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Solve the following problem using Python, implementing the functions described below, one line at a time
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Description:
 def update(self, w, offset=0):
 """Compute gradient and Hessian matrix with respect to `w`."""

        time = self.time
        x = self.x
        exp_xw = numpy.exp(offset + numpy.dot(x, w))
        n_samples, n_features = x.shape

        gradient = numpy.zeros((1, n_features), dtype=float)
        hessian = numpy.zeros((n_features, n_features), dtype=float)

        inv_n_samples = 1. / n_samples
        risk_set = 0
        risk_set_x = 0
        risk_set_xx = 0
        k = 0
        # iterate time in descending order
        for i in range(n_samples):
            ti = time[i]
            while k < n_samples and ti == time[k]:
                risk_set += exp_xw[k]

                # preserve 2D shape of row vector
                xk = x[k:k + 1]
                risk_set_x += exp_xw[k] * xk

                # outer product
                xx = numpy.dot(xk.T, xk)
                risk_set_xx += exp_xw[k] * xx

                k += 1

            if self.event[i]:
                gradient -= (x[i:i + 1] - risk_set_x / risk_set) * inv_n_samples

                a = risk_set_xx / risk_set
                z = risk_set_x / risk_set
                # outer product
                b = numpy.dot(z.T, z)

                hessian += (a - b) * inv_n_samples

        if self.alpha > 0:
            gradient += self.alpha * inv_n_samples * w

            diag_idx = numpy.diag_indices(n_features)
            hessian[diag_idx] += self.alpha * inv_n_samples

        self.gradient = gradient.ravel()
        self.hessian = hessian