解释：gnorm = np.amax(np.abs(gfk))

这行代码使用了 NumPy 库中的函数 `np.amax`，它的作用是返回给定数组中的最大值。在这个例子中，给定的数组是 `gfk`，也就是一个一维的 NumPy 数组。在计算 `gnorm` 时，它使用了 `np.abs` 函数，它返回给定数组中每个元素的绝对值，这样可以确保计算的是向量的模长而不是它们的代数和。因此这行代码的作用是计算向量 `gfk` 的模长，也就是其最大绝对值。

相关问题

将这段代码转换为伪代码：def levenberg_marquardt(fun, grad, jacobian, x0, iterations, tol): """ Minimization of scalar function of one or more variables using the Levenberg-Marquardt algorithm. Parameters ---------- fun : function Objective function. grad : function Gradient function of objective function. jacobian :function function of objective function. x0 : numpy.array, size=9 Initial value of the parameters to be estimated. iterations : int Maximum iterations of optimization algorithms. tol : float Tolerance of optimization algorithms. Returns ------- xk : numpy.array, size=9 Parameters wstimated by optimization algorithms. fval : float Objective function value at xk. grad_val : float Gradient value of objective function at xk. grad_log : numpy.array The record of gradient of objective function of each iteration. """ fval = None # y的最小值 grad_val = None # 梯度的最后一次下降的值 x_log = [] # x的迭代值的数组，n*9，9个参数 y_log = [] # y的迭代值的数组，一维 grad_log = [] # 梯度下降的迭代值的数组 x0 = asarray(x0).flatten() if x0.ndim == 0: x0.shape = (1,) # iterations = len(x0) * 200 k = 1 xk = x0 updateJ = 1 lamda = 0.01 old_fval = fun(x0) gfk = grad(x0) gnorm = np.amax(np.abs(gfk)) J = [None] H = [None] while (gnorm > tol) and (k < iterations): if updateJ == 1: x_log = np.append(x_log, xk.T) yk = fun(xk) y_log = np.append(y_log, yk) J = jacobian(x0) H = np.dot(J.T, J) H_lm = H + (lamda * np.eye(9)) gfk = grad(xk) pk = - np.linalg.inv(H_lm).dot(gfk) pk = pk.A.reshape(1, -1)[0] # 二维变一维 xk1 = xk + pk fval = fun(xk1) if fval < old_fval: lamda = lamda / 10 xk = xk1 old_fval = fval updateJ = 1 else: updateJ = 0 lamda = lamda * 10 gnorm = np.amax(np.abs(gfk)) k = k + 1 grad_log = np.append(grad_log, np.linalg.norm(xk - x_log[-1:])) fval = old_fval grad_val = grad_log[-1] return xk, fval, grad_val, x_log, y_log, grad_log

伪代码如下：

function levenberg_marquardt(fun, grad, jacobian, x0, iterations, tol):
fval = None
grad_val = None
x_log = []
y_log = []
grad_log = []
x0 = asarray(x0).flatten()
if x0.ndim == 0:
x0.shape = (1,)
k = 1
xk = x0
updateJ = 1
lamda = 0.01
old_fval = fun(x0)
gfk = grad(x0)
gnorm = np.amax(np.abs(gfk))
J = None
H = None

while (gnorm > tol) and (k < iterations):
if updateJ == 1:
x_log = np.append(x_log, xk.T)
yk = fun(xk)
y_log = np.append(y_log, yk)
J = jacobian(x0)
H = np.dot(J.T, J)
H_lm = H + (lamda * np.eye(9))
gfk = grad(xk)
pk = - np.linalg.inv(H_lm).dot(gfk)
pk = pk.A.reshape(1, -1)[0]
xk1 = xk + pk
fval = fun(xk1)

if fval < old_fval:
lamda = lamda / 10
xk = xk1
old_fval = fval
updateJ = 1
else:
updateJ = 0
lamda = lamda * 10

gnorm = np.amax(np.abs(gfk))
k = k + 1
grad_log = np.append(grad_log, np.linalg.norm(xk - x_log[-1:]))

fval = old_fval
grad_val = grad_log[-1]
return xk, fval, grad_val, x_log, y_log, grad_log

将下面这段源码转换为伪代码：def bfgs(fun, grad, x0, iterations, tol): """ Minimization of scalar function of one or more variables using the BFGS algorithm. Parameters ---------- fun : function Objective function. grad : function Gradient function of objective function. x0 : numpy.array, size=9 Initial value of the parameters to be estimated. iterations : int Maximum iterations of optimization algorithms. tol : float Tolerance of optimization algorithms. Returns ------- xk : numpy.array, size=9 Parameters wstimated by optimization algorithms. fval : float Objective function value at xk. grad_val : float Gradient value of objective function at xk. grad_log : numpy.array The record of gradient of objective function of each iteration. """ fval = None grad_val = None x_log = [] y_log = [] grad_log = [] x0 = asarray(x0).flatten() # iterations = len(x0) * 200 old_fval = fun(x0) gfk = grad(x0) k = 0 N = len(x0) I = np.eye(N, dtype=int) Hk = I old_old_fval = old_fval + np.linalg.norm(gfk) / 2 xk = x0 x_log = np.append(x_log, xk.T) y_log = np.append(y_log, fun(xk)) grad_log = np.append(grad_log, np.linalg.norm(xk - x_log[-1:])) gnorm = np.amax(np.abs(gfk)) while (gnorm > tol) and (k < iterations): pk = -np.dot(Hk, gfk) try: alpha, fc, gc, old_fval, old_old_fval, gfkp1 = _line_search_wolfe12(fun, grad, xk, pk, gfk, old_fval, old_old_fval, amin=1e-100, amax=1e100) except _LineSearchError: break x1 = xk + alpha * pk sk = x1 - xk xk = x1 if gfkp1 is None: gfkp1 = grad(x1) yk = gfkp1 - gfk gfk = gfkp1 k += 1 gnorm = np.amax(np.abs(gfk)) grad_log = np.append(grad_log, np.linalg.norm(xk - x_log[-1:])) x_log = np.append(x_log, xk.T) y_log = np.append(y_log, fun(xk)) if (gnorm <= tol): break if not np.isfinite(old_fval): break try: rhok = 1.0 / (np.dot(yk, sk)) except ZeroDivisionError: rhok = 1000.0 if isinf(rhok): rhok = 1000.0 A1 = I - sk[:, np.newaxis] * yk[np.newaxis, :] * rhok A2 = I - yk[:, np.newaxis] * sk[np.newaxis, :] * rhok Hk = np.dot(A1, np.dot(Hk, A2)) + (rhok * sk[:, np.newaxis] * sk[np.newaxis, :]) fval = old_fval grad_val = grad_log[-1] return xk, fval, grad_val, x_log, y_log, grad_log

伪代码如下：

函数 bfgs(fun, grad, x0, iterations, tol)：
fval <- 空值
grad_val <- 空值
x_log <- 空列表
y_log <- 空列表
grad_log <- 空列表
x0 <- 将 x0 转换为一维数组
old_fval <- 调用 fun(x0)
gfk <- 调用 grad(x0)
k <- 0
N <- x0 的长度
I <- 生成 N*N 的单位矩阵
Hk <- I
old_old_fval <- old_fval + 计算 gfk 的范数 / 2
xk <- x0
将 xk 添加到 x_log 列表末尾
将 fun(xk) 添加到 y_log 列表末尾
将 xk 与 x_log 最后一个元素的差的范数添加到 grad_log 列表末尾
gnorm <- 计算 gfk 的绝对值的最大值
while (gnorm > tol) and (k < iterations)：
pk <- - Hk * gfk
尝试：
alpha, fc, gc, old_fval, old_old_fval, gfkp1 <- _line_search_wolfe12(fun, grad, xk, pk, gfk, old_fval, old_old_fval, amin=1e-100, amax=1e100)
捕获 _LineSearchError：
跳出循环
x1 <- xk + alpha * pk
sk <- x1 - xk
xk <- x1
如果 gfkp1 是空值：
gfkp1 <- 调用 grad(x1)
yk <- gfkp1 - gfk
gfk <- gfkp1
k <- k + 1
gnorm <- 计算 gfk 的绝对值的最大值
将 xk 与 x_log 最后一个元素的差的范数添加到 grad_log 列表末尾
将 xk 添加到 x_log 列表末尾
将 fun(xk) 添加到 y_log 列表末尾
如果 (gnorm <= tol)：
跳出循环
如果 old_fval 不是有限数：
跳出循环
尝试：
rhok <- 1.0 / (yk · sk)
捕获 ZeroDivisionError：
rhok <- 1000.0
如果 rhok 是正无穷：
rhok <- 1000.0
A1 <- I - sk·yk.T·rhok
A2 <- I - yk·sk.T·rhok
Hk <- A1·Hk·A2 + rhok·sk·sk.T
fval <- old_fval
grad_val <- grad_log 最后一个元素
返回 xk, fval, grad_val, x_log, y_log, grad_log