func GradientDescent(X *mat64.Dense, y *mat64.Vector, alpha, tolerance float64, maxIters int) *mat64.Vector {
	// m = Number of Training Examples
	// n = Number of Features
	m, n := X.Dims()
	h := mat64.NewVector(m, nil)
	partials := mat64.NewVector(n, nil)
	new_theta := mat64.NewVector(n, nil)

Regression:
	for i := 0; i < maxIters; i++ {
		// Calculate partial derivatives
		h.MulVec(X, new_theta)
		for el := 0; el < m; el++ {
			val := (h.At(el, 0) - y.At(el, 0)) / float64(m)
			h.SetVec(el, val)
		}
		partials.MulVec(X.T(), h)

		// Update theta values
		for el := 0; el < n; el++ {
			new_val := new_theta.At(el, 0) - (alpha * partials.At(el, 0))
			new_theta.SetVec(el, new_val)
		}

		// Check the "distance" to the local minumum
		dist := math.Sqrt(mat64.Dot(partials, partials))

		if dist <= tolerance {
			break Regression
		}
	}
	return new_theta
}

// onesDotUnitary performs the equivalent of a Ddot of v with
// a ones vector of equal length. v must have have a unitary
// vector increment.
func onesDotUnitary(alpha float64, v *mat64.Vector) float64 {
	var sum float64
	for _, f := range v.RawVector().Data {
		sum += alpha * f
	}
	return sum
}

// dotUnitary performs a simplified scatter-based Ddot operations on
// v and the receiver. v must have have a unitary vector increment.
func (r compressedRow) dotUnitary(v *mat64.Vector) float64 {
	var sum float64
	vec := v.RawVector().Data
	for _, e := range r {
		sum += vec[e.index] * e.value
	}
	return sum
}

func ExampleCholesky() {
	// Construct a symmetric positive definite matrix.
	tmp := mat64.NewDense(4, 4, []float64{
		2, 6, 8, -4,
		1, 8, 7, -2,
		2, 2, 1, 7,
		8, -2, -2, 1,
	})
	var a mat64.SymDense
	a.SymOuterK(1, tmp)

	fmt.Printf("a = %0.4v\n", mat64.Formatted(&a, mat64.Prefix("    ")))

	// Compute the cholesky factorization.
	var chol mat64.Cholesky
	if ok := chol.Factorize(&a); !ok {
		fmt.Println("a matrix is not positive semi-definite.")
	}

	// Find the determinant.
	fmt.Printf("\nThe determinant of a is %0.4g\n\n", chol.Det())

	// Use the factorization to solve the system of equations a * x = b.
	b := mat64.NewVector(4, []float64{1, 2, 3, 4})
	var x mat64.Vector
	if err := x.SolveCholeskyVec(&chol, b); err != nil {
		fmt.Println("Matrix is near singular: ", err)
	}
	fmt.Println("Solve a * x = b")
	fmt.Printf("x = %0.4v\n", mat64.Formatted(&x, mat64.Prefix("    ")))

	// Extract the factorization and check that it equals the original matrix.
	var t mat64.TriDense
	t.LFromCholesky(&chol)
	var test mat64.Dense
	test.Mul(&t, t.T())
	fmt.Println()
	fmt.Printf("L * L^T = %0.4v\n", mat64.Formatted(&a, mat64.Prefix("          ")))

	// Output:
	// a = ⎡120  114   -4  -16⎤
	//     ⎢114  118   11  -24⎥
	//     ⎢ -4   11   58   17⎥
	//     ⎣-16  -24   17   73⎦
	//
	// The determinant of a is 1.543e+06
	//
	// Solve a * x = b
	// x = ⎡  -0.239⎤
	//     ⎢  0.2732⎥
	//     ⎢-0.04681⎥
	//     ⎣  0.1031⎦
	//
	// L * L^T = ⎡120  114   -4  -16⎤
	//           ⎢114  118   11  -24⎥
	//           ⎢ -4   11   58   17⎥
	//           ⎣-16  -24   17   73⎦
}

func vectorDistance(vec1, vec2 *mat.Vector) (v float64) {
	result := mat.NewVector(vec1.Len(), nil)

	result.SubVec(vec1, vec2)
	result.MulElemVec(result, result)
	v = mat.Sum(result)

	return
}

// Scatter copies the values of x into the corresponding locations in the dense
// vector y. Both vectors must have the same dimension.
func Scatter(y *mat64.Vector, x *Vector) {
	if x.N != y.Len() {
		panic("sparse: vector dimension mismatch")
	}

	raw := y.RawVector()
	for i, index := range x.Indices {
		raw.Data[index*raw.Inc] = x.Data[i]
	}
}

// findIn returns the indexes of the values in vec that match scalar
func findIn(scalar float64, vec *mat.Vector) *mat.Vector {
	var result []float64

	for i := 0; i < vec.Len(); i++ {
		if scalar == vec.At(i, 0) {
			result = append(result, float64(i))
		}
	}

	return mat.NewVector(len(result), result)
}

// Dot computes the dot product of the sparse vector x with the dense vector y.
// The vectors must have the same dimension.
func Dot(x *Vector, y *mat64.Vector) (dot float64) {
	if x.N != y.Len() {
		panic("sparse: vector dimension mismatch")
	}

	raw := y.RawVector()
	for i, index := range x.Indices {
		dot += x.Data[i] * raw.Data[index*raw.Inc]
	}
	return
}

// Gather gathers entries given by indices of the dense vector y into the sparse
// vector x. Indices must not be nil.
func Gather(x *Vector, y *mat64.Vector, indices []int) {
	if indices == nil {
		panic("sparse: slice is nil")
	}

	x.reuseAs(y.Len(), len(indices))
	copy(x.Indices, indices)
	raw := y.RawVector()
	for i, index := range x.Indices {
		x.Data[i] = raw.Data[index*raw.Inc]
	}
}

// Axpy scales the sparse vector x by alpha and adds the result to the dense
// vector y. If alpha is zero, y is not modified.
func Axpy(y *mat64.Vector, alpha float64, x *Vector) {
	if x.N != y.Len() {
		panic("sparse: vector dimension mismatch")
	}

	if alpha == 0 {
		return
	}
	raw := y.RawVector()
	for i, index := range x.Indices {
		raw.Data[index*raw.Inc] += alpha * x.Data[i]
	}
}

// rowIndexIn returns a matrix contains the rows in indexes vector
func rowIndexIn(indexes *mat.Vector, M mat.Matrix) mat.Matrix {
	m := indexes.Len()
	_, n := M.Dims()
	Res := mat.NewDense(m, n, nil)

	for i := 0; i < m; i++ {
		Res.SetRow(i, mat.Row(
			nil,
			int(indexes.At(i, 0)),
			M))
	}

	return Res
}

func Cost(x *mat64.Dense, y, theta *mat64.Vector) float64 {
	//initialize receivers
	m, _ := x.Dims()
	h := mat64.NewDense(m, 1, make([]float64, m))
	squaredErrors := mat64.NewDense(m, 1, make([]float64, m))

	//actual calculus
	h.Mul(x, theta)
	squaredErrors.Apply(func(r, c int, v float64) float64 {
		return math.Pow(h.At(r, c)-y.At(r, c), 2)
	}, h)
	j := mat64.Sum(squaredErrors) * 1.0 / (2.0 * float64(m))

	return j
}

func Solve(a sparse.Matrix, b, xInit *mat64.Vector, settings *Settings, method Method) (result Result, err error) {
	stats := Stats{
		StartTime: time.Now(),
	}

	dim, c := a.Dims()
	if dim != c {
		panic("iterative: matrix is not square")
	}
	if xInit != nil && dim != xInit.Len() {
		panic("iterative: mismatched size of the initial guess")
	}
	if b.Len() != dim {
		panic("iterative: mismatched size of the right-hand side vector")
	}

	if xInit == nil {
		xInit = mat64.NewVector(dim, nil)
	}
	if settings == nil {
		settings = DefaultSettings(dim)
	}

	ctx := Context{
		X:        mat64.NewVector(dim, nil),
		Residual: mat64.NewVector(dim, nil),
	}
	// X = xInit
	ctx.X.CopyVec(xInit)
	if mat64.Norm(ctx.X, math.Inf(1)) > 0 {
		// Residual = Ax
		sparse.MulMatVec(ctx.Residual, 1, false, a, ctx.X)
		stats.MatVecMultiplies++
	}
	// Residual = Ax - b
	ctx.Residual.SubVec(ctx.Residual, b)

	if mat64.Norm(ctx.Residual, 2) >= settings.Tolerance {
		err = iterate(method, a, b, settings, &ctx, &stats)
	}

	result = Result{
		X:       ctx.X,
		Stats:   stats,
		Runtime: time.Since(stats.StartTime),
	}
	return result, err
}

// Map produces a vector that is within the bounds of the
// rectangular manifold of toroidal space, given a vector
// that is on the torus but may be outside these bounds.
func (t Torus) Map(v *mat64.Vector) {
	x := v.At(0, 0)
	y := v.At(1, 0)

	remx := x
	right := t.W / 2
	if math.Abs(x) > right {
		remx = math.Mod(t.W, -x)
	}
	remy := y
	top := t.H / 2
	if math.Abs(y) > top {
		remy = math.Mod(t.H, -y)
	}

	v.SetVec(0, remx)
	v.SetVec(1, remy)
}

// StdDev predicts the standard deviation of the function at x.
func (g *GP) StdDev(x []float64) float64 {
	if len(x) != g.inputDim {
		panic(badInputLength)
	}
	// nu_* = k(x_*, k_*) - k_*^T * K^-1 * k_*
	n := len(g.outputs)
	kstar := mat64.NewVector(n, nil)
	for i := 0; i < n; i++ {
		v := g.kernel.Distance(g.inputs.RawRowView(i), x)
		kstar.SetVec(i, v)
	}
	self := g.kernel.Distance(x, x)
	var tmp mat64.Vector
	tmp.SolveCholeskyVec(g.cholK, kstar)
	var tmp2 mat64.Vector
	tmp2.MulVec(kstar.T(), &tmp)
	rt, ct := tmp2.Dims()
	if rt != 1 || ct != 1 {
		panic("bad size")
	}
	return math.Sqrt(self-tmp2.At(0, 0)) * g.std
}

// StdDevBatch predicts the standard deviation at a set of locations of x.
func (g *GP) StdDevBatch(std []float64, x mat64.Matrix) []float64 {
	r, c := x.Dims()
	if c != g.inputDim {
		panic(badInputLength)
	}
	if std == nil {
		std = make([]float64, r)
	}
	if len(std) != r {
		panic(badStorage)
	}
	// For a single point, the stddev is
	// 		sigma = k(x,x) - k_*^T * K^-1 * k_*
	// where k is the vector of kernels between the input points and the output points
	// For many points, the formula is:
	// 		nu_* = k(x_*, k_*) - k_*^T * K^-1 * k_*
	// This creates the full covariance matrix which is an rxr matrix. However,
	// the standard deviations are just the diagonal of this matrix. Instead, be
	// smart about it and compute the diagonal terms one at a time.
	kStar := g.formKStar(x)
	var tmp mat64.Dense
	tmp.SolveCholesky(g.cholK, kStar)

	// set k(x_*, x_*) into std then subtract k_*^T K^-1 k_* , computed one row at a time
	var tmp2 mat64.Vector
	row := make([]float64, c)
	for i := range std {
		for k := 0; k < c; k++ {
			row[k] = x.At(i, k)
		}
		std[i] = g.kernel.Distance(row, row)
		tmp2.MulVec(kStar.ColView(i).T(), tmp.ColView(i))
		rt, ct := tmp2.Dims()
		if rt != 1 && ct != 1 {
			panic("bad size")
		}
		std[i] -= tmp2.At(0, 0)
		std[i] = math.Sqrt(std[i])
	}
	// Need to scale the standard deviation to be in the same units as y.
	floats.Scale(g.std, std)
	return std
}

func BlasVec2UserVec(v *mat64.Vector) UserVec {
	u := UserVec{}
	u.X = v.At(0, 0)
	u.Y = v.At(1, 0)
	return u
}

// mulVecUnitary multiplies the receiver by the src vector, storing
// the result in dst. It assumes src and dst are the same length as m
// and that both have unitary vector increments.
func (m rowCompressedMatrix) mulVecUnitary(dst, src *mat64.Vector) {
	dMat := dst.RawVector().Data
	for i, r := range m {
		dMat[i] = r.dotUnitary(src)
	}
}

func VectorToMatrix(vector *mat64.Vector) *mat64.Dense {
	vec := vector.RawVector()
	return mat64.NewDense(1, len(vec.Data), vec.Data)
}

// ConditionNormal returns the Normal distribution that is the receiver conditioned
// on the input evidence. The returned multivariate normal has dimension
// n - len(observed), where n is the dimension of the original receiver. The updated
// mean and covariance are
//  mu = mu_un + sigma_{ob,un}^T * sigma_{ob,ob}^-1 (v - mu_ob)
//  sigma = sigma_{un,un} - sigma_{ob,un}^T * sigma_{ob,ob}^-1 * sigma_{ob,un}
// where mu_un and mu_ob are the original means of the unobserved and observed
// variables respectively, sigma_{un,un} is the unobserved subset of the covariance
// matrix, sigma_{ob,ob} is the observed subset of the covariance matrix, and
// sigma_{un,ob} are the cross terms. The elements of x_2 have been observed with
// values v. The dimension order is preserved during conditioning, so if the value
// of dimension 1 is observed, the returned normal represents dimensions {0, 2, ...}
// of the original Normal distribution.
//
// ConditionNormal returns {nil, false} if there is a failure during the update.
// Mathematically this is impossible, but can occur with finite precision arithmetic.
func (n *Normal) ConditionNormal(observed []int, values []float64, src *rand.Rand) (*Normal, bool) {
	if len(observed) == 0 {
		panic("normal: no observed value")
	}
	if len(observed) != len(values) {
		panic("normal: input slice length mismatch")
	}
	for _, v := range observed {
		if v < 0 || v >= n.Dim() {
			panic("normal: observed value out of bounds")
		}
	}

	ob := len(observed)
	unob := n.Dim() - ob
	obMap := make(map[int]struct{})
	for _, v := range observed {
		if _, ok := obMap[v]; ok {
			panic("normal: observed dimension occurs twice")
		}
		obMap[v] = struct{}{}
	}
	if len(observed) == n.Dim() {
		panic("normal: all dimensions observed")
	}
	unobserved := make([]int, 0, unob)
	for i := 0; i < n.Dim(); i++ {
		if _, ok := obMap[i]; !ok {
			unobserved = append(unobserved, i)
		}
	}
	mu1 := make([]float64, unob)
	for i, v := range unobserved {
		mu1[i] = n.mu[v]
	}
	mu2 := make([]float64, ob) // really v - mu2
	for i, v := range observed {
		mu2[i] = values[i] - n.mu[v]
	}

	n.setSigma()

	var sigma11, sigma22 mat64.SymDense
	sigma11.SubsetSym(n.sigma, unobserved)
	sigma22.SubsetSym(n.sigma, observed)

	sigma21 := mat64.NewDense(ob, unob, nil)
	for i, r := range observed {
		for j, c := range unobserved {
			v := n.sigma.At(r, c)
			sigma21.Set(i, j, v)
		}
	}

	var chol mat64.Cholesky
	ok := chol.Factorize(&sigma22)
	if !ok {
		return nil, ok
	}

	// Compute sigma_{2,1}^T * sigma_{2,2}^-1 (v - mu_2).
	v := mat64.NewVector(ob, mu2)
	var tmp, tmp2 mat64.Vector
	err := tmp.SolveCholeskyVec(&chol, v)
	if err != nil {
		return nil, false
	}
	tmp2.MulVec(sigma21.T(), &tmp)

	// Compute sigma_{2,1}^T * sigma_{2,2}^-1 * sigma_{2,1}.
	// TODO(btracey): Should this be a method of SymDense?
	var tmp3, tmp4 mat64.Dense
	err = tmp3.SolveCholesky(&chol, sigma21)
	if err != nil {
		return nil, false
	}
	tmp4.Mul(sigma21.T(), &tmp3)

	for i := range mu1 {
		mu1[i] += tmp2.At(i, 0)
	}

	// TODO(btracey): If tmp2 can constructed with a method, then this can be
	// replaced with SubSym.
//.........这里部分代码省略.........