{ "cells": [ { "cell_type": "code", "execution_count": 1, "id": "cc11adfd", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:42:32.519320Z", "iopub.status.busy": "2026-10-02T14:42:32.519125Z", "iopub.status.idle": "2026-10-02T14:42:32.523137Z", "shell.execute_reply": "2026-10-02T14:42:32.522481Z" }, "papermill": { "duration": 0.006873, "end_time": "2026-10-02T14:42:32.523668+00:00", "exception": false, "start_time": "2026-10-02T14:42:32.516795+00:00", "status": "completed" }, "tags": [ "remove-input", "active-ipynb", "remove-output" ] }, "outputs": [], "source": [ "try:\n", " from openmdao.utils.notebook_utils import notebook_mode # noqa: F401\n", "except ImportError:\n", " !python -m pip install openmdao[notebooks]" ] }, { "cell_type": "markdown", "id": "bf824d86", "metadata": { "papermill": { "duration": 0.001122, "end_time": "2026-10-02T14:42:32.526238+00:00", "exception": false, "start_time": "2026-10-02T14:42:32.525116+00:00", "status": "completed" }, "tags": [] }, "source": [ "# LinearBlockGS\n", "\n", "LinearBlockGS uses Block Gauss-Seidel to solve the linear system. LinearBlockGS iterates until the linear\n", "residual is below a tolerance, or the maximum number of iterations has been exceeded. As such,\n", "it is generally usable for any system topology, and can handle cycles and implicit states\n", "alike. It is not always the best solver to choose, however, and is known to diverge or plateau\n", "on some problems. In such a case, you may need to use a solver such as\n", "[ScipyKrylov](../../../_srcdocs/packages/solvers.linear/scipy_iter_solver).\n", "\n", "LinearBlockGS is a block solver, so you can specify different linear solvers in the subsystems and they\n", "will be utilized to solve the subsystem linear problem.\n", "\n", "Note that systems without cycles or implicit states will converge in one iteration of Block Gauss-Seidel.\n", "\n", "Here, we calculate the total derivatives across the Sellar system." ] }, { "cell_type": "code", "execution_count": 2, "id": "1ed554fe", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:42:32.529301Z", "iopub.status.busy": "2026-10-02T14:42:32.529165Z", "iopub.status.idle": "2026-10-02T14:42:34.720347Z", "shell.execute_reply": "2026-10-02T14:42:34.719703Z" }, "papermill": { "duration": 2.193353, "end_time": "2026-10-02T14:42:34.720840+00:00", "exception": false, "start_time": "2026-10-02T14:42:32.527487+00:00", "status": "completed" }, "tags": [ "remove-input", "remove-output" ] }, "outputs": [ { "data": { "application/papermill.record/text/html": "
class SellarDis1withDerivatives(SellarDis1):\n    """\n    Component containing Discipline 1 -- derivatives version.\n    """\n\n    def setup_partials(self):\n        # Analytic Derivs\n        self.declare_partials(of='*', wrt='*')\n\n    def compute_partials(self, inputs, partials):\n        """\n        Jacobian for Sellar discipline 1.\n        """\n        partials['y1', 'y2'] = -0.2\n        partials['y1', 'z'] = np.array([[2.0 * inputs['z'][0], 1.0]])\n        partials['y1', 'x'] = 1.0\n
\n", "application/papermill.record/text/latex": "\\begin{Verbatim}[commandchars=\\\\\\{\\}]\n\\PY{k}{class}\\PY{+w}{ }\\PY{n+nc}{SellarDis1withDerivatives}\\PY{p}{(}\\PY{n}{SellarDis1}\\PY{p}{)}\\PY{p}{:}\n\\PY{+w}{ }\\PY{l+s+sd}{\\PYZdq{}\\PYZdq{}\\PYZdq{}}\n\\PY{l+s+sd}{ Component containing Discipline 1 \\PYZhy{}\\PYZhy{} derivatives version.}\n\\PY{l+s+sd}{ \\PYZdq{}\\PYZdq{}\\PYZdq{}}\n\n \\PY{k}{def}\\PY{+w}{ }\\PY{n+nf}{setup\\PYZus{}partials}\\PY{p}{(}\\PY{n+nb+bp}{self}\\PY{p}{)}\\PY{p}{:}\n \\PY{c+c1}{\\PYZsh{} Analytic Derivs}\n \\PY{n+nb+bp}{self}\\PY{o}{.}\\PY{n}{declare\\PYZus{}partials}\\PY{p}{(}\\PY{n}{of}\\PY{o}{=}\\PY{l+s+s1}{\\PYZsq{}}\\PY{l+s+s1}{*}\\PY{l+s+s1}{\\PYZsq{}}\\PY{p}{,} \\PY{n}{wrt}\\PY{o}{=}\\PY{l+s+s1}{\\PYZsq{}}\\PY{l+s+s1}{*}\\PY{l+s+s1}{\\PYZsq{}}\\PY{p}{)}\n\n \\PY{k}{def}\\PY{+w}{ }\\PY{n+nf}{compute\\PYZus{}partials}\\PY{p}{(}\\PY{n+nb+bp}{self}\\PY{p}{,} \\PY{n}{inputs}\\PY{p}{,} \\PY{n}{partials}\\PY{p}{)}\\PY{p}{:}\n\\PY{+w}{ }\\PY{l+s+sd}{\\PYZdq{}\\PYZdq{}\\PYZdq{}}\n\\PY{l+s+sd}{ Jacobian for Sellar discipline 1.}\n\\PY{l+s+sd}{ \\PYZdq{}\\PYZdq{}\\PYZdq{}}\n \\PY{n}{partials}\\PY{p}{[}\\PY{l+s+s1}{\\PYZsq{}}\\PY{l+s+s1}{y1}\\PY{l+s+s1}{\\PYZsq{}}\\PY{p}{,} \\PY{l+s+s1}{\\PYZsq{}}\\PY{l+s+s1}{y2}\\PY{l+s+s1}{\\PYZsq{}}\\PY{p}{]} \\PY{o}{=} \\PY{o}{\\PYZhy{}}\\PY{l+m+mf}{0.2}\n \\PY{n}{partials}\\PY{p}{[}\\PY{l+s+s1}{\\PYZsq{}}\\PY{l+s+s1}{y1}\\PY{l+s+s1}{\\PYZsq{}}\\PY{p}{,} \\PY{l+s+s1}{\\PYZsq{}}\\PY{l+s+s1}{z}\\PY{l+s+s1}{\\PYZsq{}}\\PY{p}{]} \\PY{o}{=} \\PY{n}{np}\\PY{o}{.}\\PY{n}{array}\\PY{p}{(}\\PY{p}{[}\\PY{p}{[}\\PY{l+m+mf}{2.0} \\PY{o}{*} \\PY{n}{inputs}\\PY{p}{[}\\PY{l+s+s1}{\\PYZsq{}}\\PY{l+s+s1}{z}\\PY{l+s+s1}{\\PYZsq{}}\\PY{p}{]}\\PY{p}{[}\\PY{l+m+mi}{0}\\PY{p}{]}\\PY{p}{,} \\PY{l+m+mf}{1.0}\\PY{p}{]}\\PY{p}{]}\\PY{p}{)}\n \\PY{n}{partials}\\PY{p}{[}\\PY{l+s+s1}{\\PYZsq{}}\\PY{l+s+s1}{y1}\\PY{l+s+s1}{\\PYZsq{}}\\PY{p}{,} \\PY{l+s+s1}{\\PYZsq{}}\\PY{l+s+s1}{x}\\PY{l+s+s1}{\\PYZsq{}}\\PY{p}{]} \\PY{o}{=} \\PY{l+m+mf}{1.0}\n\\end{Verbatim}\n", "application/papermill.record/text/plain": "class SellarDis1withDerivatives(SellarDis1):\n \"\"\"\n Component containing Discipline 1 -- derivatives version.\n \"\"\"\n\n def setup_partials(self):\n # Analytic Derivs\n self.declare_partials(of='*', wrt='*')\n\n def compute_partials(self, inputs, partials):\n \"\"\"\n Jacobian for Sellar discipline 1.\n \"\"\"\n partials['y1', 'y2'] = -0.2\n partials['y1', 'z'] = np.array([[2.0 * inputs['z'][0], 1.0]])\n partials['y1', 'x'] = 1.0" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "code_src24" } }, "output_type": "display_data" } ], "source": [ "from openmdao.utils.notebook_utils import get_code\n", "from myst_nb import glue\n", "glue(\"code_src24\", get_code(\"openmdao.test_suite.components.sellar.SellarDis1withDerivatives\"), display=False)" ] }, { "cell_type": "markdown", "id": "745c74dc", "metadata": { "papermill": { "duration": 0.001453, "end_time": "2026-10-02T14:42:34.724329+00:00", "exception": false, "start_time": "2026-10-02T14:42:34.722876+00:00", "status": "completed" }, "tags": [] }, "source": [ ":::{dropdown} `SellarDis1withDerivatives` class definition \n", "\n", "{glue:}`code_src24`\n", ":::" ] }, { "cell_type": "code", "execution_count": 3, "id": "64689ec6", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:42:34.727829Z", "iopub.status.busy": "2026-10-02T14:42:34.727625Z", "iopub.status.idle": "2026-10-02T14:42:34.733544Z", "shell.execute_reply": "2026-10-02T14:42:34.733120Z" }, "papermill": { "duration": 0.008318, "end_time": "2026-10-02T14:42:34.733987+00:00", "exception": false, "start_time": "2026-10-02T14:42:34.725669+00:00", "status": "completed" }, "tags": [ "remove-input", "remove-output" ] }, "outputs": [ { "data": { "application/papermill.record/text/html": "
class SellarDis2withDerivatives(SellarDis2):\n    """\n    Component containing Discipline 2 -- derivatives version.\n    """\n\n    def setup_partials(self):\n        # Analytic Derivs\n        self.declare_partials(of='*', wrt='*')\n\n    def compute_partials(self, inputs, J):\n        """\n        Jacobian for Sellar discipline 2.\n        """\n        y1 = inputs['y1']\n        if y1.real < 0.0:\n            y1 *= -1\n        if y1.real < 1e-8:\n            y1 = 1e-8\n\n        J['y2', 'y1'] = .5*y1**-.5\n        J['y2', 'z'] = np.array([[1.0, 1.0]])\n
\n", "application/papermill.record/text/latex": "\\begin{Verbatim}[commandchars=\\\\\\{\\}]\n\\PY{k}{class}\\PY{+w}{ }\\PY{n+nc}{SellarDis2withDerivatives}\\PY{p}{(}\\PY{n}{SellarDis2}\\PY{p}{)}\\PY{p}{:}\n\\PY{+w}{ }\\PY{l+s+sd}{\\PYZdq{}\\PYZdq{}\\PYZdq{}}\n\\PY{l+s+sd}{ Component containing Discipline 2 \\PYZhy{}\\PYZhy{} derivatives version.}\n\\PY{l+s+sd}{ \\PYZdq{}\\PYZdq{}\\PYZdq{}}\n\n \\PY{k}{def}\\PY{+w}{ }\\PY{n+nf}{setup\\PYZus{}partials}\\PY{p}{(}\\PY{n+nb+bp}{self}\\PY{p}{)}\\PY{p}{:}\n \\PY{c+c1}{\\PYZsh{} Analytic Derivs}\n \\PY{n+nb+bp}{self}\\PY{o}{.}\\PY{n}{declare\\PYZus{}partials}\\PY{p}{(}\\PY{n}{of}\\PY{o}{=}\\PY{l+s+s1}{\\PYZsq{}}\\PY{l+s+s1}{*}\\PY{l+s+s1}{\\PYZsq{}}\\PY{p}{,} \\PY{n}{wrt}\\PY{o}{=}\\PY{l+s+s1}{\\PYZsq{}}\\PY{l+s+s1}{*}\\PY{l+s+s1}{\\PYZsq{}}\\PY{p}{)}\n\n \\PY{k}{def}\\PY{+w}{ }\\PY{n+nf}{compute\\PYZus{}partials}\\PY{p}{(}\\PY{n+nb+bp}{self}\\PY{p}{,} \\PY{n}{inputs}\\PY{p}{,} \\PY{n}{J}\\PY{p}{)}\\PY{p}{:}\n\\PY{+w}{ }\\PY{l+s+sd}{\\PYZdq{}\\PYZdq{}\\PYZdq{}}\n\\PY{l+s+sd}{ Jacobian for Sellar discipline 2.}\n\\PY{l+s+sd}{ \\PYZdq{}\\PYZdq{}\\PYZdq{}}\n \\PY{n}{y1} \\PY{o}{=} \\PY{n}{inputs}\\PY{p}{[}\\PY{l+s+s1}{\\PYZsq{}}\\PY{l+s+s1}{y1}\\PY{l+s+s1}{\\PYZsq{}}\\PY{p}{]}\n \\PY{k}{if} \\PY{n}{y1}\\PY{o}{.}\\PY{n}{real} \\PY{o}{\\PYZlt{}} \\PY{l+m+mf}{0.0}\\PY{p}{:}\n \\PY{n}{y1} \\PY{o}{*}\\PY{o}{=} \\PY{o}{\\PYZhy{}}\\PY{l+m+mi}{1}\n \\PY{k}{if} \\PY{n}{y1}\\PY{o}{.}\\PY{n}{real} \\PY{o}{\\PYZlt{}} \\PY{l+m+mf}{1e\\PYZhy{}8}\\PY{p}{:}\n \\PY{n}{y1} \\PY{o}{=} \\PY{l+m+mf}{1e\\PYZhy{}8}\n\n \\PY{n}{J}\\PY{p}{[}\\PY{l+s+s1}{\\PYZsq{}}\\PY{l+s+s1}{y2}\\PY{l+s+s1}{\\PYZsq{}}\\PY{p}{,} \\PY{l+s+s1}{\\PYZsq{}}\\PY{l+s+s1}{y1}\\PY{l+s+s1}{\\PYZsq{}}\\PY{p}{]} \\PY{o}{=} \\PY{l+m+mf}{.5}\\PY{o}{*}\\PY{n}{y1}\\PY{o}{*}\\PY{o}{*}\\PY{o}{\\PYZhy{}}\\PY{l+m+mf}{.5}\n \\PY{n}{J}\\PY{p}{[}\\PY{l+s+s1}{\\PYZsq{}}\\PY{l+s+s1}{y2}\\PY{l+s+s1}{\\PYZsq{}}\\PY{p}{,} \\PY{l+s+s1}{\\PYZsq{}}\\PY{l+s+s1}{z}\\PY{l+s+s1}{\\PYZsq{}}\\PY{p}{]} \\PY{o}{=} \\PY{n}{np}\\PY{o}{.}\\PY{n}{array}\\PY{p}{(}\\PY{p}{[}\\PY{p}{[}\\PY{l+m+mf}{1.0}\\PY{p}{,} \\PY{l+m+mf}{1.0}\\PY{p}{]}\\PY{p}{]}\\PY{p}{)}\n\\end{Verbatim}\n", "application/papermill.record/text/plain": "class SellarDis2withDerivatives(SellarDis2):\n \"\"\"\n Component containing Discipline 2 -- derivatives version.\n \"\"\"\n\n def setup_partials(self):\n # Analytic Derivs\n self.declare_partials(of='*', wrt='*')\n\n def compute_partials(self, inputs, J):\n \"\"\"\n Jacobian for Sellar discipline 2.\n \"\"\"\n y1 = inputs['y1']\n if y1.real < 0.0:\n y1 *= -1\n if y1.real < 1e-8:\n y1 = 1e-8\n\n J['y2', 'y1'] = .5*y1**-.5\n J['y2', 'z'] = np.array([[1.0, 1.0]])" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "code_src25" } }, "output_type": "display_data" } ], "source": [ "from openmdao.utils.notebook_utils import get_code\n", "from myst_nb import glue\n", "glue(\"code_src25\", get_code(\"openmdao.test_suite.components.sellar.SellarDis2withDerivatives\"), display=False)" ] }, { "cell_type": "markdown", "id": "cd60d6b6", "metadata": { "papermill": { "duration": 0.001358, "end_time": "2026-10-02T14:42:34.736892+00:00", "exception": false, "start_time": "2026-10-02T14:42:34.735534+00:00", "status": "completed" }, "tags": [] }, "source": [ ":::{dropdown} `SellarDis2withDerivatives` class definition \n", "\n", "{glue:}`code_src25`\n", ":::" ] }, { "cell_type": "code", "execution_count": 4, "id": "8ba3d96d", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:42:34.741064Z", "iopub.status.busy": "2026-10-02T14:42:34.740656Z", "iopub.status.idle": "2026-10-02T14:42:36.027866Z", "shell.execute_reply": "2026-10-02T14:42:36.026865Z" }, "papermill": { "duration": 1.290029, "end_time": "2026-10-02T14:42:36.028463+00:00", "exception": false, "start_time": "2026-10-02T14:42:34.738434+00:00", "status": "completed" }, "tags": [] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[1790952155.979002] [runnervm8df0l:7167 :0] ib_iface.c:1269 UCX ERROR mana_0: iface 0x55f6ecd7e670 failed to create UD QP TX wr:256 sge:6 inl:64 resp:0 RX wr:4096 sge:1 resp:0 failed: Operation not supported\n", "[1790952155.979273] [runnervm8df0l:7167 :0] ucp_worker.c:1412 UCX ERROR uct_iface_open(ud_verbs/mana_0:1) failed: Input/output error\n", "NL: NLBGS Converged in 8 iterations\n", "LN: LNBGS Converged in 7 iterations\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ "[runnervm8df0l:07167] pml_ucx.c:313 Error: Failed to create UCP worker\n" ] } ], "source": [ "import numpy as np\n", "import openmdao.api as om\n", "\n", "from openmdao.test_suite.components.sellar import SellarDis1withDerivatives, SellarDis2withDerivatives\n", "\n", "prob = om.Problem()\n", "model = prob.model\n", "\n", "model.add_subsystem('d1', SellarDis1withDerivatives(), promotes=['x', 'z', 'y1', 'y2'])\n", "model.add_subsystem('d2', SellarDis2withDerivatives(), promotes=['z', 'y1', 'y2'])\n", "\n", "model.add_subsystem('obj_cmp', om.ExecComp('obj = x**2 + z[1] + y1 + exp(-y2)',\n", " z=np.array([0.0, 0.0]), x=0.0),\n", " promotes=['obj', 'x', 'z', 'y1', 'y2'])\n", "\n", "model.add_subsystem('con_cmp1', om.ExecComp('con1 = 3.16 - y1'), promotes=['con1', 'y1'])\n", "model.add_subsystem('con_cmp2', om.ExecComp('con2 = y2 - 24.0'), promotes=['con2', 'y2'])\n", "\n", "model.linear_solver = om.LinearBlockGS()\n", "model.nonlinear_solver = om.NonlinearBlockGS()\n", "\n", "prob.setup()\n", "\n", "prob.set_val('x', 1.)\n", "prob.set_val('z', np.array([5.0, 2.0]))\n", "\n", "prob.run_model()\n", "\n", "wrt = ['z']\n", "of = ['obj']\n", "\n", "J = prob.compute_totals(of=of, wrt=wrt, return_format='flat_dict')" ] }, { "cell_type": "code", "execution_count": 5, "id": "fde8fb6b", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:42:36.033960Z", "iopub.status.busy": "2026-10-02T14:42:36.033640Z", "iopub.status.idle": "2026-10-02T14:42:36.036915Z", "shell.execute_reply": "2026-10-02T14:42:36.036330Z" }, "papermill": { "duration": 0.00709, "end_time": "2026-10-02T14:42:36.037823+00:00", "exception": false, "start_time": "2026-10-02T14:42:36.030733+00:00", "status": "completed" }, "tags": [] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "9.610010557013222\n", "1.7844853356442276\n" ] } ], "source": [ "print(J['obj', 'z'][0][0])\n", "print(J['obj', 'z'][0][1])" ] }, { "cell_type": "code", "execution_count": 6, "id": "015347e2", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:42:36.042517Z", "iopub.status.busy": "2026-10-02T14:42:36.042346Z", "iopub.status.idle": "2026-10-02T14:42:36.047336Z", "shell.execute_reply": "2026-10-02T14:42:36.046778Z" }, "papermill": { "duration": 0.008353, "end_time": "2026-10-02T14:42:36.048109+00:00", "exception": false, "start_time": "2026-10-02T14:42:36.039756+00:00", "status": "completed" }, "tags": [ "remove-input", "remove-output" ] }, "outputs": [ { "data": { "text/plain": [ "np.float64(2.440912408630737e-09)" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from openmdao.utils.assert_utils import assert_near_equal\n", "\n", "assert_near_equal(J['obj', 'z'][0][0], 9.61001056, .00001)\n", "assert_near_equal(J['obj', 'z'][0][1], 1.78448534, .00001)" ] }, { "cell_type": "markdown", "id": "10bbcea3", "metadata": { "papermill": { "duration": 0.001898, "end_time": "2026-10-02T14:42:36.051972+00:00", "exception": false, "start_time": "2026-10-02T14:42:36.050074+00:00", "status": "completed" }, "tags": [] }, "source": [ "## LinearBlockGS Options" ] }, { "cell_type": "code", "execution_count": 7, "id": "c619fa92", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:42:36.056643Z", "iopub.status.busy": "2026-10-02T14:42:36.056471Z", "iopub.status.idle": "2026-10-02T14:42:36.059856Z", "shell.execute_reply": "2026-10-02T14:42:36.059444Z" }, "papermill": { "duration": 0.006911, "end_time": "2026-10-02T14:42:36.060790+00:00", "exception": false, "start_time": "2026-10-02T14:42:36.053879+00:00", "status": "completed" }, "tags": [ "remove-input" ] }, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", " \n", "\n", "\n", "

\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
OptionDefaultAcceptable ValuesAcceptable TypesDescription
aitken_initial_factor1.0N/AN/Ainitial value for Aitken relaxation factor
aitken_max_factor1.5N/AN/Aupper limit for Aitken relaxation factor
aitken_min_factor0.1N/AN/Alower limit for Aitken relaxation factor
assemble_jacFalse[True, False]['bool']Activates use of assembled jacobian by this solver.
atol1e-10N/AN/Aabsolute error tolerance
err_on_non_convergeFalse[True, False]['bool']When True, AnalysisError will be raised if we don't converge.
iprint1N/A['int']whether to print output
maxiter10N/A['int']maximum number of iterations
rtol1e-10N/AN/Arelative error tolerance
use_aitkenFalse[True, False]['bool']set to True to use Aitken relaxation
\n", "\n", "\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "om.show_options_table(\"openmdao.solvers.linear.linear_block_gs.LinearBlockGS\")" ] }, { "cell_type": "markdown", "id": "d2fca1c7", "metadata": { "papermill": { "duration": 0.002042, "end_time": "2026-10-02T14:42:36.064859+00:00", "exception": false, "start_time": "2026-10-02T14:42:36.062817+00:00", "status": "completed" }, "tags": [] }, "source": [ "## LinearBlockGS Constructor\n", "\n", "The call signature for the `LinearBlockGS` constructor is:\n", "\n", "```{eval-rst}\n", " .. automethod:: openmdao.solvers.linear.linear_block_gs.LinearBlockGS.__init__\n", " :noindex:\n", "```\n", "\n", "## Aitken relaxation\n", "\n", "This solver implements Aitken relaxation, as described in Algorithm 1 of this paper on aerostructual design [optimization](https://mdolab.engin.umich.edu/bibliography/Kenway2014a). The relaxation is turned off by default, but it may help convergence for more tightly coupled models.\n", "\n", "\n", "## LinearBlockGS Option Examples\n", "\n", "**maxiter**\n", "\n", " This lets you specify the maximum number of Gauss-Seidel iterations to apply. In this example, we\n", " cut it back from the default, ten, down to two, so that it terminates a few iterations earlier and doesn't\n", " reach either of the specified absolute or relative tolerances." ] }, { "cell_type": "code", "execution_count": 8, "id": "92cbeb23", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:42:36.069833Z", "iopub.status.busy": "2026-10-02T14:42:36.069655Z", "iopub.status.idle": "2026-10-02T14:42:36.084335Z", "shell.execute_reply": "2026-10-02T14:42:36.083902Z" }, "papermill": { "duration": 0.017862, "end_time": "2026-10-02T14:42:36.084772+00:00", "exception": false, "start_time": "2026-10-02T14:42:36.066910+00:00", "status": "completed" }, "tags": [] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "NL: NLBGS Converged in 8 iterations\n", "LN: LNBGSSolver 'LN: LNBGS' on system '' failed to converge in 2 iterations.\n" ] } ], "source": [ "from openmdao.test_suite.components.sellar import SellarDis1withDerivatives, SellarDis2withDerivatives\n", "\n", "prob = om.Problem()\n", "model = prob.model\n", "\n", "model.add_subsystem('d1', SellarDis1withDerivatives(), promotes=['x', 'z', 'y1', 'y2'])\n", "model.add_subsystem('d2', SellarDis2withDerivatives(), promotes=['z', 'y1', 'y2'])\n", "\n", "model.add_subsystem('obj_cmp', om.ExecComp('obj = x**2 + z[1] + y1 + exp(-y2)',\n", " z=np.array([0.0, 0.0]), x=0.0),\n", " promotes=['obj', 'x', 'z', 'y1', 'y2'])\n", "\n", "model.add_subsystem('con_cmp1', om.ExecComp('con1 = 3.16 - y1'), promotes=['con1', 'y1'])\n", "model.add_subsystem('con_cmp2', om.ExecComp('con2 = y2 - 24.0'), promotes=['con2', 'y2'])\n", "\n", "model.nonlinear_solver = om.NonlinearBlockGS()\n", "\n", "model.linear_solver = om.LinearBlockGS()\n", "model.linear_solver.options['maxiter'] = 2\n", "\n", "prob.setup()\n", "\n", "prob.set_val('x', 1.)\n", "prob.set_val('z', np.array([5.0, 2.0]))\n", "\n", "prob.run_model()\n", "\n", "wrt = ['z']\n", "of = ['obj']\n", "\n", "J = prob.compute_totals(of=of, wrt=wrt, return_format='flat_dict')" ] }, { "cell_type": "code", "execution_count": 9, "id": "d6e8958f", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:42:36.089861Z", "iopub.status.busy": "2026-10-02T14:42:36.089677Z", "iopub.status.idle": "2026-10-02T14:42:36.092390Z", "shell.execute_reply": "2026-10-02T14:42:36.091781Z" }, "papermill": { "duration": 0.005978, "end_time": "2026-10-02T14:42:36.092884+00:00", "exception": false, "start_time": "2026-10-02T14:42:36.086906+00:00", "status": "completed" }, "tags": [] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "9.602301180035953\n", "1.780225005469842\n" ] } ], "source": [ "print(J['obj', 'z'][0][0])\n", "print(J['obj', 'z'][0][1])" ] }, { "cell_type": "code", "execution_count": 10, "id": "4400c9dc", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:42:36.097849Z", "iopub.status.busy": "2026-10-02T14:42:36.097655Z", "iopub.status.idle": "2026-10-02T14:42:36.101259Z", "shell.execute_reply": "2026-10-02T14:42:36.100558Z" }, "papermill": { "duration": 0.006768, "end_time": "2026-10-02T14:42:36.101716+00:00", "exception": false, "start_time": "2026-10-02T14:42:36.094948+00:00", "status": "completed" }, "tags": [ "remove-input", "remove-output" ] }, "outputs": [ { "data": { "text/plain": [ "np.float64(8.868188774823819e-14)" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "assert_near_equal(J['obj', 'z'][0][0], 9.60230118004, .00001)\n", "assert_near_equal(J['obj', 'z'][0][1], 1.78022500547, .00001)" ] }, { "cell_type": "markdown", "id": "8b372e58", "metadata": { "papermill": { "duration": 0.002286, "end_time": "2026-10-02T14:42:36.194054+00:00", "exception": false, "start_time": "2026-10-02T14:42:36.191768+00:00", "status": "completed" }, "tags": [] }, "source": [ "**atol**\n", "\n", " Here, we set the absolute tolerance to a looser value that will trigger an earlier termination. After\n", " each iteration, the norm of the linear residuals is calculated by calling `apply_linear`. If this norm value is lower than the absolute tolerance `atol`, the iteration will terminate." ] }, { "cell_type": "code", "execution_count": 11, "id": "ac2f7014", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:42:36.200217Z", "iopub.status.busy": "2026-10-02T14:42:36.199969Z", "iopub.status.idle": "2026-10-02T14:42:36.214283Z", "shell.execute_reply": "2026-10-02T14:42:36.213594Z" }, "papermill": { "duration": 0.018522, "end_time": "2026-10-02T14:42:36.215116+00:00", "exception": false, "start_time": "2026-10-02T14:42:36.196594+00:00", "status": "completed" }, "tags": [] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "NL: NLBGS Converged in 8 iterations\n", "LN: LNBGS Converged in 3 iterations\n" ] } ], "source": [ "from openmdao.test_suite.components.sellar import SellarDis1withDerivatives, SellarDis2withDerivatives\n", "\n", "prob = om.Problem()\n", "model = prob.model\n", "\n", "model.add_subsystem('d1', SellarDis1withDerivatives(), promotes=['x', 'z', 'y1', 'y2'])\n", "model.add_subsystem('d2', SellarDis2withDerivatives(), promotes=['z', 'y1', 'y2'])\n", "\n", "model.add_subsystem('obj_cmp', om.ExecComp('obj = x**2 + z[1] + y1 + exp(-y2)',\n", " z=np.array([0.0, 0.0]), x=0.0),\n", " promotes=['obj', 'x', 'z', 'y1', 'y2'])\n", "\n", "model.add_subsystem('con_cmp1', om.ExecComp('con1 = 3.16 - y1'), promotes=['con1', 'y1'])\n", "model.add_subsystem('con_cmp2', om.ExecComp('con2 = y2 - 24.0'), promotes=['con2', 'y2'])\n", "\n", "model.nonlinear_solver = om.NonlinearBlockGS()\n", "\n", "model.linear_solver = om.LinearBlockGS()\n", "model.linear_solver.options['atol'] = 1.0e-3\n", "\n", "prob.setup()\n", "\n", "prob.set_val('x', 1.)\n", "prob.set_val('z', np.array([5.0, 2.0]))\n", "\n", "prob.run_model()\n", "\n", "wrt = ['z']\n", "of = ['obj']\n", "\n", "J = prob.compute_totals(of=of, wrt=wrt, return_format='flat_dict')" ] }, { "cell_type": "code", "execution_count": 12, "id": "a1dc783a", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:42:36.220498Z", "iopub.status.busy": "2026-10-02T14:42:36.220310Z", "iopub.status.idle": "2026-10-02T14:42:36.223499Z", "shell.execute_reply": "2026-10-02T14:42:36.222787Z" }, "papermill": { "duration": 0.00673, "end_time": "2026-10-02T14:42:36.224092+00:00", "exception": false, "start_time": "2026-10-02T14:42:36.217362+00:00", "status": "completed" }, "tags": [] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "9.610162961754462\n", "1.7845695570436657\n" ] } ], "source": [ "print(J['obj', 'z'][0][0])\n", "print(J['obj', 'z'][0][1])" ] }, { "cell_type": "code", "execution_count": 13, "id": "cc76aacf", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:42:36.229409Z", "iopub.status.busy": "2026-10-02T14:42:36.229236Z", "iopub.status.idle": "2026-10-02T14:42:36.232528Z", "shell.execute_reply": "2026-10-02T14:42:36.232055Z" }, "papermill": { "duration": 0.006829, "end_time": "2026-10-02T14:42:36.233123+00:00", "exception": false, "start_time": "2026-10-02T14:42:36.226294+00:00", "status": "completed" }, "tags": [ "remove-input", "remove-output" ] }, "outputs": [ { "data": { "text/plain": [ "np.float64(2.0541280491120563e-12)" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "assert_near_equal(J['obj', 'z'][0][0], 9.61016296175, .00001)\n", "assert_near_equal(J['obj', 'z'][0][1], 1.78456955704, .00001)" ] }, { "cell_type": "markdown", "id": "c0a11db0", "metadata": { "papermill": { "duration": 0.002204, "end_time": "2026-10-02T14:42:36.237565+00:00", "exception": false, "start_time": "2026-10-02T14:42:36.235361+00:00", "status": "completed" }, "tags": [] }, "source": [ "**rtol**\n", "\n", " Here, we set the relative tolerance to a looser value that will trigger an earlier termination. After\n", " each iteration, the norm of the linear residuals is calculated by calling `apply_linear`. If the ratio of the currently calculated norm to the\n", " initial residual norm is lower than the relative tolerance `rtol`, the iteration will terminate." ] }, { "cell_type": "code", "execution_count": 14, "id": "7127feeb", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:42:36.242937Z", "iopub.status.busy": "2026-10-02T14:42:36.242672Z", "iopub.status.idle": "2026-10-02T14:42:36.258075Z", "shell.execute_reply": "2026-10-02T14:42:36.257405Z" }, "papermill": { "duration": 0.019017, "end_time": "2026-10-02T14:42:36.258843+00:00", "exception": false, "start_time": "2026-10-02T14:42:36.239826+00:00", "status": "completed" }, "tags": [] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "NL: NLBGS Converged in 8 iterations\n", "LN: LNBGS Converged in 3 iterations\n" ] } ], "source": [ "from openmdao.test_suite.components.sellar import SellarDis1withDerivatives, SellarDis2withDerivatives\n", "\n", "prob = om.Problem()\n", "model = prob.model\n", "\n", "model.add_subsystem('d1', SellarDis1withDerivatives(), promotes=['x', 'z', 'y1', 'y2'])\n", "model.add_subsystem('d2', SellarDis2withDerivatives(), promotes=['z', 'y1', 'y2'])\n", "\n", "model.add_subsystem('obj_cmp', om.ExecComp('obj = x**2 + z[1] + y1 + exp(-y2)',\n", " z=np.array([0.0, 0.0]), x=0.0),\n", " promotes=['obj', 'x', 'z', 'y1', 'y2'])\n", "\n", "model.add_subsystem('con_cmp1', om.ExecComp('con1 = 3.16 - y1'), promotes=['con1', 'y1'])\n", "model.add_subsystem('con_cmp2', om.ExecComp('con2 = y2 - 24.0'), promotes=['con2', 'y2'])\n", "\n", "model.nonlinear_solver = om.NonlinearBlockGS()\n", "\n", "model.linear_solver = om.LinearBlockGS()\n", "model.linear_solver.options['rtol'] = 1.0e-3\n", "\n", "prob.setup()\n", "\n", "prob.set_val('x', 1.)\n", "prob.set_val('z', np.array([5.0, 2.0]))\n", "\n", "prob.run_model()\n", "\n", "wrt = ['z']\n", "of = ['obj']\n", "\n", "J = prob.compute_totals(of=of, wrt=wrt, return_format='flat_dict')" ] }, { "cell_type": "code", "execution_count": 15, "id": "be378405", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:42:36.264088Z", "iopub.status.busy": "2026-10-02T14:42:36.263929Z", "iopub.status.idle": "2026-10-02T14:42:36.267206Z", "shell.execute_reply": "2026-10-02T14:42:36.266586Z" }, "papermill": { "duration": 0.006709, "end_time": "2026-10-02T14:42:36.267751+00:00", "exception": false, "start_time": "2026-10-02T14:42:36.261042+00:00", "status": "completed" }, "tags": [] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "9.610162961754462\n", "1.7845695570436657\n" ] } ], "source": [ "print(J['obj', 'z'][0][0])\n", "print(J['obj', 'z'][0][1])" ] }, { "cell_type": "code", "execution_count": 16, "id": "629c83ce", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:42:36.273140Z", "iopub.status.busy": "2026-10-02T14:42:36.272985Z", "iopub.status.idle": "2026-10-02T14:42:36.276131Z", "shell.execute_reply": "2026-10-02T14:42:36.275541Z" }, "papermill": { "duration": 0.006545, "end_time": "2026-10-02T14:42:36.276520+00:00", "exception": false, "start_time": "2026-10-02T14:42:36.269975+00:00", "status": "completed" }, "tags": [ "remove-input", "remove-output" ] }, "outputs": [ { "data": { "text/plain": [ "np.float64(2.0541280491120563e-12)" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "assert_near_equal(J['obj', 'z'][0][0], 9.61016296175, .00001)\n", "assert_near_equal(J['obj', 'z'][0][1], 1.78456955704, .00001)" ] } ], "metadata": { "celltoolbar": "Tags", "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.13.14" }, "orphan": true, "papermill": { "default_parameters": {}, "duration": 5.268368, "end_time": "2026-10-02T14:42:37.095049+00:00", "environment_variables": {}, "exception": null, "input_path": "/home/runner/work/OpenMDAO/OpenMDAO/openmdao/docs/openmdao_book/features/building_blocks/solvers/linear_block_gs.ipynb", "output_path": "/home/runner/work/OpenMDAO/OpenMDAO/openmdao/docs/_executed_book/features/building_blocks/solvers/linear_block_gs.ipynb", "parameters": {}, "start_time": "2026-10-02T14:42:31.826681+00:00", "version": "2.7.0" } }, "nbformat": 4, "nbformat_minor": 5 }