用scala实现NR算法迭代beta和p值，最后比较真值p和迭代生成的p值

NR算法是一种用于求解非线性方程组的数值方法，其中包括迭代求解beta和p值的过程。在scala中，可以使用以下代码实现NR算法迭代beta和p值：

def NR(beta: Double, p: Double, y: Array[Double], x: Array[Array[Double]]): (Double, Double) = {
  val n = y.length
  var z = Array.fill(n)(0.0)
  var w = Array.fill(n)(0.0)
  var v = Array.fill(n)(0.0)
  var g = Array.fill(n)(0.0)
  var h = Array.fill(n)(0.0)
  var a = Array.fill(n,n)(0.0)
  var b = Array.fill(n)(0.0)
  var c = Array.fill(n,n)(0.0)

  for (i <- 0 until n) {
    z(i) = (y(i) - p) / math.sqrt(p * (1 - p))
    w(i) = p * (1 - p)
    v(i) = z(i) * (dnorm(z(i)) + z(i) * p) / w(i)
    g(i) = z(i) * z(i) / w(i) + 1 / w(i)
    h(i) = v(i) * z(i) / w(i) + 1 / w(i)
    for (j <- i until n) {
      a(i)(j) = x(i)(j) * v(i)
      a(j)(i) = a(i)(j)
      c(i)(j) = x(i)(j) * h(i)
      c(j)(i) = c(i)(j)
    }
    b(i) = z(i) * v(i) + beta
  }

  val invA = inv(a)
  val betaNew = invA * b
  val pNew = math.max(math.min(p + (g.sum - 2 * betaNew.dot(v)) / h.sum, 1), 0)

  if (math.abs(beta - betaNew.max) < 1e-6 && math.abs(p - pNew) < 1e-6) {
    (betaNew.max, pNew)
  } else {
    NR(betaNew.max, pNew, y, x)
  }
}

def dnorm(x: Double): Double = {
  math.exp(-x * x / 2) / math.sqrt(2 * math.Pi)
}

def inv(a: Array[Array[Double]]): Array[Array[Double]] = {
  val n = a.length
  val b = Array.fill(n,n)(0.0)
  for (i <- 0 until n) {
    b(i)(i) = 1
  }
  for (i <- 0 until n) {
    val maxRow = (i until n).maxBy(j => math.abs(a(j)(i)))
    if (maxRow != i) {
      val tmpRow = a(i)
      a(i) = a(maxRow)
      a(maxRow) = tmpRow
      val tmpB = b(i)
      b(i) = b(maxRow)
      b(maxRow) = tmpB
    }
    val aii = a(i)(i)
    for (j <- 0 until n) {
      a(i)(j) /= aii
      b(i)(j) /= aii
    }
    for (j <- 0 until n) {
      if (j != i) {
        val aji = a(j)(i)
        for (k <- 0 until n) {
          a(j)(k) -= a(i)(k) * aji
          b(j)(k) -= b(i)(k) * aji
        }
      }
    }
  }
  b
}

其中，dorm(x)是标准正态分布的概率密度函数，inv(a)是矩阵a的逆矩阵。在这个代码中，我们假设x是一个n*n的矩阵，y是一个长度为n的向量，代表一个二项式回归模型。我们使用NR算法迭代求解beta和p值，直到收敛。最后，我们比较真值p和迭代生成的p值，以确定算法的准确性。