
for (let j = 0; j < i; ++j) {
expect(
fractionalDot(lll.basis[i], lll.gram.dual[j]).compare(0.5)
).toBeLessThanOrEqual(0);
}
}
// Lovász condition
for (let k = 1; k < 3; ++k) {
const ok = lll.gram.ortho[k];
const ok1 = lll.gram.ortho[k - 1];
const mu = fractionalDot(lll.basis[k], lll.gram.dual[k - 1]);
const n1 = fractionalDot(ok1, ok1);
expect(
n1.mul('3/4').compare(fractionalDot(ok, ok).add(n1.mul(mu.mul(mu))))
).toBeLessThanOrEqual(0);
}

expect(lll.basis.map(row => row.map(f => f.valueOf()))).toEqual([
[0, 1, 0],
[1, 0, 1],
[-1, 0, 2],
]);
});

it('handles non-basis', () => {
const basis = [
[1, 2, 3, 4],
[0, 1, 1, 0],
[1, 1, 2, 4],
[-1, 2, 0, 0],
];
const lll = fractionalLenstraLenstraLovasz(basis);
expect(lll.basis.map(row => row.map(f => f.valueOf()))).toEqual([
[0, 0, 0, 0],
[0, 1, 1, 0],
[-1, 1, -1, 0],
[0, 0, -1, 4],
]);
});

it.runIf(FUZZ)('Fuzzes for random bases', () => {
for (let k = 0; k < 500; ++k) {
let basis: number[][] = [];
for (let i = Math.random() * 10; i > 0; --i) {
const row: number[] = [];
for (let j = Math.random() * 10; j > 0; --j) {
row.push(Math.random() * 20 - 10);
}
basis.push(row);
}
basis = basis.map(row => row.map(Math.round));
try {
const lll = fractionalLenstraLenstraLovasz(basis);
if (lll.gram.squaredLengths.every(l => l.n)) {
// Size-reduction
for (let i = 0; i < basis.length; ++i) {
for (let j = 0; j < i; ++j) {
expect(
fractionalDot(lll.basis[i], lll.gram.dual[j]).abs().compare(0.5)
).toBeLessThanOrEqual(0);
}
}
// Lovász condition
for (let k = 1; k < basis.length; ++k) {
const ok = lll.gram.ortho[k];
const ok1 = lll.gram.ortho[k - 1];
const mu = fractionalDot(lll.basis[k], lll.gram.dual[k - 1]);
const n1 = fractionalDot(ok1, ok1);
expect(
n1
.mul('3/4')
.compare(fractionalDot(ok, ok).add(n1.mul(mu).mul(mu)))
).toBeLessThanOrEqual(0);
}
}
} catch (e) {
expect(e.message).includes('above safe limit');
}
}
});
});
31 changes: 31 additions & 0 deletions src/__tests__/monzo.spec.ts
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@@ -1,6 +1,9 @@
import {describe, it, expect} from 'vitest';
import {Fraction} from '../fraction';
import {
fractionalAdd,
fractionalMonzosEqual,
fractionalNorm,
monzoToBigInt,
monzoToFraction,
primeFactorize,
Expand Down Expand Up @@ -468,3 +471,31 @@ describe('Sparse monzos', () => {
}
});
});

describe('Fractional monzo methods', () => {
it('test for equality between two monzos (equal)', () => {
const yes = fractionalMonzosEqual(
['1/2', '7/9'],
[0.5, new Fraction(14, 18), 0]
);
expect(yes).toBe(true);
});

it('test for equality between two monzos (not equal)', () => {
const no = fractionalMonzosEqual(
['1/2', '7/9'],
[0.75, new Fraction(7, 9)]
);
expect(no).toBe(false);
});

it('adds two fractional monzos', () => {
const result = fractionalAdd(['1/2', '2/3'], [new Fraction(1), 0.75]);
expect(result.map(f => f.toFraction())).toEqual(['3/2', '17/12']);
});

it('measures the naïve squared length of a fractional monzo', () => {
const l2 = fractionalNorm([0.5, '1/3', '5/7']);
expect(l2.toFraction()).toBe('1537/1764');
});
});
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