Proposal for gradient wrt complex variables¶
This is a proposal to handle gradients of a scalar, real variable (usually, a cost) with respect to tensor variables, of complex (and real) type, in an optimization perspective.
Derivative of complex variables is usually studied only for so-called analytical complex functions, which have a particular structure in their partial derivatives. However, we do not want to limit ourselves to analytical functions, and we make other assumptions (that the final cost is real-valued, for instance), so we will adopt a different convention for gradients than what is usually used in the literature.
Gradient (re-)definition¶
We are interested in the case where we have a final real-valued cost,
System Message: WARNING/2 (C)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (C)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
For each real-valued variable
System Message: WARNING/2 (r)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (\Re v)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (\Im v)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (\frac{\partial C}{\partial r})
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (r)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
For any real-valued intermediate variable
System Message: WARNING/2 (t)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (\frac{\partial C}{\partial r} = \frac{\partial C}{\partial t} \frac{\partial t}{\partial r})
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
If
System Message: WARNING/2 (z)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (\Re z = x)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (\Im z = y)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (x)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (y)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (\frac{\partial C}{\partial r} = \frac{\partial C}{\partial x} \frac{\partial x}{\partial r} + \frac{\partial C}{\partial y} \frac{\partial y}{\partial r})
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
If we want to use an algorithm similar to gradient backpropagation, we can see that, here, we need to have both
System Message: WARNING/2 (\frac{\partial C}{\partial \Re t})
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (\frac{\partial C}{\partial \Im t})
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (\frac{\partial C}{\partial r})
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
For each variable
System Message: WARNING/2 (v)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (\nabla_C(v))
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (C)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (v)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (v)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (\nabla_C(v) = \frac{\partial C}{\partial \Re v} + i \frac{\partial C}{\partial \Im v})
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
This is the tensor that we are going to back-propagate through the computation graph.
Generalized chain rule¶
Using the definition above, if we have two complex variables
System Message: WARNING/2 (z = x + iy)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (t = r + is)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (x, y, r, s)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (\nabla_C(z) &= \frac{\partial C}{\partial \Re z} + i \frac{\partial C}{\partial \Im z} \\ &= \frac{\partial C}{\partial x} + i \frac{\partial C}{\partial y} \nabla_C(t) &= \frac{\partial C}{\partial \Re t} + i \frac{\partial C}{\partial \Im t} \\ &= \frac{\partial C}{\partial r} + i \frac{\partial C}{\partial s} \\ &= \left(\frac{\partial C}{\partial x} \frac{\partial x}{\partial r} + \frac{\partial C}{\partial y} \frac{\partial y}{\partial r}\right) + i \left(\frac{\partial C}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial C}{\partial y} \frac{\partial y}{\partial s}\right) \\ &= \frac{\partial C}{\partial x} \left(\frac{\partial x}{\partial r} + i \frac{\partial x}{\partial s}\right) + \frac{\partial C}{\partial y} \left(\frac{\partial y}{\partial r} + i \frac{\partial y}{\partial s}\right) \\ &= \Re \left(\nabla_C(z)\right) \left(\frac{\partial x}{\partial r} + i \frac{\partial x}{\partial s}\right) + \Im \left(\nabla_C(z)\right) \left(\frac{\partial y}{\partial r} + i \frac{\partial y}{\partial s}\right))
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
This formula can be used whether or not
System Message: WARNING/2 (C)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (z)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (t)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (z)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (t)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
Special cases¶
Real-valued input variable¶
If variable
System Message: WARNING/2 (x)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (\nabla_C(z))
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (\frac{\partial C}{\partial x})
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (y)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
Real-valued intermediate variable¶
When
System Message: WARNING/2 (x)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (C)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (t)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (y)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (\nabla_C(t) &= \frac{\partial C}{\partial r} + i \frac{\partial C}{\partial s} \\ &= \frac{\partial C}{\partial x} \frac{\partial x}{\partial r} + i \frac{\partial C}{\partial x} \frac{\partial x}{\partial s} \\ &= \Re \left(\nabla_C(z)\right) \left(\frac{\partial x}{\partial r} + i \frac{\partial x}{\partial s}\right))
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
The imaginary part of
System Message: WARNING/2 (\nabla_C(z))
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (\Im z)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
Analytic functions¶
If
System Message: WARNING/2 (z)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (t)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
Analytic (or holomorphic) functions satisfy the Cauchy-Riemann equations:
System Message: WARNING/2 (\frac{\partial \Re z}{\partial \Re t} = \frac{\partial \Im z}{\partial \Im t} \text{ and } \frac{\partial \Re z}{\partial \Im t} = - \frac{\partial \Im z}{\partial \Re t})
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
Or, in our case:
System Message: WARNING/2 (\frac{\partial x}{\partial r} = \frac{\partial y}{\partial t} \text{ and } \frac{\partial x}{\partial s} = - \frac{\partial y}{\partial r})
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
This leads to:
System Message: WARNING/2 (\nabla_C(t) &= \Re \left(\nabla_C(z)\right) \left(\frac{\partial x}{\partial r} + i \frac{\partial x}{\partial s}\right) + \Im \left(\nabla_C(z)\right) \left(\frac{\partial y}{\partial r} + i \frac{\partial y}{\partial s}\right) \\ &= \Re \left(\nabla_C(z)\right) \left(\frac{\partial x}{\partial r} + i \frac{\partial x}{\partial s}\right) + \Im \left(\nabla_C(z)\right) \left(- \frac{\partial x}{\partial s} + i \frac{\partial x}{\partial r}\right) \\ &= \Re \left(\nabla_C(z)\right) \left(\frac{\partial x}{\partial r} + i \frac{\partial x}{\partial s}\right) + i \Im \left(\nabla_C(z)\right) \left(\frac{\partial x}{\partial r} + i \frac{\partial x}{\partial s}\right) \\ \nabla_C(t) &= \nabla_C(z) \left(\frac{\partial x}{\partial r} + i \frac{\partial x}{\partial s}\right) = - i \nabla_C(z) \left(\frac{\partial y}{\partial r} + i \frac{\partial y}{\partial s}\right))
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
Finite differences¶
In order to verify that the mathematical formula for a gradient, or its implementation, is correct, we usually use a finite-differenciation approach. If
System Message: WARNING/2 (C)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (x)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (\frac{\partial C}{\partial x} \approx \frac{C(x + \varepsilon) - C(x)}{\varepsilon})
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
where
System Message: WARNING/2 (\varepsilon)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (10^{-6})
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (10^{-4})
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (x)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (x_i)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (\varepsilon_i)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (\varepsilon)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
For a complex scalar variable
System Message: WARNING/2 (z = x + iy)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
System Message: WARNING/2 (\nabla_C(z) &= \frac{\partial C}{\partial x} + i \frac{\partial C}{\partial y}\\ \nabla_C(z) &\approx \frac{C(z + \delta) - C(z)}{\delta} + i \frac{C(z + i \varepsilon) - C(z)}{\varepsilon})
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!
Both partial derivative have to be estimated independently, using generally
System Message: WARNING/2 (\delta = \varepsilon)
latex exited with error [stderr] kpathsea: Running mktexfmt latex.fmt mktexfmt: mktexfmt is using the following fmtutil.cnf files (in precedence order): mktexfmt: mktexfmt is using the following fmtutil.cnf file for writing changes: mktexfmt: /builddir/.texlive2020/texmf-config/web2c/fmtutil.cnf mktexfmt [INFO]: writing formats under /builddir/.texlive2020/texmf-var/web2c mktexfmt [INFO]: did not find entry for byfmt=latex, skipped mktexfmt [INFO]: total formats: 0 mktexfmt [INFO]: exiting with status 0 [stdout] This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=latex) restricted \write18 enabled. I can't find the format file `latex.fmt'!