Notebooks · lib/scf/mix

Mixing densities in a self-consistent field

mix_densities(x_in, x_out, hist) is a contract: the density the next pass of a self-consistent field starts from, made from the density the pass started with (x_in), the one it produced (x_out) and the earlier pairs in hist. The densities are one flat array; a solver packs whatever it mixes (lib/skyrme packs rho, tau and J of both kinds).

linear: (1 - alpha) x_in + alpha x_out. No history.
pulay: Pulay's direct inversion in the iterative subspace (P. Pulay, Chem. Phys. Lett. 73, 393 (1980)), applied to densities: the combination of the last history inputs whose residuals (output minus input) combine to the shortest residual, with the coefficients summing to one, then a linear step of alpha along that residual.
broyden: Johnson's modified Broyden method (D. D. Johnson, Phys. Rev. B 38, 12807 (1988)) with every weight 1 and w0 = BROYDEN_W0. It works on differences of consecutive pairs; with w0 = 0 its step equals Pulay's (V. Eyert, J. Comput. Phys. 124, 271 (1996)), so w0 is what makes it a different method here.

Two mixers given the same pair return different next densities on purpose; they agree only at the fixed point. The contract's agree line compares residual, the largest |x_out - x_in|, which every method computes from the same two arrays; the converged energies and densities of the three are compared in the tests of lib/skyrme/spherical.viv.

Used with from scf.mix import mix_densities, mix_start.

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[1]lines 32–233

#: MIX_ALPHA  the default fraction of the residual added in one step
MIX_ALPHA = 0.5

#: MIX_HISTORY  how many input densities Pulay and Broyden keep by default,
#:     the current one included
MIX_HISTORY: Int = 6

#: BROYDEN_W0  Johnson's w0, the weight of the first inverse-Jacobian guess;
#:     he uses 0.01
BROYDEN_W0 = 0.01

#: PULAY_CUTOFF  singular values of Pulay's bordered system below this
#:     fraction of the largest are dropped: the system is singular when the
#:     remembered residuals are linearly dependent, as they become near the
#:     fixed point
PULAY_CUTOFF = 1.0e-12

#| The densities a mixer has seen, oldest first: row i of x is an input
#| density and row i of r its residual (output minus input); `count` rows
#| are in use.
record MixHistory {
    x: Arr
    r: Arr
    count: Int
}

#| One mixing step as one value, for a solver that takes the mixer as a
#| function: the next density, the history to pass next time, and the
#| largest |x_out - x_in|.
record MixStep {
    x: Arr
    hist: MixHistory
    residual: Num
}

#| An empty history for densities of n numbers.
def mix_start(n: Int) -> MixHistory {
    return MixHistory { x: zeros(1, n), r: zeros(1, n), count: 0 }
}

#| A new history: the last keep - 1 rows of hist, then x and r.
def remember(hist: MixHistory, x: Arr, r: Arr, keep: Int) -> MixHistory {
    take: Int = min(hist.count, keep - 1)
    m: Int = take + 1
    xs = zeros(m, len(x))
    rs = zeros(m, len(x))
    first: Int = hist.count - take
    for i to take {
        xs[i, :] = hist.x[first + i, :]
        rs[i, :] = hist.r[first + i, :]
    }
    xs[take, :] = x
    rs[take, :] = r
    return MixHistory { x: xs, r: rs, count: m }
}

#| The largest |x_out - x_in|: how far this pass moved the density.
def residual_norm(x_in: Arr, x_out: Arr) -> Num {
    return max(abs(x_out - x_in))
}

#| (1 - alpha) x_in + alpha x_out.
def linear(x_in: Arr, x_out: Arr, hist: MixHistory, alpha: Num = MIX_ALPHA) {
    x_next = (1.0 - alpha) * x_in + alpha * x_out
    return x_next, hist, residual_norm(x_in, x_out)
}

#| Pulay's mixing: coefficients c over the remembered inputs minimizing
#| |sum c_i r_i| with sum c_i = 1 (a bordered linear system, solved by its
#| pseudo-inverse), then sum c_i (x_i + alpha r_i).
def pulay(x_in: Arr, x_out: Arr, hist: MixHistory, alpha: Num = MIX_ALPHA, history: Int = MIX_HISTORY) {
    h2 = remember(hist, x_in, x_out - x_in, history)
    m: Int = h2.count
    step overlaps {
        A = zeros(m + 1, m + 1)
        rhs = zeros(m + 1)
        A[0:m, 0:m] = h2.r @ transpose(h2.r)
        scale = max(abs(A[0:m, 0:m]))
        if scale > 0.0 {
            A[0:m, 0:m] = A[0:m, 0:m] / scale
        }
        for i to m {
            A[i, m] = 1.0
            A[m, i] = 1.0
        }
        rhs[m] = 1.0
    }
    step combine {
        # the pseudo-inverse: U diag(1 / s) V^T, small s dropped
        U, sv, Vt = svd(A)
        proj = transpose(U) @ rhs
        for i to m + 1 {
            proj[i] = proj[i] / sv[i] if sv[i] > PULAY_CUTOFF * sv[0] else 0.0
        }
        c = transpose(Vt) @ proj
        x_next = c[0:m] @ (h2.x + alpha * h2.r)
    }
    return x_next, h2, residual_norm(x_in, x_out)
}

#| Johnson's modified Broyden mixing, every weight 1: from the normalized
#| differences dF, dV of consecutive residuals and inputs,
#| x_next = x_in + alpha r - sum_l gamma_l (alpha dF_l + dV_l), with
#| gamma = (w0^2 + dF dF^T)^-1 dF r.
def broyden(x_in: Arr, x_out: Arr, hist: MixHistory, alpha: Num = MIX_ALPHA, history: Int = MIX_HISTORY, w0: Num = BROYDEN_W0) {
    r = x_out - x_in
    h2 = remember(hist, x_in, r, history)
    nd: Int = h2.count - 1
    x_next = x_in + alpha * r
    if nd > 0 {
        step differences {
            dF = zeros(nd, len(x_in))
            dV = zeros(nd, len(x_in))
            for i to nd {
                df = h2.r[i + 1, :] - h2.r[i, :]
                nrm = sqrt(dot(df, df))
                dF[i, :] = df / nrm
                dV[i, :] = (h2.x[i + 1, :] - h2.x[i, :]) / nrm
            }
        }
        step gamma {
            a = dF @ transpose(dF) + w0 * w0 * eye(nd)
            gam = lin_solve(a, dF @ r)
        }
        x_next = x_next - gam @ (alpha * dF + dV)
    }
    return x_next, h2, residual_norm(x_in, x_out)
}

contract mix_densities(x_in: Arr, x_out: Arr, hist: MixHistory) -> x_next: Arr, hist_next: MixHistory, residual: Num {
    intent "the density the next pass of a self-consistent field starts from, made from this pass's input and output densities and the earlier ones"
    methods
        simple: linear
        accelerated: pulay, broyden
    default pulay
    agree residual exactly
}

# ---- tests: a linear fixed-point map x = G x + b, whose fixed point solves (1 - G) x = b

MIX_N: Int = 4
MIX_TOL = 1.0e-12
MIX_PASSES: Int = 5000

#| G x + b for a fixed contraction G (spectral radius below 1) and b.
def probe_map(x: Arr) -> Arr {
    G = [[0.6, 0.1, 0.0, 0.05], [0.1, 0.5, 0.2, 0.0], [0.0, 0.2, 0.7, 0.1], [0.05, 0.0, 0.1, 0.8]]
    b = [1.0, -2.0, 0.5, 3.0]
    return G @ x + b
}

def by_linear(x_in: Arr, x_out: Arr, hist: MixHistory) -> MixStep {
    x, hs, res = mix_densities(x_in, x_out, hist) using linear
    return MixStep { x: x, hist: hs, residual: res }
}
def by_pulay(x_in: Arr, x_out: Arr, hist: MixHistory) -> MixStep {
    x, hs, res = mix_densities(x_in, x_out, hist) using pulay
    return MixStep { x: x, hist: hs, residual: res }
}
def by_broyden(x_in: Arr, x_out: Arr, hist: MixHistory) -> MixStep {
    x, hs, res = mix_densities(x_in, x_out, hist) using broyden
    return MixStep { x: x, hist: hs, residual: res }
}

#| Iterate x -> mix(x, probe_map(x)) until the residual is under MIX_TOL;
#| the passes taken and the fixed point.
def probe_solve(mixer) {
    x = zeros(MIX_N)
    hist = mix_start(MIX_N)
    passes: Int = 0
    res = 1.0
    while res > MIX_TOL and passes < MIX_PASSES {
        st = mixer(x, probe_map(x), hist)
        x = st.x
        hist = st.hist
        res = st.residual
        passes += 1
    }
    return passes, x
}

test "every mixer reaches the fixed point of a linear contraction" {
    n_lin, x_lin = probe_solve(by_linear)
    n_pul, x_pul = probe_solve(by_pulay)
    n_bro, x_bro = probe_solve(by_broyden)
    assert_true(n_lin < MIX_PASSES and n_pul < MIX_PASSES and n_bro < MIX_PASSES)
    assert_near(max(abs(probe_map(x_lin) - x_lin)), 0.0, 1.0e-11)
    assert_near(max(abs(x_pul - x_lin)), 0.0, 1.0e-10)
    assert_near(max(abs(x_bro - x_lin)), 0.0, 1.0e-10)
    # on a linear map Pulay's subspace holds the solution after MIX_N + 1 inputs
    assert_true(n_pul < n_lin and n_bro < n_lin)
}

test "every method measures the same residual" {
    x = zeros(MIX_N)
    hist = mix_start(MIX_N)
    x, hist, res = mix_densities(x, probe_map(x), hist) using every method
    assert_near(res, 3.0, 0.0)
}

# SPDX-License-Identifier: MIT OR Apache-2.0
# SPDX-FileCopyrightText: 2026 The Viv authors

#: MIX_ALPHA  the default fraction of the residual added in one step
MIX_ALPHA = 0.5

#: MIX_HISTORY  how many input densities Pulay and Broyden keep by default,
#:     the current one included
MIX_HISTORY: Int = 6

#: BROYDEN_W0  Johnson's w0, the weight of the first inverse-Jacobian guess;
#:     he uses 0.01
BROYDEN_W0 = 0.01

#: PULAY_CUTOFF  singular values of Pulay's bordered system below this
#:     fraction of the largest are dropped: the system is singular when the
#:     remembered residuals are linearly dependent, as they become near the
#:     fixed point
PULAY_CUTOFF = 1.0e-12

#| The densities a mixer has seen, oldest first: row i of x is an input
#| density and row i of r its residual (output minus input); `count` rows
#| are in use.
record MixHistory {
    x: Arr
    r: Arr
    count: Int
}

#| One mixing step as one value, for a solver that takes the mixer as a
#| function: the next density, the history to pass next time, and the
#| largest |x_out - x_in|.
record MixStep {
    x: Arr
    hist: MixHistory
    residual: Num
}

#| An empty history for densities of n numbers.
def mix_start(n: Int) -> MixHistory {
    return MixHistory { x: zeros(1, n), r: zeros(1, n), count: 0 }
}

#| A new history: the last keep - 1 rows of hist, then x and r.
def remember(hist: MixHistory, x: Arr, r: Arr, keep: Int) -> MixHistory {
    take: Int = min(hist.count, keep - 1)
    m: Int = take + 1
    xs = zeros(m, len(x))
    rs = zeros(m, len(x))
    first: Int = hist.count - take
    for i to take {
        xs[i, :] = hist.x[first + i, :]
        rs[i, :] = hist.r[first + i, :]
    }
    xs[take, :] = x
    rs[take, :] = r
    return MixHistory { x: xs, r: rs, count: m }
}

#| The largest |x_out - x_in|: how far this pass moved the density.
def residual_norm(x_in: Arr, x_out: Arr) -> Num {
    return max(abs(x_out - x_in))
}

#| (1 - alpha) x_in + alpha x_out.
def linear(x_in: Arr, x_out: Arr, hist: MixHistory, alpha: Num = MIX_ALPHA) {
    x_next = (1.0 - alpha) * x_in + alpha * x_out
    return x_next, hist, residual_norm(x_in, x_out)
}

#| Pulay's mixing: coefficients c over the remembered inputs minimizing
#| |sum c_i r_i| with sum c_i = 1 (a bordered linear system, solved by its
#| pseudo-inverse), then sum c_i (x_i + alpha r_i).
def pulay(x_in: Arr, x_out: Arr, hist: MixHistory, alpha: Num = MIX_ALPHA, history: Int = MIX_HISTORY) {
    h2 = remember(hist, x_in, x_out - x_in, history)
    m: Int = h2.count
    step overlaps {
        A = zeros(m + 1, m + 1)
        rhs = zeros(m + 1)
        A[0:m, 0:m] = h2.r @ transpose(h2.r)
        scale = max(abs(A[0:m, 0:m]))
        if scale > 0.0 {
            A[0:m, 0:m] = A[0:m, 0:m] / scale
        }
        for i to m {
            A[i, m] = 1.0
            A[m, i] = 1.0
        }
        rhs[m] = 1.0
    }
    step combine {
        # the pseudo-inverse: U diag(1 / s) V^T, small s dropped
        U, sv, Vt = svd(A)
        proj = transpose(U) @ rhs
        for i to m + 1 {
            proj[i] = proj[i] / sv[i] if sv[i] > PULAY_CUTOFF * sv[0] else 0.0
        }
        c = transpose(Vt) @ proj
        x_next = c[0:m] @ (h2.x + alpha * h2.r)
    }
    return x_next, h2, residual_norm(x_in, x_out)
}

#| Johnson's modified Broyden mixing, every weight 1: from the normalized
#| differences dF, dV of consecutive residuals and inputs,
#| x_next = x_in + alpha r - sum_l gamma_l (alpha dF_l + dV_l), with
#| gamma = (w0^2 + dF dF^T)^-1 dF r.
def broyden(x_in: Arr, x_out: Arr, hist: MixHistory, alpha: Num = MIX_ALPHA, history: Int = MIX_HISTORY, w0: Num = BROYDEN_W0) {
    r = x_out - x_in
    h2 = remember(hist, x_in, r, history)
    nd: Int = h2.count - 1
    x_next = x_in + alpha * r
    if nd > 0 {
        step differences {
            dF = zeros(nd, len(x_in))
            dV = zeros(nd, len(x_in))
            for i to nd {
                df = h2.r[i + 1, :] - h2.r[i, :]
                nrm = sqrt(dot(df, df))
                dF[i, :] = df / nrm
                dV[i, :] = (h2.x[i + 1, :] - h2.x[i, :]) / nrm
            }
        }
        step gamma {
            a = dF @ transpose(dF) + w0 * w0 * eye(nd)
            gam = lin_solve(a, dF @ r)
        }
        x_next = x_next - gam @ (alpha * dF + dV)
    }
    return x_next, h2, residual_norm(x_in, x_out)
}

contract mix_densities(x_in: Arr, x_out: Arr, hist: MixHistory) -> x_next: Arr, hist_next: MixHistory, residual: Num {
    intent "the density the next pass of a self-consistent field starts from, made from this pass's input and output densities and the earlier ones"
    methods
        simple: linear
        accelerated: pulay, broyden
    default pulay
    agree residual exactly
}

# ---- tests: a linear fixed-point map x = G x + b, whose fixed point solves (1 - G) x = b

MIX_N: Int = 4
MIX_TOL = 1.0e-12
MIX_PASSES: Int = 5000

#| G x + b for a fixed contraction G (spectral radius below 1) and b.
def probe_map(x: Arr) -> Arr {
    G = [[0.6, 0.1, 0.0, 0.05], [0.1, 0.5, 0.2, 0.0], [0.0, 0.2, 0.7, 0.1], [0.05, 0.0, 0.1, 0.8]]
    b = [1.0, -2.0, 0.5, 3.0]
    return G @ x + b
}

def by_linear(x_in: Arr, x_out: Arr, hist: MixHistory) -> MixStep {
    x, hs, res = mix_densities(x_in, x_out, hist) using linear
    return MixStep { x: x, hist: hs, residual: res }
}
def by_pulay(x_in: Arr, x_out: Arr, hist: MixHistory) -> MixStep {
    x, hs, res = mix_densities(x_in, x_out, hist) using pulay
    return MixStep { x: x, hist: hs, residual: res }
}
def by_broyden(x_in: Arr, x_out: Arr, hist: MixHistory) -> MixStep {
    x, hs, res = mix_densities(x_in, x_out, hist) using broyden
    return MixStep { x: x, hist: hs, residual: res }
}

#| Iterate x -> mix(x, probe_map(x)) until the residual is under MIX_TOL;
#| the passes taken and the fixed point.
def probe_solve(mixer) {
    x = zeros(MIX_N)
    hist = mix_start(MIX_N)
    passes: Int = 0
    res = 1.0
    while res > MIX_TOL and passes < MIX_PASSES {
        st = mixer(x, probe_map(x), hist)
        x = st.x
        hist = st.hist
        res = st.residual
        passes += 1
    }
    return passes, x
}

test "every mixer reaches the fixed point of a linear contraction" {
    n_lin, x_lin = probe_solve(by_linear)
    n_pul, x_pul = probe_solve(by_pulay)
    n_bro, x_bro = probe_solve(by_broyden)
    assert_true(n_lin < MIX_PASSES and n_pul < MIX_PASSES and n_bro < MIX_PASSES)
    assert_near(max(abs(probe_map(x_lin) - x_lin)), 0.0, 1.0e-11)
    assert_near(max(abs(x_pul - x_lin)), 0.0, 1.0e-10)
    assert_near(max(abs(x_bro - x_lin)), 0.0, 1.0e-10)
    # on a linear map Pulay's subspace holds the solution after MIX_N + 1 inputs
    assert_true(n_pul < n_lin and n_bro < n_lin)
}

test "every method measures the same residual" {
    x = zeros(MIX_N)
    hist = mix_start(MIX_N)
    x, hist, res = mix_densities(x, probe_map(x), hist) using every method
    assert_near(res, 3.0, 0.0)
}

# SPDX-License-Identifier: MIT OR Apache-2.0
# SPDX-FileCopyrightText: 2026 The Viv authors