Example: Solving Nonlinear Systems of Equations

Solve Blocks > Example: Solving Nonlinear Systems of Equations

Example: Solving Nonlinear Systems of Equations

Use solve blocks to solve a system of n equations in n unknowns.

1. Insert a solve block and define the guess values, the constraints and the solver function Find.

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<region id="ID0E3EELB" top="192" left="24" height="238.475485342492" width="300" xmlns="http://schemas.mathsoft.com/worksheet50">
  <solveblock>
    <regions>
      <region id="" top="24" left="24" solve-block-category="guess-value">
        <math disable-calc="false">
          <define xmlns="http://schemas.mathsoft.com/math50">
            <id label="Variable">x</id>
            <real>1</real>
          </define>
        </math>
      </region>
      <region id="" top="48" left="24" solve-block-category="guess-value">
        <math disable-calc="false">
          <define xmlns="http://schemas.mathsoft.com/math50">
            <id label="Variable">y</id>
            <real>1</real>
          </define>
        </math>
      </region>
      <region id="" top="72" left="24" solve-block-category="guess-value">
        <math disable-calc="false">
          <define xmlns="http://schemas.mathsoft.com/math50">
            <id label="Variable">z</id>
            <real>0</real>
          </define>
        </math>
      </region>
      <region id="" top="120" left="24" solve-block-category="constraint">
        <math disable-calc="false">
          <apply xmlns="http://schemas.mathsoft.com/math50">
            <equal />
            <apply>
              <plus />
              <apply>
                <mult />
                <real>2</real>
                <id>x</id>
              </apply>
              <id>y</id>
            </apply>
            <apply>
              <minus />
              <real>5</real>
              <apply>
                <mult />
                <real>2</real>
                <apply>
                  <pow />
                  <id>z</id>
                  <real>2</real>
                </apply>
              </apply>
            </apply>
          </apply>
        </math>
      </region>
      <region id="" top="144" left="24" solve-block-category="constraint">
        <math disable-calc="false">
          <apply xmlns="http://schemas.mathsoft.com/math50">
            <equal />
            <apply>
              <plus />
              <apply>
                <pow />
                <id>y</id>
                <real>3</real>
              </apply>
              <apply>
                <mult />
                <real>4</real>
                <id>z</id>
              </apply>
            </apply>
            <real>4</real>
          </apply>
        </math>
      </region>
      <region id="" top="168" left="24" solve-block-category="constraint">
        <math disable-calc="false">
          <apply xmlns="http://schemas.mathsoft.com/math50">
            <equal />
            <apply>
              <plus />
              <apply>
                <mult />
                <id>x</id>
                <id>y</id>
              </apply>
              <id>z</id>
            </apply>
            <apply>
              <pow />
              <id>e</id>
              <id>z</id>
            </apply>
          </apply>
        </math>
      </region>
      <region id="" top="216" left="24" solve-block-category="solver">
        <math disable-calc="false">
          <define xmlns="http://schemas.mathsoft.com/math50">
            <id label="Variable">vec</id>
            <apply>
              <id label="Keyword">Find</id>
              <sequence>
                <id>x</id>
                <id>y</id>
                <id>z</id>
              </sequence>
            </apply>
          </define>
        </math>
      </region>
    </regions>
  </solveblock>
</region>

2. Evaluate vec:

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Solving Equations in Matrix Form

You can solve matrix equations, using Eigenanalysis or a solve block.

1. Solve the following non-linear equation:

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2. Use the eigenvecs function to calculate the eigenvectors of matrix M:

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3. Use the eigenvals to calculate the eigenvalues of matrix M, and the diag function to place these values along the diagonal of matrix Vals:

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4. Use the Vectorization and Transpose operators to find X:

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5. Use the solver function Find in a Solve Block to find X:

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<region id="ID0ECTELB" top="1320" left="24" height="210.573851031992" width="300" xmlns="http://schemas.mathsoft.com/worksheet50">
  <solveblock>
    <regions>
      <region id="" top="24" left="24" solve-block-category="guess-value">
        <math disable-calc="false">
          <define xmlns="http://schemas.mathsoft.com/math50">
            <id label="Variable">Xsb</id>
            <id>M</id>
          </define>
        </math>
      </region>
      <region id="" top="120" left="24" solve-block-category="constraint">
        <math disable-calc="false">
          <apply xmlns="http://schemas.mathsoft.com/math50">
            <equal />
            <apply>
              <pow />
              <id>Xsb</id>
              <real>2</real>
            </apply>
            <id>M</id>
          </apply>
        </math>
      </region>
      <region id="" top="192" left="24" solve-block-category="solver">
        <math disable-calc="false">
          <define xmlns="http://schemas.mathsoft.com/math50">
            <id label="Variable">Xsb</id>
            <apply>
              <id label="Keyword">Find</id>
              <id>Xsb</id>
            </apply>
          </define>
        </math>
      </region>
    </regions>
  </solveblock>
</region>

6. Evaluate Xsb2 or M:

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Solving for Quantities with Units

You can use solve blocks to solve for quantities with units. This example determines the velocity of flow through a straight length of pipe.

1. Define the density and permeability of the flowing substance:

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2. Define the diameter and length of the pipe:

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3. Define the pressure and gravitational force within the pipe:

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<region id="ID0EC3ELB" top="2616" left="24" height="411" width="300" xmlns="http://schemas.mathsoft.com/worksheet50">
  <solveblock>
    <regions>
      <region id="" top="24" left="24" solve-block-category="guess-value">
        <math disable-calc="false">
          <define xmlns="http://schemas.mathsoft.com/math50">
            <id label="Variable">f</id>
            <apply>
              <mult />
              <real>1</real>
              <apply>
                <pow />
                <real>10</real>
                <real>-3</real>
              </apply>
            </apply>
          </define>
        </math>
      </region>
      <region id="" top="72" left="24" solve-block-category="guess-value">
        <math disable-calc="false">
          <define xmlns="http://schemas.mathsoft.com/math50">
            <id label="Variable">v</id>
            <apply>
              <mult />
              <real>10</real>
              <apply>
                <div />
                <id>ft</id>
                <id>s</id>
              </apply>
            </apply>
          </define>
        </math>
      </region>
      <region id="" top="120" left="24" solve-block-category="guess-value">
        <math disable-calc="false">
          <define xmlns="http://schemas.mathsoft.com/math50">
            <id label="Variable">NRe</id>
            <apply>
              <mult />
              <real>1</real>
              <apply>
                <pow />
                <real>10</real>
                <real>4</real>
              </apply>
            </apply>
          </define>
        </math>
      </region>
      <region id="" top="168" left="24" solve-block-category="constraint">
        <math disable-calc="false">
          <apply xmlns="http://schemas.mathsoft.com/math50">
            <equal />
            <id>NRe</id>
            <apply>
              <div />
              <apply>
                <mult />
                <apply>
                  <mult />
                  <id>ρ</id>
                  <id>v</id>
                </apply>
                <id>D</id>
              </apply>
              <id>μ</id>
            </apply>
          </apply>
        </math>
      </region>
      <region id="" top="216" left="24" solve-block-category="constraint">
        <math disable-calc="false">
          <apply xmlns="http://schemas.mathsoft.com/math50">
            <equal />
            <apply>
              <div />
              <id label="Variable">ΔP</id>
              <id>ρ</id>
            </apply>
            <apply>
              <mult />
              <apply>
                <mult />
                <real>4</real>
                <id>f</id>
              </apply>
              <apply>
                <div />
                <apply>
                  <mult />
                  <apply>
                    <pow />
                    <id>v</id>
                    <real>2</real>
                  </apply>
                  <id label="Variable">ΔL</id>
                </apply>
                <apply>
                  <mult />
                  <apply>
                    <mult />
                    <real>2</real>
                    <id>D</id>
                  </apply>
                  <id>gc</id>
                </apply>
              </apply>
            </apply>
          </apply>
        </math>
      </region>
      <region id="" top="264" left="24" solve-block-category="constraint">
        <math disable-calc="false">
          <apply xmlns="http://schemas.mathsoft.com/math50">
            <equal />
            <apply>
              <div />
              <real>1</real>
              <apply>
                <nthRoot />
                <placeholder />
                <apply>
                  <div />
                  <id>f</id>
                  <real>2</real>
                </apply>
              </apply>
            </apply>
            <apply>
              <plus />
              <apply>
                <mult />
                <real>2.5</real>
                <apply>
                  <id>ln</id>
                  <apply>
                    <mult />
                    <id>NRe</id>
                    <apply>
                      <nthRoot />
                      <placeholder />
                      <apply>
                        <div />
                        <id>f</id>
                        <real>8</real>
                      </apply>
                    </apply>
                  </apply>
                </apply>
              </apply>
              <real>1.75</real>
            </apply>
          </apply>
        </math>
      </region>
      <region id="" top="360" left="24" solve-block-category="solver">
        <math disable-calc="false">
          <define xmlns="http://schemas.mathsoft.com/math50">
            <matrix rows="3" cols="1">
              <id label="Variable">vsol</id>
              <id label="Variable">fsol</id>
              <id label="Variable">NRe_sol</id>
            </matrix>
            <apply>
              <id label="Keyword">Find</id>
              <sequence>
                <id>v</id>
                <id>f</id>
                <id>NRe</id>
              </sequence>
            </apply>
          </define>
        </math>
      </region>
    </regions>
  </solveblock>
</region>

4. Find the solution for v, f, and NRe

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