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Jacobi eigenvalue algorithm #461

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ivandev0 merged 5 commits from kylchik/jacobi into dev 2022-02-20 10:18:06 +03:00
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3 changed files with 149 additions and 75 deletions
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5
benchmarks/build.gradle.kts
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@ -124,6 +124,11 @@ benchmark {
include("JafamaBenchmark")
}
configurations.register("tensorAlgebra") {
commonConfiguration()
include("TensorAlgebraBenchmark")
}
configurations.register("viktor") {
commonConfiguration()
include("ViktorBenchmark")

37
benchmarks/src/jvmMain/kotlin/space/kscience/kmath/benchmarks/TensorAlgebraBenchmark.kt Normal file
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@ -0,0 +1,37 @@
/*
* Copyright 2018-2021 KMath contributors.
* Use of this source code is governed by the Apache 2.0 license that can be found in the license/LICENSE.txt file.
*/
package space.kscience.kmath.benchmarks
import kotlinx.benchmark.Benchmark
import kotlinx.benchmark.Blackhole
import kotlinx.benchmark.Scope
import kotlinx.benchmark.State
import space.kscience.kmath.linear.linearSpace
import space.kscience.kmath.linear.matrix
import space.kscience.kmath.linear.symmetric
import space.kscience.kmath.operations.DoubleField
import space.kscience.kmath.tensors.core.tensorAlgebra
import kotlin.random.Random
@State(Scope.Benchmark)
internal class TensorAlgebraBenchmark {
companion object {
private val random = Random(12224)
private const val dim = 30
private val matrix = DoubleField.linearSpace.matrix(dim, dim).symmetric { _, _ -> random.nextDouble() }
}
@Benchmark
fun tensorSymEigSvd(blackhole: Blackhole) = with(Double.tensorAlgebra) {
blackhole.consume(matrix.symEigSvd(1e-10))
}
@Benchmark
fun tensorSymEigJacobi(blackhole: Blackhole) = with(Double.tensorAlgebra) {
blackhole.consume(matrix.symEigJacobi(50, 1e-10))
}
}

180
kmath-tensors/src/commonMain/kotlin/space/kscience/kmath/tensors/core/DoubleTensorAlgebra.kt
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@ -9,11 +9,7 @@
package space.kscience.kmath.tensors.core
import space.kscience.kmath.misc.PerformancePitfall
import space.kscience.kmath.nd.MutableStructure2D
import space.kscience.kmath.nd.Structure2D
import space.kscience.kmath.nd.StructureND
import space.kscience.kmath.nd.as1D
import space.kscience.kmath.nd.as2D
import space.kscience.kmath.nd.*
import space.kscience.kmath.operations.DoubleField
import space.kscience.kmath.structures.MutableBuffer
import space.kscience.kmath.structures.indices
@ -886,7 +882,7 @@ public open class DoubleTensorAlgebra :
return Triple(uTensor.transpose(), sTensor, vTensor.transpose())
}
override fun StructureND<Double>.symEig(): Pair<DoubleTensor, DoubleTensor> = symEigJacobi(epsilon = 1e-10)
override fun StructureND<Double>.symEig(): Pair<DoubleTensor, DoubleTensor> = symEigJacobi(maxIteration = 50, epsilon = 1e-15)
/**
* Returns eigenvalues and eigenvectors of a real symmetric matrix input or a batch of real symmetric matrices,
@ -923,103 +919,139 @@ public open class DoubleTensorAlgebra :
return eig to v
}
public fun StructureND<Double>.symEigJacobi(epsilon: Double): Pair<DoubleTensor, DoubleTensor> {
public fun StructureND<Double>.symEigJacobi(maxIteration: Int, epsilon: Double): Pair<DoubleTensor, DoubleTensor> {
checkSymmetric(tensor, epsilon)
val size = this.dimension
val s = zeros(this.shape)
val eigenvectors = zeros(this.shape)
val eigenvalues = zeros(this.shape.sliceArray(0 until size - 1))
var eigenvalueStart = 0
var eigenvectorStart = 0
for (matrix in tensor.matrixSequence()) {
matrix.as2D().jacobiHelper(eigenvalues, s, eigenvalueStart, eigenvectorStart, epsilon)
val matrix2D = matrix.as2D()
val (d, v) = matrix2D.jacobiHelper(maxIteration, epsilon)
for (i in 0 until matrix2D.rowNum) {
for (j in 0 until matrix2D.colNum) {
eigenvectors.mutableBuffer.array()[eigenvectorStart + i * matrix2D.rowNum + j] = v[i, j]
}
}
for (i in 0 until matrix2D.rowNum) {
eigenvalues.mutableBuffer.array()[eigenvalueStart + i] = d[i]
}
eigenvalueStart += this.shape.last()
eigenvectorStart += this.shape.last() * this.shape.last()
}
// TODO sort eigenvalues
return eigenvalues to s
return eigenvalues to eigenvectors
}
private fun MutableStructure2D<Double>.jacobiHelper(
eigenvalues: DoubleTensor,
eigenvectors: DoubleTensor,
eigenvalueStart: Int,
eigenvectorStart: Int,
maxIteration: Int,
epsilon: Double
) {
var d = this
var s = eye(this.shape[0])
): Pair<Structure1D<Double>, Structure2D<Double>> {
val A_ = this.copy().as2D()
val V = eye(this.shape[0]).as2D()
val D = DoubleTensor(intArrayOf(this.shape[0]), (0 until this.rowNum).map { this[it, it] }.toDoubleArray()).as1D()
val B = DoubleTensor(intArrayOf(this.shape[0]), (0 until this.rowNum).map { this[it, it] }.toDoubleArray()).as1D()
val Z = zeros(intArrayOf(this.shape[0])).as1D()
// TODO implement cyclic method
do {
// 1. Find max element by abs value that is not on diagonal
fun maxOffDiagonal(matrix: MutableStructure2D<Double>): Double {
var maxOffDiagonalElement = 0.0
var maxElementIndex = Pair(0, 0)
for (i in 0 until this.rowNum) {
for (j in 0 until this.colNum) {
if (i == j) continue
if (abs(d[i, j]) > maxOffDiagonalElement) {
maxOffDiagonalElement = abs(d[i, j])
maxElementIndex = i to j
for (i in 0 until matrix.rowNum - 1) {
for (j in i + 1 until matrix.colNum) {
maxOffDiagonalElement = max(maxOffDiagonalElement, abs(matrix[i, j]))
}
}
return maxOffDiagonalElement
}
fun rotate(a: MutableStructure2D<Double>, s: Double, tau: Double, i: Int, j: Int, k: Int, l: Int) {
val g = a[i, j]
val h = a[k, l]
a[i, j] = g - s * (h + g * tau)
a[k, l] = h + s * (g - h * tau)
}
fun jacobiIteration(
a: MutableStructure2D<Double>,
v: MutableStructure2D<Double>,
d: MutableStructure1D<Double>,
z: MutableStructure1D<Double>,
) {
for (ip in 0 until a.rowNum - 1) {
for (iq in ip + 1 until a.colNum) {
val g = 100.0 * abs(a[ip, iq])
if (g <= epsilon * abs(d[ip]) && g <= epsilon * abs(d[iq])) {
a[ip, iq] = 0.0
continue
}
var h = d[iq] - d[ip]
val t = when {
g <= epsilon * abs(h) -> (a[ip, iq]) / h
else -> {
val theta = 0.5 * h / (a[ip, iq])
val denominator = abs(theta) + sqrt(1.0 + theta * theta)
if (theta < 0.0) -1.0 / denominator else 1.0 / denominator
}
}
val c = 1.0 / sqrt(1 + t * t)
val s = t * c
val tau = s / (1.0 + c)
h = t * a[ip, iq]
z[ip] -= h
z[iq] += h
d[ip] -= h
d[iq] += h
a[ip, iq] = 0.0
for (j in 0 until ip) {
rotate(a, s, tau, j, ip, j, iq)
}
for (j in (ip + 1) until iq) {
rotate(a, s, tau, ip, j, j, iq)
}
for (j in (iq + 1) until a.rowNum) {
rotate(a, s, tau, ip, j, iq, j)
}
for (j in 0 until a.rowNum) {
rotate(v, s, tau, j, ip, j, iq)
}
}
}
}
// 2. Evaluate "rotation" angle
val dIJ = d[maxElementIndex.first, maxElementIndex.second]
val dII = d[maxElementIndex.first, maxElementIndex.first]
val dJJ = d[maxElementIndex.second, maxElementIndex.second]
val angle = if (dII == dJJ) {
if (dIJ > 0) PI / 4 else -PI / 4
} else {
0.5 * atan(2 * dIJ / (dJJ - dII))
}
// 3. Build rotation tensor `s1`
val s1 = eye(this.rowNum)
for (i in 0 until this.rowNum) {
for (j in 0 until this.colNum) {
s1.mutableBuffer.array()[i * this.rowNum + j] = when {
maxElementIndex.first == i && maxElementIndex.first == j -> cos(angle)
maxElementIndex.second == i && maxElementIndex.second == j -> cos(angle)
maxElementIndex.first == i && maxElementIndex.second == j -> sin(angle)
maxElementIndex.first == j && maxElementIndex.second == i -> -sin(angle)
else -> s1.mutableBuffer.array()[i * this.rowNum + j]
}
}
}
// 4. Evaluate new tensor
d = ((s1.transpose() dot d) dot s1).as2D()
s = s dot s1
if (d.isDiagonal(epsilon)) break
} while(true)
// 5. Copy result
for (i in 0 until this.rowNum) {
for (j in 0 until this.colNum) {
eigenvectors.mutableBuffer.array()[eigenvectorStart + i * this.rowNum + j] = s.mutableBuffer.array()[i * this.rowNum + j]
fun updateDiagonal(
d: MutableStructure1D<Double>,
z: MutableStructure1D<Double>,
b: MutableStructure1D<Double>,
) {
for (ip in 0 until d.size) {
b[ip] += z[ip]
d[ip] = b[ip]
z[ip] = 0.0
}
}
for (i in 0 until this.rowNum) {
eigenvalues.mutableBuffer.array()[eigenvalueStart + i] = d[i, i]
}
var sm = maxOffDiagonal(A_)
for (iteration in 0 until maxIteration) {
if (sm < epsilon) {
break
}
public fun Structure2D<Double>.isDiagonal(epsilon: Double = 1e-9): Boolean {
for (i in 0 until this.rowNum) {
for (j in 0 until this.colNum) {
if (i != j && abs(this[i, j]) > epsilon) {
return false
}
}
jacobiIteration(A_, V, D, Z)
updateDiagonal(D, Z, B)
sm = maxOffDiagonal(A_)
}
return true
// TODO sort eigenvalues
return D to V
}
/**