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Implement Jacobi algorithm to find eigenvalues

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This commit is contained in:
Ivan Kylchik 2022-02-13 13:16:35 +03:00
parent ac3adfa644
commit 7a72a0b979
1 changed files with 97 additions and 1 deletions
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kmath-tensors/src/commonMain/kotlin/space/kscience/kmath/tensors/core/DoubleTensorAlgebra.kt
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@ -895,7 +895,7 @@ public open class DoubleTensorAlgebra :
* and when the cosine approaches 1 in the SVD algorithm.
* @return a pair `eigenvalues to eigenvectors`.
*/
public fun StructureND<Double>.symEig(epsilon: Double): Pair<DoubleTensor, DoubleTensor> {
public fun StructureND<Double>.symEigSvd(epsilon: Double): Pair<DoubleTensor, DoubleTensor> {
checkSymmetric(tensor, epsilon)
fun MutableStructure2D<Double>.cleanSym(n: Int) {
@ -922,6 +922,102 @@ 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> {
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)
var d = this.copy()
var s = diagonalEmbedding(ones(this.shape.sliceArray(0 until size - 1)))
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])
}
}
}
// 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.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) {
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]
}
}
// 4. Evaluate new tensor
d = (s1.transpose() dot d) dot s1
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]
}
}
return eigenvalues to s
}
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) {
return false
}
}
return true
}
/**
* Computes the determinant of a square matrix input, or of each square matrix in a batched input
* using LU factorization algorithm.