Mixing
What is the vectorized form of this expression?
$begingroup$
I want to figure out a general formula for vectorizing computations.
For example, if I am given matrices, A,B such that A is $m times q$, B is $n times q$. And I want to compute the matrix $X$, where each component of $X$ is
$$
X_{ij} = sum_{r = 1}^q A_{ir} B_{jr}
$$
Note that:
$i = 1 ldots m$
$j = 1 ldots n$
This one is simple. By vectorizing, I mean we get rid of the summation and the individual component computations. So for this example, the vectorized form is $X = A*B^T$.
But what if instead we add another multplication term, $c$ that is $qtimes1$
$$
X_{ij} = sum_{r = 1}^q A_{ir} * B_{jr} * c_r
$$
How would we vectorize this? If $q$ is small, say q = 2, we could simply expand out the terms like $X = A*B^T*c[1] + A*B^T*c[2]$.
But this is obviously not feasible if q is large. How would you vectorize something like this?
Is there a general procedure to follow to a problem like this, or is the vectorization very problem-specific that no general procedure can be followed?
matrices vectors vectorization
asked Jan 29 at 19:12
IamanonIamanon
1328
$endgroup$
add a comment |
$begingroup$
I want to figure out a general formula for vectorizing computations.
For example, if I am given matrices, A,B such that A is $m times q$, B is $n times q$. And I want to compute the matrix $X$, where each component of $X$ is
$$
X_{ij} = sum_{r = 1}^q A_{ir} B_{jr}
$$
Note that:
$i = 1 ldots m$
$j = 1 ldots n$
This one is simple. By vectorizing, I mean we get rid of the summation and the individual component computations. So for this example, the vectorized form is $X = A*B^T$.
But what if instead we add another multplication term, $c$ that is $qtimes1$
$$
X_{ij} = sum_{r = 1}^q A_{ir} * B_{jr} * c_r
$$
How would we vectorize this? If $q$ is small, say q = 2, we could simply expand out the terms like $X = A*B^T*c[1] + A*B^T*c[2]$.
But this is obviously not feasible if q is large. How would you vectorize something like this?
Is there a general procedure to follow to a problem like this, or is the vectorization very problem-specific that no general procedure can be followed?
matrices vectors vectorization
asked Jan 29 at 19:12
IamanonIamanon
1328
$endgroup$
1
$begingroup$
Let $$C = begin{bmatrix}c_1 & 0 & dots & 0\0 & c_2 & dots & 0 \vdots & vdots &ddots& vdots\0&0&dots&c_qend{bmatrix}$$ Then $$X = ACB^T$$
$endgroup$
– Paul Sinclair
Jan 30 at 3:20
add a comment |
$begingroup$
I want to figure out a general formula for vectorizing computations.
For example, if I am given matrices, A,B such that A is $m times q$, B is $n times q$. And I want to compute the matrix $X$, where each component of $X$ is
$$
X_{ij} = sum_{r = 1}^q A_{ir} B_{jr}
$$
Note that:
$i = 1 ldots m$
$j = 1 ldots n$
This one is simple. By vectorizing, I mean we get rid of the summation and the individual component computations. So for this example, the vectorized form is $X = A*B^T$.
But what if instead we add another multplication term, $c$ that is $qtimes1$
$$
X_{ij} = sum_{r = 1}^q A_{ir} * B_{jr} * c_r
$$
How would we vectorize this? If $q$ is small, say q = 2, we could simply expand out the terms like $X = A*B^T*c[1] + A*B^T*c[2]$.
But this is obviously not feasible if q is large. How would you vectorize something like this?
Is there a general procedure to follow to a problem like this, or is the vectorization very problem-specific that no general procedure can be followed?
matrices vectors vectorization
asked Jan 29 at 19:12
IamanonIamanon
1328
$endgroup$
I want to figure out a general formula for vectorizing computations.
For example, if I am given matrices, A,B such that A is $m times q$, B is $n times q$. And I want to compute the matrix $X$, where each component of $X$ is
$$
X_{ij} = sum_{r = 1}^q A_{ir} B_{jr}
$$
Note that:
$i = 1 ldots m$
$j = 1 ldots n$
This one is simple. By vectorizing, I mean we get rid of the summation and the individual component computations. So for this example, the vectorized form is $X = A*B^T$.
But what if instead we add another multplication term, $c$ that is $qtimes1$
$$
X_{ij} = sum_{r = 1}^q A_{ir} * B_{jr} * c_r
$$
How would we vectorize this? If $q$ is small, say q = 2, we could simply expand out the terms like $X = A*B^T*c[1] + A*B^T*c[2]$.
But this is obviously not feasible if q is large. How would you vectorize something like this?
Is there a general procedure to follow to a problem like this, or is the vectorization very problem-specific that no general procedure can be followed?
matrices vectors vectorization
matrices vectors vectorization
asked Jan 29 at 19:12
IamanonIamanon
1328
asked Jan 29 at 19:12
IamanonIamanon
1328
asked Jan 29 at 19:12
IamanonIamanon
1328
asked Jan 29 at 19:12
IamanonIamanon
1328
asked Jan 29 at 19:12
IamanonIamanon
1328
1328
1
$begingroup$
Let $$C = begin{bmatrix}c_1 & 0 & dots & 0\0 & c_2 & dots & 0 \vdots & vdots &ddots& vdots\0&0&dots&c_qend{bmatrix}$$ Then $$X = ACB^T$$
$endgroup$
– Paul Sinclair
Jan 30 at 3:20
add a comment |
1
$begingroup$
Let $$C = begin{bmatrix}c_1 & 0 & dots & 0\0 & c_2 & dots & 0 \vdots & vdots &ddots& vdots\0&0&dots&c_qend{bmatrix}$$ Then $$X = ACB^T$$
$endgroup$
– Paul Sinclair
Jan 30 at 3:20
1
1
$begingroup$
Let $$C = begin{bmatrix}c_1 & 0 & dots & 0\0 & c_2 & dots & 0 \vdots & vdots &ddots& vdots\0&0&dots&c_qend{bmatrix}$$ Then $$X = ACB^T$$
$endgroup$
– Paul Sinclair
Jan 30 at 3:20
$begingroup$
Let $$C = begin{bmatrix}c_1 & 0 & dots & 0\0 & c_2 & dots & 0 \vdots & vdots &ddots& vdots\0&0&dots&c_qend{bmatrix}$$ Then $$X = ACB^T$$
$endgroup$
– Paul Sinclair
Jan 30 at 3:20
add a comment |
0
active
oldest
votes
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3092601%2fwhat-is-the-vectorized-form-of-this-expression%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3092601%2fwhat-is-the-vectorized-form-of-this-expression%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
1
$begingroup$
Let $$C = begin{bmatrix}c_1 & 0 & dots & 0\0 & c_2 & dots & 0 \vdots & vdots &ddots& vdots\0&0&dots&c_qend{bmatrix}$$ Then $$X = ACB^T$$
$endgroup$
– Paul Sinclair
Jan 30 at 3:20