From 8e4948fb2a0e4167c7ef00f7947accbc9a1749d0 Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Tue, 27 Aug 2024 04:56:38 +0200 Subject: [PATCH] update --- doc/Textbooks/.DS_Store | Bin 10244 -> 10244 bytes doc/pub/week35/html/._week35-bs010.html | 6 +- doc/pub/week35/html/._week35-bs014.html | 7 +- doc/pub/week35/html/week35-reveal.html | 14 +- doc/pub/week35/html/week35-solarized.html | 13 +- doc/pub/week35/html/week35.html | 13 +- doc/pub/week35/ipynb/ipynb-week35-src.tar.gz | Bin 192 -> 192 bytes doc/pub/week35/ipynb/week35.ipynb | 832 ++++++++++--------- doc/src/week35/week35.do.txt | 14 +- 9 files changed, 469 insertions(+), 430 deletions(-) diff --git a/doc/Textbooks/.DS_Store b/doc/Textbooks/.DS_Store index 3a6051c5d00afec0aa00abd1384750ba46bb3d45..75c5352a283f74b9d27fdbda6c3f7a08e973b10c 100644 GIT binary patch delta 54 zcmZn(XbG6$&nU7nU^hRb$Yve^ZnnvbCFLe(3oO{YSTcoeW5YS7&Fl&S$fEj;0PZ*u A`~Uy| delta 241 zcmZn(XbG6$&nU4mU^hRb#AY4=ZZ<}(Ns{u6zLV=D#3$EF96;qNOzxHBU=#tW;PqwD zV<=!qVMt?OV2}g~aQiZpG86%YG8if+CkU)&1PRI`3BuGDlqC6;7MB!3^$BB9GWmy? W+~f>_#hb5+rLb*gR}i3which means that (using our previous example) we have

+

which means that (using our previous example and keeping track of our definition of the derivative of a scalar) we have

$$ -\frac{\partial \alpha}{\partial \boldsymbol{x}} = \boldsymbol{z}=\boldsymbol{A}^T\boldsymbol{y}. +\frac{\partial \alpha}{\partial \boldsymbol{x}} = \frac{\partial \boldsymbol{z}^T\boldsymbol{x}}{\partial \boldsymbol{x}}=\boldsymbol{z}^T=\boldsymbol{A}^T\boldsymbol{y}. $$

Note that the resulting vector elements are the same for \( \boldsymbol{z}^T \) and \( \boldsymbol{z} \), the only difference is that one is just the transpose of the other.

-

Since \( \alpha \) is a scalar we have \( \alpha =\alpha^T=\boldsymbol{x}^T\boldsymbol{A}^T\boldsymbol{y} \). Defining now \( \boldsymbol{z}=\boldsymbol{x}^T\boldsymbol{A}^T \) we find that

+

Since \( \alpha \) is a scalar we have \( \alpha =\alpha^T=\boldsymbol{x}^T\boldsymbol{A}^T\boldsymbol{y} \). Defining now \( \boldsymbol{z}^T=\boldsymbol{x}^T\boldsymbol{A}^T \) we find that

$$ \frac{\partial \alpha}{\partial \boldsymbol{y}} = \boldsymbol{z}^T=\boldsymbol{x}^T\boldsymbol{A}^T. $$ diff --git a/doc/pub/week35/html/._week35-bs014.html b/doc/pub/week35/html/._week35-bs014.html index 3e251c1f6..9ebd409ce 100644 --- a/doc/pub/week35/html/._week35-bs014.html +++ b/doc/pub/week35/html/._week35-bs014.html @@ -364,9 +364,14 @@ MathJax.Hub.Config({

We list here some other useful relations we may encounter (recall that vectors are defined by boldfaced low-key letters)

$$ -\frac{\partial (\boldsymbol{b}^T\boldsymbol{a})}{\partial \boldsymbol{a}} = \boldsymbol{b}, +\frac{\partial (\boldsymbol{x}^T\boldsymbol{a})}{\partial \boldsymbol{x}} = \boldsymbol{a}^T, $$ +$$ +\frac{\partial (\boldsymbol{a}^T\boldsymbol{x})}{\partial \boldsymbol{x}} = \boldsymbol{a}^T, +$$ + + $$ \frac{\partial tr(\boldsymbol{B}\boldsymbol{A})}{\partial \boldsymbol{A}} = \boldsymbol{B}^T, $$ diff --git a/doc/pub/week35/html/week35-reveal.html b/doc/pub/week35/html/week35-reveal.html index fb3474d30..78deaf60f 100644 --- a/doc/pub/week35/html/week35-reveal.html +++ b/doc/pub/week35/html/week35-reveal.html @@ -537,16 +537,16 @@ $$ $$

 
-

which means that (using our previous example) we have

+

which means that (using our previous example and keeping track of our definition of the derivative of a scalar) we have

 
$$ -\frac{\partial \alpha}{\partial \boldsymbol{x}} = \boldsymbol{z}=\boldsymbol{A}^T\boldsymbol{y}. +\frac{\partial \alpha}{\partial \boldsymbol{x}} = \frac{\partial \boldsymbol{z}^T\boldsymbol{x}}{\partial \boldsymbol{x}}=\boldsymbol{z}^T=\boldsymbol{A}^T\boldsymbol{y}. $$

 

Note that the resulting vector elements are the same for \( \boldsymbol{z}^T \) and \( \boldsymbol{z} \), the only difference is that one is just the transpose of the other.

-

Since \( \alpha \) is a scalar we have \( \alpha =\alpha^T=\boldsymbol{x}^T\boldsymbol{A}^T\boldsymbol{y} \). Defining now \( \boldsymbol{z}=\boldsymbol{x}^T\boldsymbol{A}^T \) we find that

+

Since \( \alpha \) is a scalar we have \( \alpha =\alpha^T=\boldsymbol{x}^T\boldsymbol{A}^T\boldsymbol{y} \). Defining now \( \boldsymbol{z}^T=\boldsymbol{x}^T\boldsymbol{A}^T \) we find that

 
$$ \frac{\partial \alpha}{\partial \boldsymbol{y}} = \boldsymbol{z}^T=\boldsymbol{x}^T\boldsymbol{A}^T. @@ -715,7 +715,13 @@ $$

We list here some other useful relations we may encounter (recall that vectors are defined by boldfaced low-key letters)

 
$$ -\frac{\partial (\boldsymbol{b}^T\boldsymbol{a})}{\partial \boldsymbol{a}} = \boldsymbol{b}, +\frac{\partial (\boldsymbol{x}^T\boldsymbol{a})}{\partial \boldsymbol{x}} = \boldsymbol{a}^T, +$$ +

 
+ +

 
+$$ +\frac{\partial (\boldsymbol{a}^T\boldsymbol{x})}{\partial \boldsymbol{x}} = \boldsymbol{a}^T, $$

 
diff --git a/doc/pub/week35/html/week35-solarized.html b/doc/pub/week35/html/week35-solarized.html index eabfc92d3..9310fb771 100644 --- a/doc/pub/week35/html/week35-solarized.html +++ b/doc/pub/week35/html/week35-solarized.html @@ -602,14 +602,14 @@ $$ \alpha = \boldsymbol{z}^T\boldsymbol{x}, $$ -

which means that (using our previous example) we have

+

which means that (using our previous example and keeping track of our definition of the derivative of a scalar) we have

$$ -\frac{\partial \alpha}{\partial \boldsymbol{x}} = \boldsymbol{z}=\boldsymbol{A}^T\boldsymbol{y}. +\frac{\partial \alpha}{\partial \boldsymbol{x}} = \frac{\partial \boldsymbol{z}^T\boldsymbol{x}}{\partial \boldsymbol{x}}=\boldsymbol{z}^T=\boldsymbol{A}^T\boldsymbol{y}. $$

Note that the resulting vector elements are the same for \( \boldsymbol{z}^T \) and \( \boldsymbol{z} \), the only difference is that one is just the transpose of the other.

-

Since \( \alpha \) is a scalar we have \( \alpha =\alpha^T=\boldsymbol{x}^T\boldsymbol{A}^T\boldsymbol{y} \). Defining now \( \boldsymbol{z}=\boldsymbol{x}^T\boldsymbol{A}^T \) we find that

+

Since \( \alpha \) is a scalar we have \( \alpha =\alpha^T=\boldsymbol{x}^T\boldsymbol{A}^T\boldsymbol{y} \). Defining now \( \boldsymbol{z}^T=\boldsymbol{x}^T\boldsymbol{A}^T \) we find that

$$ \frac{\partial \alpha}{\partial \boldsymbol{y}} = \boldsymbol{z}^T=\boldsymbol{x}^T\boldsymbol{A}^T. $$ @@ -739,9 +739,14 @@ $$

We list here some other useful relations we may encounter (recall that vectors are defined by boldfaced low-key letters)

$$ -\frac{\partial (\boldsymbol{b}^T\boldsymbol{a})}{\partial \boldsymbol{a}} = \boldsymbol{b}, +\frac{\partial (\boldsymbol{x}^T\boldsymbol{a})}{\partial \boldsymbol{x}} = \boldsymbol{a}^T, $$ +$$ +\frac{\partial (\boldsymbol{a}^T\boldsymbol{x})}{\partial \boldsymbol{x}} = \boldsymbol{a}^T, +$$ + + $$ \frac{\partial tr(\boldsymbol{B}\boldsymbol{A})}{\partial \boldsymbol{A}} = \boldsymbol{B}^T, $$ diff --git a/doc/pub/week35/html/week35.html b/doc/pub/week35/html/week35.html index ce93085af..64d36e6e1 100644 --- a/doc/pub/week35/html/week35.html +++ b/doc/pub/week35/html/week35.html @@ -679,14 +679,14 @@ $$ \alpha = \boldsymbol{z}^T\boldsymbol{x}, $$ -

which means that (using our previous example) we have

+

which means that (using our previous example and keeping track of our definition of the derivative of a scalar) we have

$$ -\frac{\partial \alpha}{\partial \boldsymbol{x}} = \boldsymbol{z}=\boldsymbol{A}^T\boldsymbol{y}. +\frac{\partial \alpha}{\partial \boldsymbol{x}} = \frac{\partial \boldsymbol{z}^T\boldsymbol{x}}{\partial \boldsymbol{x}}=\boldsymbol{z}^T=\boldsymbol{A}^T\boldsymbol{y}. $$

Note that the resulting vector elements are the same for \( \boldsymbol{z}^T \) and \( \boldsymbol{z} \), the only difference is that one is just the transpose of the other.

-

Since \( \alpha \) is a scalar we have \( \alpha =\alpha^T=\boldsymbol{x}^T\boldsymbol{A}^T\boldsymbol{y} \). Defining now \( \boldsymbol{z}=\boldsymbol{x}^T\boldsymbol{A}^T \) we find that

+

Since \( \alpha \) is a scalar we have \( \alpha =\alpha^T=\boldsymbol{x}^T\boldsymbol{A}^T\boldsymbol{y} \). Defining now \( \boldsymbol{z}^T=\boldsymbol{x}^T\boldsymbol{A}^T \) we find that

$$ \frac{\partial \alpha}{\partial \boldsymbol{y}} = \boldsymbol{z}^T=\boldsymbol{x}^T\boldsymbol{A}^T. $$ @@ -816,9 +816,14 @@ $$

We list here some other useful relations we may encounter (recall that vectors are defined by boldfaced low-key letters)

$$ -\frac{\partial (\boldsymbol{b}^T\boldsymbol{a})}{\partial \boldsymbol{a}} = \boldsymbol{b}, +\frac{\partial (\boldsymbol{x}^T\boldsymbol{a})}{\partial \boldsymbol{x}} = \boldsymbol{a}^T, $$ +$$ +\frac{\partial (\boldsymbol{a}^T\boldsymbol{x})}{\partial \boldsymbol{x}} = \boldsymbol{a}^T, +$$ + + $$ \frac{\partial tr(\boldsymbol{B}\boldsymbol{A})}{\partial \boldsymbol{A}} = \boldsymbol{B}^T, $$ diff --git a/doc/pub/week35/ipynb/ipynb-week35-src.tar.gz b/doc/pub/week35/ipynb/ipynb-week35-src.tar.gz index 2a9642d24f53d26324aafb355f3d552d03943356..49dc5e02d46b13322cd939ab8275d24fddca5228 100644 GIT binary patch literal 192 zcmV;x06+g9iwFQ#K+R?V1MSaC3c@fD2H>uHia9|^(j;9Ax^N+gc!89rHr6IJNzvZk zK0sHBn<7HK&Cf8yFmu?f*1JvO?><@#LWoleV`iL9iO5_}FlK-$=P98G6AlQcjARj@ zy#y`?R5@? uk#1~(mDf&L2}1WEib8p1w7A5rH76??m&Ct*CJ2Hc_}T+p*}Fmj2mk=U<5bT8 literal 192 zcmV;x06+g9iwFSHVa#R#1MSbv3c@f92k@Qu6nTP?u3L8&^x#1d@dY~8xjNU*wnO*! z?gR9sco`z}cli?%LUP!w*1JvQ?k-piBC;fbF*BA