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| + | ====== Product ====== | ||
| + | ---- | ||
| + | |||
| + | =====Basics===== | ||
| + | ''Product'' represents a product of terms. Like any other expression, it inherits all properties of [[Tensor]]. ''Product'' adopts the following convention on its indices: the indices of product are sorted concatenated indices of its multipliers: | ||
| + | <sxh groovy; gutter: false> | ||
| + | def t = 'f_ba*t^c_c*h^a'.t | ||
| + | println t.indice | ||
| + | </sxh> | ||
| + | <sxh plain; gutter: false> | ||
| + | > ^{ac}_{abc} | ||
| + | </sxh> | ||
| + | |||
| + | ''Product'' sorts its arguments and collapses equal scalar terms into powers. | ||
| + | |||
| + | |||
| + | ''Product'' is one of the most sophisticated entities in Redberry: contractions of indices brings additional structure of mathematical //graph// which is reflected in ''Product'' by introducing several methods to access graph structure of product. | ||
| + | |||
| + | The data structure used to represent product contains three main blocks: numerical factor, subproduct of scalar factors (with ''indices.size() == 0''), and tensorial content (with ''indices.size() != 0''). Each block can be obtained via special additional methods introduced in product. | ||
| + | =====Additional features===== | ||
| + | There are several features in addition to those defined in [[Tensor]] that are inherent only in [[Product]]: | ||
| + | |||
| + | <html> | ||
| + | <table style="width:600px;" cellpadding="10"> | ||
| + | <tr> | ||
| + | <td valign="top" style="padding: 15px;margin-top: 35px;border: 1px solid gray;"><code>.select(positions)</code></td> | ||
| + | <td valign="top" style="padding: 15px;margin-top: 35px;border: 1px solid gray;">returns a product of elements at specified positions</td> | ||
| + | </tr> | ||
| + | <tr> | ||
| + | <td valign="top" style="padding: 15px;margin-top: 35px;border: 1px solid gray;"><code>.remove(positions)</code></td> | ||
| + | <td valign="top" style="padding: 15px;margin-top: 35px;border: 1px solid gray;">returns a product with removed elements at specified positions</td> | ||
| + | </tr> | ||
| + | <tr> | ||
| + | <td valign="top" style="padding: 15px;margin-top: 35px;border: 1px solid gray;"><code>.factor</code></td> | ||
| + | <td valign="top" style="padding: 15px;margin-top: 35px;border: 1px solid gray;">returns a numerical factor of product</td> | ||
| + | </tr> | ||
| + | <tr> | ||
| + | <td valign="top" style="padding: 15px;margin-top: 35px;border: 1px solid gray;"><code>.indexlessSubProduct</code></td> | ||
| + | <td valign="top" style="padding: 15px;margin-top: 35px;border: 1px solid gray;">returns a subproduct of indexless terms, i.e. product of those terms which <code>indices.size() == 0</code></td> | ||
| + | </tr> | ||
| + | <tr> | ||
| + | <td valign="top" style="padding: 15px;margin-top: 35px;border: 1px solid gray;"><code>.dataSubProduct</code></td> | ||
| + | <td valign="top" style="padding: 15px;margin-top: 35px;border: 1px solid gray;">returns a subproduct of indexed terms, i.e. product of those terms which <code>indices.size() != 0</code></td> | ||
| + | </tr> | ||
| + | <tr> | ||
| + | <td valign="top" style="padding: 15px;margin-top: 35px;border: 1px solid gray;"><code>.content</code></td> | ||
| + | <td valign="top" style="padding: 15px;margin-top: 35px;border: 1px solid gray;">returns a data structure that allows to access <i>graph structure</i> of product</td> | ||
| + | </tr> | ||
| + | </table> | ||
| + | </html> | ||
| + | |||
| + | Consider examples: | ||
| + | <sxh groovy; gutter: true> | ||
| + | def p = '2*a*b*f_ba*t^c_c*h^a'.t | ||
| + | println p.select(2, 3, 5) | ||
| + | </sxh> | ||
| + | <sxh plain; gutter: false> | ||
| + | > a*t^{c}_{c}*h^{a} | ||
| + | </sxh> | ||
| + | <sxh groovy; gutter: true;first-line: 3> | ||
| + | println p.remove(2, 3, 5) | ||
| + | </sxh> | ||
| + | <sxh plain; gutter: false> | ||
| + | > 2*b*f_{ba} | ||
| + | </sxh> | ||
| + | <sxh groovy; gutter: true;first-line: 4> | ||
| + | println p.factor | ||
| + | </sxh> | ||
| + | <sxh plain; gutter: false> | ||
| + | > 2 | ||
| + | </sxh> | ||
| + | <sxh groovy; gutter: true;first-line: 5> | ||
| + | println p.indexlessSubProduct | ||
| + | </sxh> | ||
| + | <sxh plain; gutter: false> | ||
| + | > 2*b*a | ||
| + | </sxh> | ||
| + | <sxh groovy; gutter: true;first-line: 6> | ||
| + | println p.dataSubProduct | ||
| + | </sxh> | ||
| + | <sxh plain; gutter: false> | ||
| + | > t^{c}_{c}*f_{ba}*h^{a} | ||
| + | </sxh> | ||
| + | =====Advanced features: product content and graph structure ===== | ||
| + | The presence of indices bring a //graph structure// of Product: each multiplier represents a graph vertex, while contractions between indices are edges. One can check whether two graph isomorphic, i.e. that two tensors are equal to within free and dummy indices relabelling using [[documentation:guide:mappings_of_indices]]. | ||
| + | |||
| + | There is a special property ''.content'', which allows to access graph structure of ''Product''; the object returned by ''.content'' called ''ProductContent''. It holds tensorial part of product and additional low-level data structures used in Redberry to represent tensorial graphs. Besides, ''ProductContent'' provides several useful methods like ''.scalars'' and ''.nonScalar'' which return array of scalar sub-products and non-scalar part of tensor respectively. Consider examples: | ||
| + | <sxh groovy; gutter: true> | ||
| + | def p = '2*a*f_a*t^ba*g_b*t^i_i*t^mn*f_nmk'.t | ||
| + | println p.size() | ||
| + | </sxh> | ||
| + | <sxh plain; gutter: false> | ||
| + | > 8 | ||
| + | </sxh> | ||
| + | <sxh groovy; gutter: true;first-line: 3> | ||
| + | def content = p.content | ||
| + | println content.size() | ||
| + | </sxh> | ||
| + | <sxh plain; gutter: false> | ||
| + | > 6 | ||
| + | </sxh> | ||
| + | <sxh groovy; gutter: true;first-line: 5> | ||
| + | println content[0..content.size()] | ||
| + | </sxh> | ||
| + | <sxh plain; gutter: false> | ||
| + | > [f_a, t^ba, g_b, t^i_i, t^mn, f_nmk] | ||
| + | </sxh> | ||
| + | <sxh groovy; gutter: true;first-line: 6> | ||
| + | println content.scalars | ||
| + | </sxh> | ||
| + | <sxh plain; gutter: false> | ||
| + | > [t^{i}_{i}, t^{ba}*g_{b}*f_{a}] | ||
| + | </sxh> | ||
| + | <sxh groovy; gutter: true;first-line: 7> | ||
| + | println content.nonScalar | ||
| + | </sxh> | ||
| + | <sxh plain; gutter: false> | ||
| + | > t^{mn}*f_{nmk} | ||
| + | </sxh> | ||
| + | |||
| + | In order to access low-level structure that encodes ''Product'' graph there is ''.structureOfContractions'' property in ''ProductContent''. This structure allows to check connected components of graph and check for each particular tensor with which tensors it is contracted. Without going into low-level details, let's illustrate this features by the examples: | ||
| + | <sxh groovy; gutter: true> | ||
| + | def p = '3*c*f_i*g_b*t^i_i*t^pq*f_qpk*t^bie'.t | ||
| + | def content = p.content | ||
| + | //get graph representation | ||
| + | def graph = content.structureOfContractions | ||
| + | //array of connected components | ||
| + | println graph.components | ||
| + | </sxh> | ||
| + | <sxh plain; gutter: false> | ||
| + | > [0, 1, 0, 1, 2, 0] | ||
| + | </sxh> | ||
| + | The array of //components// indicates whether two tensors belongs to the same connected component of graph: two tensors at //i//-th and //j//-th positions belongs to the same component if | ||
| + | ''components[i] == components[j]'', and ''components[i]'' is the number of corresponding component. | ||
| + | <sxh groovy; gutter: true;first-line: 7> | ||
| + | //get position of tensor t^bie | ||
| + | def i = content.findIndexOf { it.equals('t^bie'.t) } | ||
| + | //contractions of tensor t^bie | ||
| + | def contractions = graph.getContractedWith(i) | ||
| + | for (def c in contractions) { | ||
| + | if (c.tensor == -1)//free index | ||
| + | println "Free indices:" + content[i].indices[c.indicesFrom] | ||
| + | else { | ||
| + | println "With tensor: " + content[c.tensor] | ||
| + | println "This indices:" + content[i].indices[c.indicesFrom] | ||
| + | println "With indices:" + content[c.tensor].indices[c.indicesTo] | ||
| + | } | ||
| + | } | ||
| + | </sxh> | ||
| + | <sxh plain; gutter: false> | ||
| + | > Free indices:^{e} | ||
| + | > With tensor: g_{b} | ||
| + | > This indices:^{b} | ||
| + | > With indices:_{b} | ||
| + | > With tensor: f_{i} | ||
| + | > This indices:^{i} | ||
| + | > With indices:_{i} | ||
| + | </sxh> | ||
| + | =====See also===== | ||
| + | * Related guides: [[documentation:guide:tensors_and_indices]], [[documentation:guide:tree_traversal]], [[documentation:guide:mappings_of_indices]] | ||
| + | * Reference: [[documentation:ref:tensor]], [[documentation:ref:sum]], [[documentation:ref:indices]], [[documentation:ref:simpletensor]], [[documentation:ref:tensorfield]] | ||
| + | * JavaDocs: [[http://api.redberry.cc/redberry/1.1.9/java-api/cc/redberry/core/tensor/Product.html| Product]] | ||
| + | * Source code: [[https://bitbucket.org/redberry/redberry/src/tip/core/src/main/java/cc/redberry/core/tensor/Product.java|Product.java]] | ||