{ "cells": [ { "cell_type": "code", "execution_count": 1, "id": "2fa2fc44", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:42:47.462336Z", "iopub.status.busy": "2026-10-02T14:42:47.462153Z", "iopub.status.idle": "2026-10-02T14:42:47.466719Z", "shell.execute_reply": "2026-10-02T14:42:47.465625Z" }, "papermill": { "duration": 0.0083, "end_time": "2026-10-02T14:42:47.467428+00:00", "exception": false, "start_time": "2026-10-02T14:42:47.459128+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": "16dd799b", "metadata": { "papermill": { "duration": 0.001425, "end_time": "2026-10-02T14:42:47.470458+00:00", "exception": false, "start_time": "2026-10-02T14:42:47.469033+00:00", "status": "completed" }, "tags": [] }, "source": [ "# NonlinearBlockJac\n", "\n", "NonlinearBlockJac is a nonlinear solver that uses the block Jacobi method to solve\n", "the system. When to choose this solver over [NonlinearBlockGS](../../../_srcdocs/packages/solvers.nonlinear/nonlinear_block_gs) is an advanced topic, but it is valid for systems that satisfy the same conditions:\n", "\n", "1. System (or subsystem) contains a cycle, though subsystems may.\n", "2. System does not contain any implicit states, though subsystems may.\n", "\n", "Note that you may not know if you satisfy the second condition, so choosing a solver can be \"trial and error.\" If\n", "NonlinearBlockJac doesn't work, then you will need to use [NewtonSolver](../../../_srcdocs/packages/solvers.nonlinear/newton)\n", "\n", "The main difference over `NonlinearBlockGS` is that data passing is delayed until after all subsystems have been\n", "executed.\n", "\n", "Here, we choose NonlinearBlockJac to solve the Sellar problem, which has two components with a\n", "cyclic dependency, has no implicit states, and works with Jacobi, although it fails to converge before\n", "hitting the default iteration limit of 10. In the next section we'll show how to increase the number\n", "of allowed iterations so that it will converge." ] }, { "cell_type": "code", "execution_count": 2, "id": "ba6ed75c", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:42:47.474027Z", "iopub.status.busy": "2026-10-02T14:42:47.473875Z", "iopub.status.idle": "2026-10-02T14:42:49.873021Z", "shell.execute_reply": "2026-10-02T14:42:49.872369Z" }, "papermill": { "duration": 2.40264, "end_time": "2026-10-02T14:42:49.874297+00:00", "exception": false, "start_time": "2026-10-02T14:42:47.471657+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_src36" } }, "output_type": "display_data" } ], "source": [ "from openmdao.utils.notebook_utils import get_code\n", "from myst_nb import glue\n", "glue(\"code_src36\", get_code(\"openmdao.test_suite.components.sellar.SellarDis1withDerivatives\"), display=False)" ] }, { "cell_type": "markdown", "id": "f2f146e5", "metadata": { "papermill": { "duration": 0.001922, "end_time": "2026-10-02T14:42:49.878653+00:00", "exception": false, "start_time": "2026-10-02T14:42:49.876731+00:00", "status": "completed" }, "tags": [] }, "source": [ ":::{dropdown} `SellarDis1withDerivatives` class definition \n", "\n", "{glue:}`code_src36`\n", ":::" ] }, { "cell_type": "code", "execution_count": 3, "id": "2625ccf6", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:42:49.883304Z", "iopub.status.busy": "2026-10-02T14:42:49.883059Z", "iopub.status.idle": "2026-10-02T14:42:49.892287Z", "shell.execute_reply": "2026-10-02T14:42:49.891357Z" }, "papermill": { "duration": 0.012222, "end_time": "2026-10-02T14:42:49.892843+00:00", "exception": false, "start_time": "2026-10-02T14:42:49.880621+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_src37" } }, "output_type": "display_data" } ], "source": [ "from openmdao.utils.notebook_utils import get_code\n", "from myst_nb import glue\n", "glue(\"code_src37\", get_code(\"openmdao.test_suite.components.sellar.SellarDis2withDerivatives\"), display=False)" ] }, { "cell_type": "markdown", "id": "ceb48696", "metadata": { "papermill": { "duration": 0.004051, "end_time": "2026-10-02T14:42:49.898613+00:00", "exception": false, "start_time": "2026-10-02T14:42:49.894562+00:00", "status": "completed" }, "tags": [] }, "source": [ ":::{dropdown} `SellarDis2withDerivatives` class definition \n", "\n", "{glue:}`code_src37`\n", ":::" ] }, { "cell_type": "code", "execution_count": 4, "id": "1f5a43a9", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:42:49.902940Z", "iopub.status.busy": "2026-10-02T14:42:49.902757Z", "iopub.status.idle": "2026-10-02T14:42:51.178862Z", "shell.execute_reply": "2026-10-02T14:42:51.177863Z" }, "papermill": { "duration": 1.279154, "end_time": "2026-10-02T14:42:51.179525+00:00", "exception": false, "start_time": "2026-10-02T14:42:49.900371+00:00", "status": "completed" }, "tags": [] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[1790952171.158910] [runnervm8df0l:7323 :0] ib_iface.c:1269 UCX ERROR mana_0: iface 0x558e7e920570 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", "[1790952171.159169] [runnervm8df0l:7323 :0] ucp_worker.c:1412 UCX ERROR uct_iface_open(ud_verbs/mana_0:1) failed: Input/output error\n", "NL: NLBJSolver 'NL: NLBJ' on system '' failed to converge in 10 iterations.\n", "[25.58830249]\n", "[12.05848818]\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ "[runnervm8df0l:07323] 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.NonlinearBlockJac()\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", "print(prob['y1'])\n", "print(prob['y2'])" ] }, { "cell_type": "code", "execution_count": 5, "id": "98070b3a", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:42:51.184523Z", "iopub.status.busy": "2026-10-02T14:42:51.184339Z", "iopub.status.idle": "2026-10-02T14:42:51.189749Z", "shell.execute_reply": "2026-10-02T14:42:51.189150Z" }, "papermill": { "duration": 0.008817, "end_time": "2026-10-02T14:42:51.190480+00:00", "exception": false, "start_time": "2026-10-02T14:42:51.181663+00:00", "status": "completed" }, "tags": [ "remove-input", "remove-output" ] }, "outputs": [ { "data": { "text/plain": [ "np.float64(5.538631102517653e-10)" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from openmdao.utils.assert_utils import assert_near_equal\n", "\n", "assert_near_equal(prob['y1'], 25.58830273, .00001)\n", "assert_near_equal(prob['y2'], 12.05848819, .00001)" ] }, { "cell_type": "markdown", "id": "796aa1a1", "metadata": { "papermill": { "duration": 0.001852, "end_time": "2026-10-02T14:42:51.194237+00:00", "exception": false, "start_time": "2026-10-02T14:42:51.192385+00:00", "status": "completed" }, "tags": [] }, "source": [ "This solver runs all of the subsystems each iteration, but just passes the data along all connections\n", "simultaneously once per iteration. After each iteration, the iteration count and the residual norm are\n", "checked to see if termination has been satisfied.\n", "\n", "You can control the termination criteria for the solver using the following options:\n", "\n", "## NonlinearBlockJac Options" ] }, { "cell_type": "code", "execution_count": 6, "id": "abc054d8", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:42:51.201535Z", "iopub.status.busy": "2026-10-02T14:42:51.201348Z", "iopub.status.idle": "2026-10-02T14:42:51.205027Z", "shell.execute_reply": "2026-10-02T14:42:51.204431Z" }, "papermill": { "duration": 0.009378, "end_time": "2026-10-02T14:42:51.205518+00:00", "exception": false, "start_time": "2026-10-02T14:42:51.196140+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
atol1e-10N/AN/Aabsolute error tolerance
debug_printFalse[True, False]['bool']If True, the values of input and output variables at the start of iteration are printed and written to a file after a failure to converge or when encountering aninvalid value in the residual.
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
restart_from_successfulFalse[True, False]['bool']If True, the states are cached after a successful solve and used to restart the solver in the case of a failed solve.
rtol1e-10N/AN/Arelative error tolerance
stall_limit0N/AN/ANumber of iterations after which, if the residual norms are identical within the stall_tol, then terminate as if max iterations were reached. Default is 0, which disables this feature.
stall_tol1e-12N/AN/AWhen stall checking is enabled, the threshold below which the residual norm is considered unchanged.
stall_tol_typerel['abs', 'rel']N/ASpecifies whether the absolute or relative norm of the residual is used for stall detection.
\n", "\n", "\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "om.show_options_table(\"openmdao.solvers.nonlinear.nonlinear_block_jac.NonlinearBlockJac\")" ] }, { "cell_type": "markdown", "id": "1d995bb8", "metadata": { "papermill": { "duration": 0.002006, "end_time": "2026-10-02T14:42:51.209509+00:00", "exception": false, "start_time": "2026-10-02T14:42:51.207503+00:00", "status": "completed" }, "tags": [] }, "source": [ "## NonlinearBlockJac Constructor\n", "\n", "The call signature for the `NonlinearBlockJac` constructor is:\n", "\n", "```{eval-rst}\n", " .. automethod:: openmdao.solvers.nonlinear.nonlinear_block_jac.NonlinearBlockJac.__init__\n", " :noindex:\n", "```\n", "\n", "## NonlinearBlockJac Option Examples\n", "\n", "**maxiter**\n", "\n", " This lets you specify the maximum number of allowed Jacobi iterations during a solve. In this example, we\n", " increase it from the default, 10, up to 20, so that it successfully converges." ] }, { "cell_type": "code", "execution_count": 7, "id": "ccd24a06", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:42:51.214190Z", "iopub.status.busy": "2026-10-02T14:42:51.214026Z", "iopub.status.idle": "2026-10-02T14:42:51.226289Z", "shell.execute_reply": "2026-10-02T14:42:51.225848Z" }, "papermill": { "duration": 0.015283, "end_time": "2026-10-02T14:42:51.226720+00:00", "exception": false, "start_time": "2026-10-02T14:42:51.211437+00:00", "status": "completed" }, "tags": [] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "NL: NLBJ Converged in 13 iterations\n", "[25.58830237]\n", "[12.05848815]\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.linear_solver = om.LinearBlockGS()\n", "\n", "nlbgs = model.nonlinear_solver = om.NonlinearBlockJac()\n", "nlbgs.options['maxiter'] = 20\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", "print(prob['y1'])\n", "print(prob['y2'])" ] }, { "cell_type": "code", "execution_count": 8, "id": "891f4d0c", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:42:51.231512Z", "iopub.status.busy": "2026-10-02T14:42:51.231269Z", "iopub.status.idle": "2026-10-02T14:42:51.234289Z", "shell.execute_reply": "2026-10-02T14:42:51.233864Z" }, "papermill": { "duration": 0.006033, "end_time": "2026-10-02T14:42:51.234670+00:00", "exception": false, "start_time": "2026-10-02T14:42:51.228637+00:00", "status": "completed" }, "tags": [ "remove-input", "remove-output" ] }, "outputs": [ { "data": { "text/plain": [ "np.float64(3.078240795202139e-11)" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "assert_near_equal(prob['y1'], 25.58830237, .00001)\n", "assert_near_equal(prob['y2'], 12.05848815, .00001)" ] }, { "cell_type": "markdown", "id": "f4fab2b0", "metadata": { "papermill": { "duration": 0.001924, "end_time": "2026-10-02T14:42:51.238641+00:00", "exception": false, "start_time": "2026-10-02T14:42:51.236717+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 residuals is calculated by calling `apply_nonlinear` on all of the components.\n", " If this norm value is lower than the absolute\n", " tolerance `atol`, the iteration will terminate." ] }, { "cell_type": "code", "execution_count": 9, "id": "94fe527b", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:42:51.243719Z", "iopub.status.busy": "2026-10-02T14:42:51.243444Z", "iopub.status.idle": "2026-10-02T14:42:51.254159Z", "shell.execute_reply": "2026-10-02T14:42:51.253727Z" }, "papermill": { "duration": 0.013907, "end_time": "2026-10-02T14:42:51.254628+00:00", "exception": false, "start_time": "2026-10-02T14:42:51.240721+00:00", "status": "completed" }, "tags": [] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "NL: NLBJ Converged in 6 iterations\n", "[25.58861716]\n", "[12.05857185]\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.linear_solver = om.LinearBlockGS()\n", "\n", "nlbgs = model.nonlinear_solver = om.NonlinearBlockJac()\n", "nlbgs.options['atol'] = 1e-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", "print(prob['y1'])\n", "print(prob['y2'])" ] }, { "cell_type": "code", "execution_count": 10, "id": "b2357978", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:42:51.259513Z", "iopub.status.busy": "2026-10-02T14:42:51.259352Z", "iopub.status.idle": "2026-10-02T14:42:51.262432Z", "shell.execute_reply": "2026-10-02T14:42:51.262034Z" }, "papermill": { "duration": 0.006179, "end_time": "2026-10-02T14:42:51.262894+00:00", "exception": false, "start_time": "2026-10-02T14:42:51.256715+00:00", "status": "completed" }, "tags": [ "remove-input", "remove-output" ] }, "outputs": [ { "data": { "text/plain": [ "np.float64(6.937777327317264e-06)" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "assert_near_equal(prob['y1'], 25.5886171567, .00001)\n", "assert_near_equal(prob['y2'], 12.05848819, .00001)" ] }, { "cell_type": "markdown", "id": "5b2d319f", "metadata": { "papermill": { "duration": 0.002072, "end_time": "2026-10-02T14:42:51.266997+00:00", "exception": false, "start_time": "2026-10-02T14:42:51.264925+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 residuals is calculated by calling `apply_nonlinear` on all of the components.\n", " 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": 11, "id": "016c5423", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:42:51.271981Z", "iopub.status.busy": "2026-10-02T14:42:51.271816Z", "iopub.status.idle": "2026-10-02T14:42:51.281955Z", "shell.execute_reply": "2026-10-02T14:42:51.281434Z" }, "papermill": { "duration": 0.013273, "end_time": "2026-10-02T14:42:51.282358+00:00", "exception": false, "start_time": "2026-10-02T14:42:51.269085+00:00", "status": "completed" }, "tags": [] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "NL: NLBJ Converged in 5 iterations\n", "[25.58914915]\n", "[12.05691422]\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.linear_solver = om.LinearBlockGS()\n", "\n", "nlbgs = model.nonlinear_solver = om.NonlinearBlockJac()\n", "nlbgs.options['rtol'] = 1e-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", "print(prob.get_val('y1'))\n", "print(prob.get_val('y2'))" ] }, { "cell_type": "code", "execution_count": 12, "id": "d1c0a459", "metadata": { "execution": { "iopub.execute_input": "2026-10-02T14:42:51.301457Z", "iopub.status.busy": "2026-10-02T14:42:51.301150Z", "iopub.status.idle": "2026-10-02T14:42:51.304506Z", "shell.execute_reply": "2026-10-02T14:42:51.304023Z" }, "papermill": { "duration": 0.020486, "end_time": "2026-10-02T14:42:51.304961+00:00", "exception": false, "start_time": "2026-10-02T14:42:51.284475+00:00", "status": "completed" }, "tags": [ "remove-input", "remove-output" ] }, "outputs": [ { "data": { "text/plain": [ "np.float64(2.8243346514333536e-12)" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "assert_near_equal(prob.get_val('y1'), 25.5891491526, .00001)\n", "assert_near_equal(prob.get_val('y2'), 12.0569142166, .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": 5.434899, "end_time": "2026-10-02T14:42:52.222721+00:00", "environment_variables": {}, "exception": null, "input_path": "/home/runner/work/OpenMDAO/OpenMDAO/openmdao/docs/openmdao_book/features/building_blocks/solvers/nonlinear_block_jac.ipynb", "output_path": "/home/runner/work/OpenMDAO/OpenMDAO/openmdao/docs/_executed_book/features/building_blocks/solvers/nonlinear_block_jac.ipynb", "parameters": {}, "start_time": "2026-10-02T14:42:46.787822+00:00", "version": "2.7.0" } }, "nbformat": 4, "nbformat_minor": 5 }