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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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kmath-tensors/src/commonMain/kotlin/space/kscience/kmath/tensors/core/DoubleTensorAlgebra.kt
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@ -10,6 +10,7 @@ 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
@ -885,7 +886,7 @@ public open class DoubleTensorAlgebra :
return Triple(uTensor.transpose(), sTensor, vTensor.transpose())
}
override fun StructureND<Double>.symEig(): Pair<DoubleTensor, DoubleTensor> = symEig(epsilon = 1e-15)
override fun StructureND<Double>.symEig(): Pair<DoubleTensor, DoubleTensor> = symEigJacobi(epsilon = 1e-10)
/**
* Returns eigenvalues and eigenvectors of a real symmetric matrix input or a batch of real symmetric matrices,
@ -922,98 +923,101 @@ public open class DoubleTensorAlgebra :
return eig to v
}
// TODO
// 1. Cyclic method
// 2. Sort eigenvalues
public fun StructureND<Double>.symEig(epsilon: Double): Pair<DoubleTensor, DoubleTensor> {
public fun StructureND<Double>.symEigJacobi(epsilon: Double): Pair<DoubleTensor, DoubleTensor> {
checkSymmetric(tensor, epsilon)
val ii = tensor.minusIndex(-2)
val jj = tensor.minusIndex(-1)
val n = tensor.numElements
val size = this.dimension
val commonShape = this.shape.sliceArray(0 until size - 2) + intArrayOf(1, 1)
val s = zeros(this.shape)
val eigenvalues = zeros(this.shape.sliceArray(0 until size - 1))
var d = this.copy()
var s = diagonalEmbedding(ones(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)
eigenvalueStart += this.shape.last()
eigenvectorStart += this.shape.last() * this.shape.last()
}
// TODO sort eigenvalues
return eigenvalues to s
}
private fun MutableStructure2D<Double>.jacobiHelper(
eigenvalues: DoubleTensor,
eigenvectors: DoubleTensor,
eigenvalueStart: Int,
eigenvectorStart: Int,
epsilon: Double
) {
var d = this
var s = eye(this.shape[0])
// TODO implement cyclic method
do {
// 1. Find max element by abs value that is not on diagonal
val buffer = MutableBuffer.boxing(commonShape.reduce(Int::times)) { Triple(0.0, 0, 0) }
val maxOffDiagonalElements = BufferedTensor(commonShape, buffer, 0)
for (offset in 0 until n) {
val multiIndex = d.linearStructure.index(offset)
if (multiIndex[ii] != multiIndex[jj]) {
val value = d.mutableBuffer.array()[offset]
val commonIndex = multiIndex.sliceArray(0 until size - 2) + intArrayOf(0, 0)
if (abs(value) > maxOffDiagonalElements[commonIndex].first) {
maxOffDiagonalElements[commonIndex] = Triple(abs(value), multiIndex[ii], multiIndex[jj])
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
}
}
}
// 2. Evaluate "rotation" angle
val angles = zeros(commonShape)
for (offset in 0 until maxOffDiagonalElements.numElements) {
val (_, i, j) = maxOffDiagonalElements.mutableBuffer[offset]
val multiIndex = maxOffDiagonalElements.linearStructure.index(offset)
val dIJ = d[maxElementIndex.first, maxElementIndex.second]
val dII = d[maxElementIndex.first, maxElementIndex.first]
val dJJ = d[maxElementIndex.second, maxElementIndex.second]
val dIJ = d.mutableBuffer[d.linearStructure.offset(multiIndex.also { it[ii] = i; it[jj] = j })]
val dII = d.mutableBuffer[d.linearStructure.offset(multiIndex.also { it[ii] = i; it[jj] = i })]
val dJJ = d.mutableBuffer[d.linearStructure.offset(multiIndex.also { it[ii] = j; it[jj] = j })]
angles.mutableBuffer.array()[offset] = if (dII == dJJ) {
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 = diagonalEmbedding(ones(this.shape.sliceArray(0 until size - 1)))
for (offset in 0 until n) {
val multiIndex = d.linearStructure.index(offset)
val commonIndex = multiIndex.sliceArray(0 until size - 2) + intArrayOf(0, 0)
val (_, i, j) = maxOffDiagonalElements[commonIndex]
val angleValue = angles[commonIndex]
s1.mutableBuffer.array()[offset] = when {
multiIndex[ii] == i && multiIndex[jj] == i || multiIndex[ii] == j && multiIndex[jj] == j -> cos(angleValue)
multiIndex[ii] == i && multiIndex[jj] == j -> sin(angleValue)
multiIndex[ii] == j && multiIndex[jj] == i -> -sin(angleValue)
else -> s1.mutableBuffer.array()[offset]
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
d = ((s1.transpose() dot d) dot s1).as2D()
s = s dot s1
if (d.isDiagonal(epsilon)) break
} while(true)
val eigenvalues = zeros(d.shape.sliceArray(0 until size - 1))
for (offset in 0 until n) {
val multiIndex = d.linearStructure.index(offset)
if (multiIndex[ii] == multiIndex[jj]) {
eigenvalues[multiIndex.sliceArray(0 until size - 1)] = d.mutableBuffer.array()[offset]
// 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]
}
}
return eigenvalues to s
for (i in 0 until this.rowNum) {
eigenvalues.mutableBuffer.array()[eigenvalueStart + i] = d[i, i]
}
}
public fun StructureND<Double>.isDiagonal(epsilon: Double = 1e-9): Boolean {
val ii = tensor.minusIndex(-2)
val jj = tensor.minusIndex(-1)
for (offset in 0 until tensor.numElements) {
val multiIndex = tensor.linearStructure.index(offset)
if (multiIndex[ii] != multiIndex[jj] && abs(tensor.mutableBuffer.array()[offset]) > epsilon) {
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
}
}
}
return true
}