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< div class = "title" > qr_decompose.h< / div > < / div >
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< a href = "../../d4/d68/qr__decompose_8h.html" > Go to the documentation of this file.< / a > < div class = "fragment" > < div class = "line" > < a name = "l00001" > < / a > < span class = "lineno" > 1< / span >   < span class = "comment" > /**< / span > < / div >
< div class = "line" > < a name = "l00002" > < / a > < span class = "lineno" > 2< / span >   < span class = "comment" > * @file< / span > < / div >
< div class = "line" > < a name = "l00003" > < / a > < span class = "lineno" > 3< / span >   < span class = "comment" > * \brief Library functions to compute [QR< / span > < / div >
< div class = "line" > < a name = "l00004" > < / a > < span class = "lineno" > 4< / span >   < span class = "comment" > * decomposition](https://en.wikipedia.org/wiki/QR_decomposition) of a given< / span > < / div >
< div class = "line" > < a name = "l00005" > < / a > < span class = "lineno" > 5< / span >   < span class = "comment" > * matrix.< / span > < / div >
< div class = "line" > < a name = "l00006" > < / a > < span class = "lineno" > 6< / span >   < span class = "comment" > * \author [Krishna Vedala](https://github.com/kvedala)< / span > < / div >
< div class = "line" > < a name = "l00007" > < / a > < span class = "lineno" > 7< / span >   < span class = "comment" > */< / span > < / div >
< div class = "line" > < a name = "l00008" > < / a > < span class = "lineno" > 8< / span >   < / div >
< div class = "line" > < a name = "l00009" > < / a > < span class = "lineno" > 9< / span >   < span class = "preprocessor" > #ifndef QR_DECOMPOSE_H< / span > < / div >
< div class = "line" > < a name = "l00010" > < / a > < span class = "lineno" > 10< / span >   < span class = "preprocessor" > #define QR_DECOMPOSE_H< / span > < / div >
< div class = "line" > < a name = "l00011" > < / a > < span class = "lineno" > 11< / span >   < / div >
< div class = "line" > < a name = "l00012" > < / a > < span class = "lineno" > 12< / span >   < span class = "preprocessor" > #include < math.h> < / span > < / div >
< div class = "line" > < a name = "l00013" > < / a > < span class = "lineno" > 13< / span >   < span class = "preprocessor" > #include < stdio.h> < / span > < / div >
< div class = "line" > < a name = "l00014" > < / a > < span class = "lineno" > 14< / span >   < span class = "preprocessor" > #include < stdlib.h> < / span > < / div >
< div class = "line" > < a name = "l00015" > < / a > < span class = "lineno" > 15< / span >   < span class = "preprocessor" > #ifdef _OPENMP< / span > < / div >
< div class = "line" > < a name = "l00016" > < / a > < span class = "lineno" > 16< / span >   < span class = "preprocessor" > #include < omp.h> < / span > < / div >
< div class = "line" > < a name = "l00017" > < / a > < span class = "lineno" > 17< / span >   < span class = "preprocessor" > #endif< / span > < / div >
< div class = "line" > < a name = "l00018" > < / a > < span class = "lineno" > 18< / span >   < span class = "comment" > < / span > < / div >
< div class = "line" > < a name = "l00019" > < / a > < span class = "lineno" > 19< / span >   < span class = "comment" > /**< / span > < / div >
< div class = "line" > < a name = "l00020" > < / a > < span class = "lineno" > 20< / span >   < span class = "comment" > * function to display matrix on stdout< / span > < / div >
< div class = "line" > < a name = "l00021" > < / a > < span class = "lineno" > 21< / span >   < span class = "comment" > */< / span > < / div >
< div class = "line" > < a name = "l00022" > < / a > < span class = "lineno" > < a class = "line" href = "../../d4/d68/qr__decompose_8h.html#a90562ce8c3707401e9c5809dece68d6a" > 22< / a > < / span >   < span class = "keywordtype" > void< / span > < a class = "code" href = "../../d4/d68/qr__decompose_8h.html#a90562ce8c3707401e9c5809dece68d6a" > print_matrix< / a > (< span class = "keywordtype" > double< / span > **A, < span class = "comment" > /**< matrix to print */< / span > < / div >
< div class = "line" > < a name = "l00023" > < / a > < span class = "lineno" > 23< / span >   < span class = "keywordtype" > int< / span > M, < span class = "comment" > /**< number of rows of matrix */< / span > < / div >
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< div class = "line" > < a name = "l00024" > < / a > < span class = "lineno" > 24< / span >   < span class = "keywordtype" > int< / span > N) < span class = "comment" > /**< number of columns of matrix */< / span > < / div >
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< div class = "line" > < a name = "l00025" > < / a > < span class = "lineno" > 25< / span >   {< / div >
< div class = "line" > < a name = "l00026" > < / a > < span class = "lineno" > 26< / span >   < span class = "keywordflow" > for< / span > (< span class = "keywordtype" > int< / span > row = 0; row < M; row++)< / div >
< div class = "line" > < a name = "l00027" > < / a > < span class = "lineno" > 27< / span >   {< / div >
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< div class = "line" > < a name = "l00028" > < / a > < span class = "lineno" > 28< / span >   < span class = "keywordflow" > for< / span > (< span class = "keywordtype" > int< / span > col = 0; col < N; col++) printf(< span class = "stringliteral" > " % 9.3g\t" < / span > , A[row][col]);< / div >
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< div class = "line" > < a name = "l00029" > < / a > < span class = "lineno" > 29< / span >   putchar(< span class = "charliteral" > ' \n' < / span > );< / div >
< div class = "line" > < a name = "l00030" > < / a > < span class = "lineno" > 30< / span >   }< / div >
< div class = "line" > < a name = "l00031" > < / a > < span class = "lineno" > 31< / span >   putchar(< span class = "charliteral" > ' \n' < / span > );< / div >
< div class = "line" > < a name = "l00032" > < / a > < span class = "lineno" > 32< / span >   }< / div >
< div class = "line" > < a name = "l00033" > < / a > < span class = "lineno" > 33< / span >   < span class = "comment" > < / span > < / div >
< div class = "line" > < a name = "l00034" > < / a > < span class = "lineno" > 34< / span >   < span class = "comment" > /**< / span > < / div >
< div class = "line" > < a name = "l00035" > < / a > < span class = "lineno" > 35< / span >   < span class = "comment" > * Compute dot product of two vectors of equal lengths< / span > < / div >
< div class = "line" > < a name = "l00036" > < / a > < span class = "lineno" > 36< / span >   < span class = "comment" > *< / span > < / div >
< div class = "line" > < a name = "l00037" > < / a > < span class = "lineno" > 37< / span >   < span class = "comment" > * If \f$\vec{a}=\left[a_0,a_1,a_2,...,a_L\right]\f$ and< / span > < / div >
< div class = "line" > < a name = "l00038" > < / a > < span class = "lineno" > 38< / span >   < span class = "comment" > * \f$\vec{b}=\left[b_0,b_1,b_1,...,b_L\right]\f$ then< / span > < / div >
< div class = "line" > < a name = "l00039" > < / a > < span class = "lineno" > 39< / span >   < span class = "comment" > * \f$\vec{a}\cdot\vec{b}=\displaystyle\sum_{i=0}^L a_i\times b_i\f$< / span > < / div >
< div class = "line" > < a name = "l00040" > < / a > < span class = "lineno" > 40< / span >   < span class = "comment" > *< / span > < / div >
< div class = "line" > < a name = "l00041" > < / a > < span class = "lineno" > 41< / span >   < span class = "comment" > * \returns \f$\vec{a}\cdot\vec{b}\f$< / span > < / div >
< div class = "line" > < a name = "l00042" > < / a > < span class = "lineno" > 42< / span >   < span class = "comment" > */< / span > < / div >
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< div class = "line" > < a name = "l00043" > < / a > < span class = "lineno" > < a class = "line" href = "../../d4/d68/qr__decompose_8h.html#a3a584b79941a43d775f9d4ce446dbe05" > 43< / a > < / span >   < span class = "keywordtype" > double< / span > < a class = "code" href = "../../d4/d68/qr__decompose_8h.html#a3a584b79941a43d775f9d4ce446dbe05" > vector_dot< / a > (< span class = "keywordtype" > double< / span > *a, < span class = "keywordtype" > double< / span > *b, < span class = "keywordtype" > int< / span > < a class = "code" href = "../../df/db3/struct_l.html" > L< / a > )< / div >
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< div class = "line" > < a name = "l00044" > < / a > < span class = "lineno" > 44< / span >   {< / div >
< div class = "line" > < a name = "l00045" > < / a > < span class = "lineno" > 45< / span >   < span class = "keywordtype" > double< / span > mag = 0.f;< / div >
< div class = "line" > < a name = "l00046" > < / a > < span class = "lineno" > 46< / span >   < span class = "keywordtype" > int< / span > i;< / div >
< div class = "line" > < a name = "l00047" > < / a > < span class = "lineno" > 47< / span >   < span class = "preprocessor" > #ifdef _OPENMP< / span > < / div >
< div class = "line" > < a name = "l00048" > < / a > < span class = "lineno" > 48< / span >   < span class = "comment" > // parallelize on threads< / span > < / div >
< div class = "line" > < a name = "l00049" > < / a > < span class = "lineno" > 49< / span >   < span class = "preprocessor" > #pragma omp parallel for reduction(+ : mag)< / span > < / div >
< div class = "line" > < a name = "l00050" > < / a > < span class = "lineno" > 50< / span >   < span class = "preprocessor" > #endif< / span > < / div >
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< div class = "line" > < a name = "l00051" > < / a > < span class = "lineno" > 51< / span >   < span class = "keywordflow" > for< / span > (i = 0; i < < a class = "code" href = "../../df/db3/struct_l.html" > L< / a > ; i++) mag += a[i] * b[i];< / div >
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< div class = "line" > < a name = "l00052" > < / a > < span class = "lineno" > 52< / span >   < / div >
< div class = "line" > < a name = "l00053" > < / a > < span class = "lineno" > 53< / span >   < span class = "keywordflow" > return< / span > mag;< / div >
< div class = "line" > < a name = "l00054" > < / a > < span class = "lineno" > 54< / span >   }< / div >
< div class = "line" > < a name = "l00055" > < / a > < span class = "lineno" > 55< / span >   < span class = "comment" > < / span > < / div >
< div class = "line" > < a name = "l00056" > < / a > < span class = "lineno" > 56< / span >   < span class = "comment" > /**< / span > < / div >
< div class = "line" > < a name = "l00057" > < / a > < span class = "lineno" > 57< / span >   < span class = "comment" > * Compute magnitude of vector.< / span > < / div >
< div class = "line" > < a name = "l00058" > < / a > < span class = "lineno" > 58< / span >   < span class = "comment" > *< / span > < / div >
< div class = "line" > < a name = "l00059" > < / a > < span class = "lineno" > 59< / span >   < span class = "comment" > * If \f$\vec{a}=\left[a_0,a_1,a_2,...,a_L\right]\f$ then< / span > < / div >
< div class = "line" > < a name = "l00060" > < / a > < span class = "lineno" > 60< / span >   < span class = "comment" > * \f$\left|\vec{a}\right|=\sqrt{\displaystyle\sum_{i=0}^L a_i^2}\f$< / span > < / div >
< div class = "line" > < a name = "l00061" > < / a > < span class = "lineno" > 61< / span >   < span class = "comment" > *< / span > < / div >
< div class = "line" > < a name = "l00062" > < / a > < span class = "lineno" > 62< / span >   < span class = "comment" > * \returns \f$\left|\vec{a}\right|\f$< / span > < / div >
< div class = "line" > < a name = "l00063" > < / a > < span class = "lineno" > 63< / span >   < span class = "comment" > */< / span > < / div >
< div class = "line" > < a name = "l00064" > < / a > < span class = "lineno" > < a class = "line" href = "../../d4/d68/qr__decompose_8h.html#abeec1f78a7a7e7251687e75340331212" > 64< / a > < / span >   < span class = "keywordtype" > double< / span > < a class = "code" href = "../../d4/d68/qr__decompose_8h.html#abeec1f78a7a7e7251687e75340331212" > vector_mag< / a > (< span class = "keywordtype" > double< / span > *vector, < span class = "keywordtype" > int< / span > < a class = "code" href = "../../df/db3/struct_l.html" > L< / a > )< / div >
< div class = "line" > < a name = "l00065" > < / a > < span class = "lineno" > 65< / span >   {< / div >
< div class = "line" > < a name = "l00066" > < / a > < span class = "lineno" > 66< / span >   < span class = "keywordtype" > double< / span > dot = < a class = "code" href = "../../d4/d68/qr__decompose_8h.html#a3a584b79941a43d775f9d4ce446dbe05" > vector_dot< / a > (vector, vector, < a class = "code" href = "../../df/db3/struct_l.html" > L< / a > );< / div >
< div class = "line" > < a name = "l00067" > < / a > < span class = "lineno" > 67< / span >   < span class = "keywordflow" > return< / span > sqrt(dot);< / div >
< div class = "line" > < a name = "l00068" > < / a > < span class = "lineno" > 68< / span >   }< / div >
< div class = "line" > < a name = "l00069" > < / a > < span class = "lineno" > 69< / span >   < span class = "comment" > < / span > < / div >
< div class = "line" > < a name = "l00070" > < / a > < span class = "lineno" > 70< / span >   < span class = "comment" > /**< / span > < / div >
< div class = "line" > < a name = "l00071" > < / a > < span class = "lineno" > 71< / span >   < span class = "comment" > * Compute projection of vector \f$\vec{a}\f$ on \f$\vec{b}\f$ defined as< / span > < / div >
< div class = "line" > < a name = "l00072" > < / a > < span class = "lineno" > 72< / span >   < span class = "comment" > * \f[\text{proj}_\vec{b}\vec{a}=\frac{\vec{a}\cdot\vec{b}}{\left|\vec{b}\right|^2}\vec{b}\f]< / span > < / div >
< div class = "line" > < a name = "l00073" > < / a > < span class = "lineno" > 73< / span >   < span class = "comment" > *< / span > < / div >
< div class = "line" > < a name = "l00074" > < / a > < span class = "lineno" > 74< / span >   < span class = "comment" > * \returns NULL if error, otherwise pointer to output< / span > < / div >
< div class = "line" > < a name = "l00075" > < / a > < span class = "lineno" > 75< / span >   < span class = "comment" > */< / span > < / div >
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< div class = "line" > < a name = "l00076" > < / a > < span class = "lineno" > < a class = "line" href = "../../d4/d68/qr__decompose_8h.html#a82b20e027437df768d7e994cf4cae29f" > 76< / a > < / span >   < span class = "keywordtype" > double< / span > *< a class = "code" href = "../../d4/d68/qr__decompose_8h.html#a82b20e027437df768d7e994cf4cae29f" > vector_proj< / a > (< span class = "keywordtype" > double< / span > *a, < span class = "keywordtype" > double< / span > *b, < span class = "keywordtype" > double< / span > *out, < span class = "keywordtype" > int< / span > < a class = "code" href = "../../df/db3/struct_l.html" > L< / a > )< / div >
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< div class = "line" > < a name = "l00077" > < / a > < span class = "lineno" > 77< / span >   {< / div >
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< div class = "line" > < a name = "l00078" > < / a > < span class = "lineno" > 78< / span >   < span class = "keyword" > const< / span > < span class = "keywordtype" > double< / span > num = < a class = "code" href = "../../d4/d68/qr__decompose_8h.html#a3a584b79941a43d775f9d4ce446dbe05" > vector_dot< / a > (a, b, < a class = "code" href = "../../df/db3/struct_l.html" > L< / a > );< / div >
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< div class = "line" > < a name = "l00079" > < / a > < span class = "lineno" > 79< / span >   < span class = "keyword" > const< / span > < span class = "keywordtype" > double< / span > deno = < a class = "code" href = "../../d4/d68/qr__decompose_8h.html#a3a584b79941a43d775f9d4ce446dbe05" > vector_dot< / a > (b, b, < a class = "code" href = "../../df/db3/struct_l.html" > L< / a > );< / div >
< div class = "line" > < a name = "l00080" > < / a > < span class = "lineno" > 80< / span >   < span class = "keywordflow" > if< / span > (deno == 0) < span class = "comment" > /*! check for division by zero */< / span > < / div >
< div class = "line" > < a name = "l00081" > < / a > < span class = "lineno" > 81< / span >   < span class = "keywordflow" > return< / span > NULL;< / div >
< div class = "line" > < a name = "l00082" > < / a > < span class = "lineno" > 82< / span >   < / div >
< div class = "line" > < a name = "l00083" > < / a > < span class = "lineno" > 83< / span >   < span class = "keyword" > const< / span > < span class = "keywordtype" > double< / span > scalar = num / deno;< / div >
< div class = "line" > < a name = "l00084" > < / a > < span class = "lineno" > 84< / span >   < span class = "keywordtype" > int< / span > i;< / div >
< div class = "line" > < a name = "l00085" > < / a > < span class = "lineno" > 85< / span >   < span class = "preprocessor" > #ifdef _OPENMP< / span > < / div >
< div class = "line" > < a name = "l00086" > < / a > < span class = "lineno" > 86< / span >   < span class = "comment" > // parallelize on threads< / span > < / div >
< div class = "line" > < a name = "l00087" > < / a > < span class = "lineno" > 87< / span >   < span class = "preprocessor" > #pragma omp for< / span > < / div >
< div class = "line" > < a name = "l00088" > < / a > < span class = "lineno" > 88< / span >   < span class = "preprocessor" > #endif< / span > < / div >
< div class = "line" > < a name = "l00089" > < / a > < span class = "lineno" > 89< / span >   < span class = "keywordflow" > for< / span > (i = 0; i < < a class = "code" href = "../../df/db3/struct_l.html" > L< / a > ; i++) out[i] = scalar * b[i];< / div >
< div class = "line" > < a name = "l00090" > < / a > < span class = "lineno" > 90< / span >   < / div >
< div class = "line" > < a name = "l00091" > < / a > < span class = "lineno" > 91< / span >   < span class = "keywordflow" > return< / span > out;< / div >
< div class = "line" > < a name = "l00092" > < / a > < span class = "lineno" > 92< / span >   }< / div >
< div class = "line" > < a name = "l00093" > < / a > < span class = "lineno" > 93< / span >   < span class = "comment" > < / span > < / div >
< div class = "line" > < a name = "l00094" > < / a > < span class = "lineno" > 94< / span >   < span class = "comment" > /**< / span > < / div >
< div class = "line" > < a name = "l00095" > < / a > < span class = "lineno" > 95< / span >   < span class = "comment" > * Compute vector subtraction< / span > < / div >
< div class = "line" > < a name = "l00096" > < / a > < span class = "lineno" > 96< / span >   < span class = "comment" > *< / span > < / div >
< div class = "line" > < a name = "l00097" > < / a > < span class = "lineno" > 97< / span >   < span class = "comment" > * \f$\vec{c}=\vec{a}-\vec{b}\f$< / span > < / div >
< div class = "line" > < a name = "l00098" > < / a > < span class = "lineno" > 98< / span >   < span class = "comment" > *< / span > < / div >
< div class = "line" > < a name = "l00099" > < / a > < span class = "lineno" > 99< / span >   < span class = "comment" > * \returns pointer to output vector< / span > < / div >
< div class = "line" > < a name = "l00100" > < / a > < span class = "lineno" > 100< / span >   < span class = "comment" > */< / span > < / div >
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< div class = "line" > < a name = "l00101" > < / a > < span class = "lineno" > < a class = "line" href = "../../d4/d68/qr__decompose_8h.html#a6b6a0e75e75ff7919057dd275bb69145" > 101< / a > < / span >   < span class = "keywordtype" > double< / span > *< a class = "code" href = "../../d4/d68/qr__decompose_8h.html#a6b6a0e75e75ff7919057dd275bb69145" > vector_sub< / a > (< span class = "keywordtype" > double< / span > *a, < span class = "comment" > /**< minuend */< / span > < / div >
2020-06-28 23:30:37 +08:00
< div class = "line" > < a name = "l00102" > < / a > < span class = "lineno" > 102< / span >   < span class = "keywordtype" > double< / span > *b, < span class = "comment" > /**< subtrahend */< / span > < / div >
< div class = "line" > < a name = "l00103" > < / a > < span class = "lineno" > 103< / span >   < span class = "keywordtype" > double< / span > *out, < span class = "comment" > /**< resultant vector */< / span > < / div >
< div class = "line" > < a name = "l00104" > < / a > < span class = "lineno" > 104< / span >   < span class = "keywordtype" > int< / span > < a class = "code" href = "../../df/db3/struct_l.html" > L< / a > < span class = "comment" > /**< length of vectors */< / span > < / div >
< div class = "line" > < a name = "l00105" > < / a > < span class = "lineno" > 105< / span >   )< / div >
< div class = "line" > < a name = "l00106" > < / a > < span class = "lineno" > 106< / span >   {< / div >
< div class = "line" > < a name = "l00107" > < / a > < span class = "lineno" > 107< / span >   < span class = "keywordtype" > int< / span > i;< / div >
< div class = "line" > < a name = "l00108" > < / a > < span class = "lineno" > 108< / span >   < span class = "preprocessor" > #ifdef _OPENMP< / span > < / div >
< div class = "line" > < a name = "l00109" > < / a > < span class = "lineno" > 109< / span >   < span class = "comment" > // parallelize on threads< / span > < / div >
< div class = "line" > < a name = "l00110" > < / a > < span class = "lineno" > 110< / span >   < span class = "preprocessor" > #pragma omp for< / span > < / div >
< div class = "line" > < a name = "l00111" > < / a > < span class = "lineno" > 111< / span >   < span class = "preprocessor" > #endif< / span > < / div >
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< div class = "line" > < a name = "l00112" > < / a > < span class = "lineno" > 112< / span >   < span class = "keywordflow" > for< / span > (i = 0; i < < a class = "code" href = "../../df/db3/struct_l.html" > L< / a > ; i++) out[i] = a[i] - b[i];< / div >
2020-06-28 23:30:37 +08:00
< div class = "line" > < a name = "l00113" > < / a > < span class = "lineno" > 113< / span >   < / div >
< div class = "line" > < a name = "l00114" > < / a > < span class = "lineno" > 114< / span >   < span class = "keywordflow" > return< / span > out;< / div >
< div class = "line" > < a name = "l00115" > < / a > < span class = "lineno" > 115< / span >   }< / div >
< div class = "line" > < a name = "l00116" > < / a > < span class = "lineno" > 116< / span >   < span class = "comment" > < / span > < / div >
< div class = "line" > < a name = "l00117" > < / a > < span class = "lineno" > 117< / span >   < span class = "comment" > /**< / span > < / div >
< div class = "line" > < a name = "l00118" > < / a > < span class = "lineno" > 118< / span >   < span class = "comment" > * Decompose matrix \f$A\f$ using [Gram-Schmidt< / span > < / div >
< div class = "line" > < a name = "l00119" > < / a > < span class = "lineno" > 119< / span >   < span class = "comment" > *process](https://en.wikipedia.org/wiki/QR_decomposition).< / span > < / div >
< div class = "line" > < a name = "l00120" > < / a > < span class = "lineno" > 120< / span >   < span class = "comment" > *< / span > < / div >
< div class = "line" > < a name = "l00121" > < / a > < span class = "lineno" > 121< / span >   < span class = "comment" > * \f{eqnarray*}{< / span > < / div >
< div class = "line" > < a name = "l00122" > < / a > < span class = "lineno" > 122< / span >   < span class = "comment" > * \text{given that}\quad A & =& < / span > < / div >
< div class = "line" > < a name = "l00123" > < / a > < span class = "lineno" > 123< / span >   < span class = "comment" > *\left[\mathbf{a}_1,\mathbf{a}_2,\ldots,\mathbf{a}_{N-1},\right]\\< / span > < / div >
< div class = "line" > < a name = "l00124" > < / a > < span class = "lineno" > 124< / span >   < span class = "comment" > * \text{where}\quad\mathbf{a}_i & =& < / span > < / div >
< div class = "line" > < a name = "l00125" > < / a > < span class = "lineno" > 125< / span >   < span class = "comment" > *\left[a_{0i},a_{1i},a_{2i},\ldots,a_{(M-1)i}\right]^T\quad\ldots\mbox{(column< / span > < / div >
< div class = "line" > < a name = "l00126" > < / a > < span class = "lineno" > 126< / span >   < span class = "comment" > *vectors)}\\< / span > < / div >
< div class = "line" > < a name = "l00127" > < / a > < span class = "lineno" > 127< / span >   < span class = "comment" > * \text{then}\quad\mathbf{u}_i & =& \mathbf{a}_i< / span > < / div >
< div class = "line" > < a name = "l00128" > < / a > < span class = "lineno" > 128< / span >   < span class = "comment" > *-\sum_{j=0}^{i-1}\text{proj}_{\mathbf{u}_j}\mathbf{a}_i\\< / span > < / div >
< div class = "line" > < a name = "l00129" > < / a > < span class = "lineno" > 129< / span >   < span class = "comment" > * \mathbf{e}_i & =& \frac{\mathbf{u}_i}{\left|\mathbf{u}_i\right|}\\< / span > < / div >
< div class = "line" > < a name = "l00130" > < / a > < span class = "lineno" > 130< / span >   < span class = "comment" > * Q & =& \begin{bmatrix}\mathbf{e}_0 & \mathbf{e}_1 & \mathbf{e}_2 & \dots & < / span > < / div >
< div class = "line" > < a name = "l00131" > < / a > < span class = "lineno" > 131< / span >   < span class = "comment" > *\mathbf{e}_{N-1}\end{bmatrix}\\< / span > < / div >
< div class = "line" > < a name = "l00132" > < / a > < span class = "lineno" > 132< / span >   < span class = "comment" > * R & =& \begin{bmatrix}\langle\mathbf{e}_0\,,\mathbf{a}_0\rangle & < / span > < / div >
< div class = "line" > < a name = "l00133" > < / a > < span class = "lineno" > 133< / span >   < span class = "comment" > *\langle\mathbf{e}_1\,,\mathbf{a}_1\rangle & < / span > < / div >
< div class = "line" > < a name = "l00134" > < / a > < span class = "lineno" > 134< / span >   < span class = "comment" > *\langle\mathbf{e}_2\,,\mathbf{a}_2\rangle & \dots \\< / span > < / div >
< div class = "line" > < a name = "l00135" > < / a > < span class = "lineno" > 135< / span >   < span class = "comment" > * 0 & \langle\mathbf{e}_1\,,\mathbf{a}_1\rangle & < / span > < / div >
< div class = "line" > < a name = "l00136" > < / a > < span class = "lineno" > 136< / span >   < span class = "comment" > *\langle\mathbf{e}_2\,,\mathbf{a}_2\rangle & \dots\\< / span > < / div >
< div class = "line" > < a name = "l00137" > < / a > < span class = "lineno" > 137< / span >   < span class = "comment" > * 0 & 0 & \langle\mathbf{e}_2\,,\mathbf{a}_2\rangle & \dots\\< / span > < / div >
< div class = "line" > < a name = "l00138" > < / a > < span class = "lineno" > 138< / span >   < span class = "comment" > * \vdots & \vdots & \vdots & \ddots< / span > < / div >
< div class = "line" > < a name = "l00139" > < / a > < span class = "lineno" > 139< / span >   < span class = "comment" > * \end{bmatrix}\\< / span > < / div >
< div class = "line" > < a name = "l00140" > < / a > < span class = "lineno" > 140< / span >   < span class = "comment" > * \f}< / span > < / div >
< div class = "line" > < a name = "l00141" > < / a > < span class = "lineno" > 141< / span >   < span class = "comment" > */< / span > < / div >
< div class = "line" > < a name = "l00142" > < / a > < span class = "lineno" > < a class = "line" href = "../../d4/d68/qr__decompose_8h.html#a45c7640d9d22c89c11beb1f567843c56" > 142< / a > < / span >   < span class = "keywordtype" > void< / span > < a class = "code" href = "../../d4/d68/qr__decompose_8h.html#a45c7640d9d22c89c11beb1f567843c56" > qr_decompose< / a > (< span class = "keywordtype" > double< / span > **A, < span class = "comment" > /**< input matrix to decompose */< / span > < / div >
< div class = "line" > < a name = "l00143" > < / a > < span class = "lineno" > 143< / span >   < span class = "keywordtype" > double< / span > **Q, < span class = "comment" > /**< output decomposed matrix */< / span > < / div >
< div class = "line" > < a name = "l00144" > < / a > < span class = "lineno" > 144< / span >   < span class = "keywordtype" > double< / span > **R, < span class = "comment" > /**< output decomposed matrix */< / span > < / div >
< div class = "line" > < a name = "l00145" > < / a > < span class = "lineno" > 145< / span >   < span class = "keywordtype" > int< / span > M, < span class = "comment" > /**< number of rows of matrix A */< / span > < / div >
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< div class = "line" > < a name = "l00146" > < / a > < span class = "lineno" > 146< / span >   < span class = "keywordtype" > int< / span > N < span class = "comment" > /**< number of columns of matrix A */< / span > < / div >
2020-06-28 23:30:37 +08:00
< div class = "line" > < a name = "l00147" > < / a > < span class = "lineno" > 147< / span >   )< / div >
< div class = "line" > < a name = "l00148" > < / a > < span class = "lineno" > 148< / span >   {< / div >
2021-02-18 06:41:16 +08:00
< div class = "line" > < a name = "l00149" > < / a > < span class = "lineno" > 149< / span >   < span class = "keywordtype" > double< / span > *col_vector = (< span class = "keywordtype" > double< / span > *)< a class = "code" href = "../../d2/ddd/malloc__dbg_8h.html#a725f50ecaf1959d96de79b36b4788fee" > malloc< / a > (M * < span class = "keyword" > sizeof< / span > (< span class = "keywordtype" > double< / span > ));< / div >
< div class = "line" > < a name = "l00150" > < / a > < span class = "lineno" > 150< / span >   < span class = "keywordtype" > double< / span > *col_vector2 = (< span class = "keywordtype" > double< / span > *)< a class = "code" href = "../../d2/ddd/malloc__dbg_8h.html#a725f50ecaf1959d96de79b36b4788fee" > malloc< / a > (M * < span class = "keyword" > sizeof< / span > (< span class = "keywordtype" > double< / span > ));< / div >
< div class = "line" > < a name = "l00151" > < / a > < span class = "lineno" > 151< / span >   < span class = "keywordtype" > double< / span > *tmp_vector = (< span class = "keywordtype" > double< / span > *)< a class = "code" href = "../../d2/ddd/malloc__dbg_8h.html#a725f50ecaf1959d96de79b36b4788fee" > malloc< / a > (M * < span class = "keyword" > sizeof< / span > (< span class = "keywordtype" > double< / span > ));< / div >
2021-02-06 03:52:57 +08:00
< div class = "line" > < a name = "l00152" > < / a > < span class = "lineno" > 152< / span >   < span class = "keywordflow" > for< / span > (< span class = "keywordtype" > int< / span > i = 0; i < N;< / div >
2020-06-28 23:30:37 +08:00
< div class = "line" > < a name = "l00153" > < / a > < span class = "lineno" > 153< / span >   i++) < span class = "comment" > /* for each column => R is a square matrix of NxN */< / span > < / div >
< div class = "line" > < a name = "l00154" > < / a > < span class = "lineno" > 154< / span >   {< / div >
< div class = "line" > < a name = "l00155" > < / a > < span class = "lineno" > 155< / span >   < span class = "keywordtype" > int< / span > j;< / div >
< div class = "line" > < a name = "l00156" > < / a > < span class = "lineno" > 156< / span >   < span class = "preprocessor" > #ifdef _OPENMP< / span > < / div >
< div class = "line" > < a name = "l00157" > < / a > < span class = "lineno" > 157< / span >   < span class = "comment" > // parallelize on threads< / span > < / div >
< div class = "line" > < a name = "l00158" > < / a > < span class = "lineno" > 158< / span >   < span class = "preprocessor" > #pragma omp for< / span > < / div >
< div class = "line" > < a name = "l00159" > < / a > < span class = "lineno" > 159< / span >   < span class = "preprocessor" > #endif< / span > < / div >
< div class = "line" > < a name = "l00160" > < / a > < span class = "lineno" > 160< / span >   < span class = "keywordflow" > for< / span > (j = 0; j < i; j++) < span class = "comment" > /* second dimension of column */< / span > < / div >
< div class = "line" > < a name = "l00161" > < / a > < span class = "lineno" > 161< / span >   R[i][j] = 0.; < span class = "comment" > /* make R upper triangular */< / span > < / div >
< div class = "line" > < a name = "l00162" > < / a > < span class = "lineno" > 162< / span >   < / div >
< div class = "line" > < a name = "l00163" > < / a > < span class = "lineno" > 163< / span >   < span class = "comment" > /* get corresponding Q vector */< / span > < / div >
< div class = "line" > < a name = "l00164" > < / a > < span class = "lineno" > 164< / span >   < span class = "preprocessor" > #ifdef _OPENMP< / span > < / div >
< div class = "line" > < a name = "l00165" > < / a > < span class = "lineno" > 165< / span >   < span class = "comment" > // parallelize on threads< / span > < / div >
< div class = "line" > < a name = "l00166" > < / a > < span class = "lineno" > 166< / span >   < span class = "preprocessor" > #pragma omp for< / span > < / div >
< div class = "line" > < a name = "l00167" > < / a > < span class = "lineno" > 167< / span >   < span class = "preprocessor" > #endif< / span > < / div >
< div class = "line" > < a name = "l00168" > < / a > < span class = "lineno" > 168< / span >   < span class = "keywordflow" > for< / span > (j = 0; j < M; j++)< / div >
< div class = "line" > < a name = "l00169" > < / a > < span class = "lineno" > 169< / span >   {< / div >
< div class = "line" > < a name = "l00170" > < / a > < span class = "lineno" > 170< / span >   tmp_vector[j] = A[j][i]; < span class = "comment" > /* accumulator for uk */< / span > < / div >
< div class = "line" > < a name = "l00171" > < / a > < span class = "lineno" > 171< / span >   col_vector[j] = A[j][i];< / div >
< div class = "line" > < a name = "l00172" > < / a > < span class = "lineno" > 172< / span >   }< / div >
< div class = "line" > < a name = "l00173" > < / a > < span class = "lineno" > 173< / span >   < span class = "keywordflow" > for< / span > (j = 0; j < i; j++)< / div >
< div class = "line" > < a name = "l00174" > < / a > < span class = "lineno" > 174< / span >   {< / div >
< div class = "line" > < a name = "l00175" > < / a > < span class = "lineno" > 175< / span >   < span class = "keywordflow" > for< / span > (< span class = "keywordtype" > int< / span > k = 0; k < M; k++) col_vector2[k] = Q[k][j];< / div >
< div class = "line" > < a name = "l00176" > < / a > < span class = "lineno" > 176< / span >   < a class = "code" href = "../../d4/d68/qr__decompose_8h.html#a82b20e027437df768d7e994cf4cae29f" > vector_proj< / a > (col_vector, col_vector2, col_vector2, M);< / div >
< div class = "line" > < a name = "l00177" > < / a > < span class = "lineno" > 177< / span >   < a class = "code" href = "../../d4/d68/qr__decompose_8h.html#a6b6a0e75e75ff7919057dd275bb69145" > vector_sub< / a > (tmp_vector, col_vector2, tmp_vector, M);< / div >
< div class = "line" > < a name = "l00178" > < / a > < span class = "lineno" > 178< / span >   }< / div >
< div class = "line" > < a name = "l00179" > < / a > < span class = "lineno" > 179< / span >   < span class = "keywordtype" > double< / span > mag = < a class = "code" href = "../../d4/d68/qr__decompose_8h.html#abeec1f78a7a7e7251687e75340331212" > vector_mag< / a > (tmp_vector, M);< / div >
< div class = "line" > < a name = "l00180" > < / a > < span class = "lineno" > 180< / span >   < / div >
< div class = "line" > < a name = "l00181" > < / a > < span class = "lineno" > 181< / span >   < span class = "preprocessor" > #ifdef _OPENMP< / span > < / div >
< div class = "line" > < a name = "l00182" > < / a > < span class = "lineno" > 182< / span >   < span class = "comment" > // parallelize on threads< / span > < / div >
< div class = "line" > < a name = "l00183" > < / a > < span class = "lineno" > 183< / span >   < span class = "preprocessor" > #pragma omp for< / span > < / div >
< div class = "line" > < a name = "l00184" > < / a > < span class = "lineno" > 184< / span >   < span class = "preprocessor" > #endif< / span > < / div >
< div class = "line" > < a name = "l00185" > < / a > < span class = "lineno" > 185< / span >   < span class = "keywordflow" > for< / span > (j = 0; j < M; j++) Q[j][i] = tmp_vector[j] / mag;< / div >
< div class = "line" > < a name = "l00186" > < / a > < span class = "lineno" > 186< / span >   < / div >
< div class = "line" > < a name = "l00187" > < / a > < span class = "lineno" > 187< / span >   < span class = "comment" > /* compute upper triangular values of R */< / span > < / div >
< div class = "line" > < a name = "l00188" > < / a > < span class = "lineno" > 188< / span >   < span class = "keywordflow" > for< / span > (< span class = "keywordtype" > int< / span > kk = 0; kk < M; kk++) col_vector[kk] = Q[kk][i];< / div >
2021-02-06 03:52:57 +08:00
< div class = "line" > < a name = "l00189" > < / a > < span class = "lineno" > 189< / span >   < span class = "keywordflow" > for< / span > (< span class = "keywordtype" > int< / span > k = i; k < N; k++)< / div >
2020-06-28 23:30:37 +08:00
< div class = "line" > < a name = "l00190" > < / a > < span class = "lineno" > 190< / span >   {< / div >
< div class = "line" > < a name = "l00191" > < / a > < span class = "lineno" > 191< / span >   < span class = "keywordflow" > for< / span > (< span class = "keywordtype" > int< / span > kk = 0; kk < M; kk++) col_vector2[kk] = A[kk][k];< / div >
< div class = "line" > < a name = "l00192" > < / a > < span class = "lineno" > 192< / span >   R[i][k] = < a class = "code" href = "../../d4/d68/qr__decompose_8h.html#a3a584b79941a43d775f9d4ce446dbe05" > vector_dot< / a > (col_vector, col_vector2, M);< / div >
< div class = "line" > < a name = "l00193" > < / a > < span class = "lineno" > 193< / span >   }< / div >
< div class = "line" > < a name = "l00194" > < / a > < span class = "lineno" > 194< / span >   }< / div >
< div class = "line" > < a name = "l00195" > < / a > < span class = "lineno" > 195< / span >   < / div >
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< div class = "line" > < a name = "l00196" > < / a > < span class = "lineno" > 196< / span >   < a class = "code" href = "../../d2/ddd/malloc__dbg_8h.html#a9cc854374299a1dd933bf62029761768" > free< / a > (col_vector);< / div >
< div class = "line" > < a name = "l00197" > < / a > < span class = "lineno" > 197< / span >   < a class = "code" href = "../../d2/ddd/malloc__dbg_8h.html#a9cc854374299a1dd933bf62029761768" > free< / a > (col_vector2);< / div >
< div class = "line" > < a name = "l00198" > < / a > < span class = "lineno" > 198< / span >   < a class = "code" href = "../../d2/ddd/malloc__dbg_8h.html#a9cc854374299a1dd933bf62029761768" > free< / a > (tmp_vector);< / div >
2020-06-28 23:30:37 +08:00
< div class = "line" > < a name = "l00199" > < / a > < span class = "lineno" > 199< / span >   }< / div >
< div class = "line" > < a name = "l00200" > < / a > < span class = "lineno" > 200< / span >   < / div >
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< div class = "line" > < a name = "l00201" > < / a > < span class = "lineno" > 201< / span >   < span class = "preprocessor" > #endif < / span > < span class = "comment" > // QR_DECOMPOSE_H< / span > < / div >
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< div class = "ttc" id = "amalloc__dbg_8h_html_a725f50ecaf1959d96de79b36b4788fee" > < div class = "ttname" > < a href = "../../d2/ddd/malloc__dbg_8h.html#a725f50ecaf1959d96de79b36b4788fee" > malloc< / a > < / div > < div class = "ttdeci" > #define malloc(bytes)< / div > < div class = "ttdoc" > This macro replace the standard malloc function with malloc_dbg.< / div > < div class = "ttdef" > < b > Definition:< / b > malloc_dbg.h:18< / div > < / div >
< div class = "ttc" id = "amalloc__dbg_8h_html_a9cc854374299a1dd933bf62029761768" > < div class = "ttname" > < a href = "../../d2/ddd/malloc__dbg_8h.html#a9cc854374299a1dd933bf62029761768" > free< / a > < / div > < div class = "ttdeci" > #define free(ptr)< / div > < div class = "ttdoc" > This macro replace the standard free function with free_dbg.< / div > < div class = "ttdef" > < b > Definition:< / b > malloc_dbg.h:26< / div > < / div >
2020-06-29 04:40:24 +08:00
< div class = "ttc" id = "aqr__decompose_8h_html_a3a584b79941a43d775f9d4ce446dbe05" > < div class = "ttname" > < a href = "../../d4/d68/qr__decompose_8h.html#a3a584b79941a43d775f9d4ce446dbe05" > vector_dot< / a > < / div > < div class = "ttdeci" > double vector_dot(double *a, double *b, int L)< / div > < div class = "ttdoc" > Compute dot product of two vectors of equal lengths.< / div > < div class = "ttdef" > < b > Definition:< / b > qr_decompose.h:43< / div > < / div >
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< div class = "ttc" id = "aqr__decompose_8h_html_a45c7640d9d22c89c11beb1f567843c56" > < div class = "ttname" > < a href = "../../d4/d68/qr__decompose_8h.html#a45c7640d9d22c89c11beb1f567843c56" > qr_decompose< / a > < / div > < div class = "ttdeci" > void qr_decompose(double **A, double **Q, double **R, int M, int N)< / div > < div class = "ttdoc" > Decompose matrix using Gram-Schmidt process.< / div > < div class = "ttdef" > < b > Definition:< / b > qr_decompose.h:142< / div > < / div >
2020-06-29 04:40:24 +08:00
< div class = "ttc" id = "aqr__decompose_8h_html_a6b6a0e75e75ff7919057dd275bb69145" > < div class = "ttname" > < a href = "../../d4/d68/qr__decompose_8h.html#a6b6a0e75e75ff7919057dd275bb69145" > vector_sub< / a > < / div > < div class = "ttdeci" > double * vector_sub(double *a, double *b, double *out, int L)< / div > < div class = "ttdoc" > Compute vector subtraction.< / div > < div class = "ttdef" > < b > Definition:< / b > qr_decompose.h:101< / div > < / div >
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< div class = "ttc" id = "aqr__decompose_8h_html_a82b20e027437df768d7e994cf4cae29f" > < div class = "ttname" > < a href = "../../d4/d68/qr__decompose_8h.html#a82b20e027437df768d7e994cf4cae29f" > vector_proj< / a > < / div > < div class = "ttdeci" > double * vector_proj(double *a, double *b, double *out, int L)< / div > < div class = "ttdoc" > Compute projection of vector on defined as.< / div > < div class = "ttdef" > < b > Definition:< / b > qr_decompose.h:76< / div > < / div >
< div class = "ttc" id = "aqr__decompose_8h_html_a90562ce8c3707401e9c5809dece68d6a" > < div class = "ttname" > < a href = "../../d4/d68/qr__decompose_8h.html#a90562ce8c3707401e9c5809dece68d6a" > print_matrix< / a > < / div > < div class = "ttdeci" > void print_matrix(double **A, int M, int N)< / div > < div class = "ttdoc" > function to display matrix on stdout< / div > < div class = "ttdef" > < b > Definition:< / b > qr_decompose.h:22< / div > < / div >
< div class = "ttc" id = "aqr__decompose_8h_html_abeec1f78a7a7e7251687e75340331212" > < div class = "ttname" > < a href = "../../d4/d68/qr__decompose_8h.html#abeec1f78a7a7e7251687e75340331212" > vector_mag< / a > < / div > < div class = "ttdeci" > double vector_mag(double *vector, int L)< / div > < div class = "ttdoc" > Compute magnitude of vector.< / div > < div class = "ttdef" > < b > Definition:< / b > qr_decompose.h:64< / div > < / div >
< div class = "ttc" id = "astruct_l_html" > < div class = "ttname" > < a href = "../../df/db3/struct_l.html" > L< / a > < / div > < div class = "ttdef" > < b > Definition:< / b > list.h:8< / div > < / div >
< / div > <!-- fragment --> < / div > <!-- contents -->
< / div > <!-- doc - content -->
2020-06-28 23:07:29 +08:00
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< li class = "navelem" > < a class = "el" href = "../../dir_9c6faab82c22511b50177aa2e38e2780.html" > numerical_methods< / a > < / li > < li class = "navelem" > < a class = "el" href = "../../d4/d68/qr__decompose_8h.html" > qr_decompose.h< / a > < / li >
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