From d21a7de801b4a326001e45c0d26826e9ab53589b Mon Sep 17 00:00:00 2001 From: Elizabeth Hunt Date: Fri, 13 Oct 2023 21:00:07 -0600 Subject: [PATCH] hw 4 --- .gitignore | 1 + Makefile | 4 +- doc/software_manual.org | 727 ++++++++++++++++++++++++++ doc/software_manual.pdf | Bin 0 -> 129228 bytes doc/software_manual.tex | 867 +++++++++++++++++++++++++++++++ homeworks/hw-4.org | 42 +- homeworks/hw-4.pdf | Bin 0 -> 215081 bytes homeworks/hw-4.tex | 70 +++ homeworks/img/make_run.png | Bin 0 -> 73679 bytes homeworks/img/test_routine_1.png | Bin 0 -> 217677 bytes homeworks/img/test_routine_2.png | Bin 0 -> 145961 bytes inc/lizfcm.h | 16 +- lib/lizfcm.a | Bin 15472 -> 0 bytes src/lin.c | 4 +- src/matrix.c | 37 +- src/vector.c | 84 +-- test/main.c | 31 +- 17 files changed, 1805 insertions(+), 78 deletions(-) create mode 100644 doc/software_manual.org create mode 100644 doc/software_manual.pdf create mode 100644 doc/software_manual.tex create mode 100644 homeworks/hw-4.pdf create mode 100644 homeworks/hw-4.tex create mode 100644 homeworks/img/make_run.png create mode 100644 homeworks/img/test_routine_1.png create mode 100644 homeworks/img/test_routine_2.png delete mode 100644 lib/lizfcm.a diff --git a/.gitignore b/.gitignore index 9d0b71a..8f0345c 100644 --- a/.gitignore +++ b/.gitignore @@ -1,2 +1,3 @@ build dist +lib diff --git a/Makefile b/Makefile index 176e8cd..aed2e52 100644 --- a/Makefile +++ b/Makefile @@ -21,7 +21,7 @@ all: $(TEST_EXE) $(TEST_EXE): $(BIN_DIR) | $(LIBRARY) $(CC) $(CPPFLAGS) $(CFLAGS) $(LDFLAGS) $(TEST_SRC) $(LIBRARY) -o $@ -$(LIBRARY): $(OBJ) +$(LIBRARY): $(OBJ) | $(LIB_DIR) ar rcs $(LIBRARY) $(OBJ_DIR)/*.o ranlib $(LIBRARY) @@ -32,6 +32,6 @@ $(BIN_DIR) $(OBJ_DIR) $(LIB_DIR): mkdir -p $@ clean: - @$(RM) -r $(BIN_DIR) $(OBJ_DIR) + @$(RM) -r $(BIN_DIR) $(OBJ_DIR) $(LIB_DIR) -include $(OBJ:.o=.d) diff --git a/doc/software_manual.org b/doc/software_manual.org new file mode 100644 index 0000000..08a554e --- /dev/null +++ b/doc/software_manual.org @@ -0,0 +1,727 @@ +#+TITLE: LIZFCM Software Manual (v0.1) +#+AUTHOR: Elizabeth Hunt +#+LATEX_HEADER: \notindent \notag \usepackage{amsmath} \usepackage[a4paper,margin=1in,portrait]{geometry} +#+LATEX: \setlength\parindent{0pt} + +* Design +The LIZFCM static library (at [[https://github.com/Simponic/math-4610][[https://github.com/Simponic/math-4610]] is a successor to my +attempt at writing codes for the Fundamentals of Computational Mathematics course in Common +Lisp, but the effort required to meet the requirement of creating a static library became +too difficult to integrate outside of the ~ASDF~ solution that Common Lisp already brings +to the table. + +All of the work established in ~deprecated-cl~ has been painstakingly translated into +the C programming language. I have a couple tenets for its design: + ++ Implemntations of routines should all be done immutably in respect to arguments. ++ Functional programming is good (it's... rough in C though). ++ Routines are separated into "module" c files, and not individual files per function. + +* Compilation +A provided ~Makefile~ is added for convencience. It has been tested on an M1 machine running MacOS as +well as Arch Linux. + +1. ~cd~ into the root of the repo +2. ~make~ + +Then, as of homework 4, the testing routine in ~test/main.c~ can be run via +~./dist/lizfcm.test~. + +Execution of the Makefile will perform compilation of individual routines. + +But, in the requirement of manual intervention (should the little alien workers +inside the computer fail to do their job), one can use the following command to +produce an object file: + +\begin{verbatim} + gcc -Iinc/ -lm -Wall -c src/.c -o build/.o +\end{verbatim} + +Which is then bundled into a static library in ~lib/lizfcm.a~ which can be linked +in the standard method. + +* The LIZFCM API +** Simple Routines +*** ~smaceps~ ++ Author: Elizabeth Hunt ++ Name: ~smaceps~ ++ Location: ~src/maceps.c~ ++ Input: none ++ Output: a ~float~ returning the specific "Machine Epsilon" of a machine on a + single precision floating point number at which it becomes "indistinguishable". + +#+BEGIN_SRC c +float smaceps() { + float one = 1.0; + float machine_epsilon = 1.0; + float one_approx = one + machine_epsilon; + + while (fabsf(one_approx - one) > 0) { + machine_epsilon /= 2; + one_approx = one + machine_epsilon; + } + + return machine_epsilon; +} +#+END_SRC + +*** ~dmaceps~ ++ Author: Elizabeth Hunt ++ Name: ~dmaceps~ ++ Location: ~src/maceps.c~ ++ Input: none ++ Output: a ~double~ returning the specific "Machine Epsilon" of a machine on a + double precision floating point number at which it becomes "indistinguishable". + +#+BEGIN_SRC c +double dmaceps() { + double one = 1.0; + double machine_epsilon = 1.0; + double one_approx = one + machine_epsilon; + + while (fabs(one_approx - one) > 0) { + machine_epsilon /= 2; + one_approx = one + machine_epsilon; + } + + return machine_epsilon; +} +#+END_SRC + +** Derivative Routines +*** ~central_derivative_at~ ++ Author: Elizabeth Hunt ++ Name: ~central_derivative_at~ ++ Location: ~src/approx_derivative.c~ ++ Input: + - ~f~ is a pointer to a one-ary function that takes a double as input and produces + a double as output + - ~a~ is the domain value at which we approximate ~f'~ + - ~h~ is the step size ++ Output: a ~double~ of the approximate value of ~f'(a)~ via the central difference + method. + +#+BEGIN_SRC c +double central_derivative_at(double (*f)(double), double a, double h) { + assert(h > 0); + + double x2 = a + h; + double x1 = a - h; + + double y2 = (*f)(x2); + double y1 = (*f)(x1); + + return (y2 - y1) / (x2 - x1); +} +#+END_SRC + +*** ~forward_derivative_at~ ++ Author: Elizabeth Hunt ++ Name: ~forward_derivative_at~ ++ Location: ~src/approx_derivative.c~ ++ Input: + - ~f~ is a pointer to a one-ary function that takes a double as input and produces + a double as output + - ~a~ is the domain value at which we approximate ~f'~ + - ~h~ is the step size ++ Output: a ~double~ of the approximate value of ~f'(a)~ via the forward difference + method. + +#+BEGIN_SRC c +double forward_derivative_at(double (*f)(double), double a, double h) { + assert(h > 0); + + double x2 = a + h; + double x1 = a; + + double y2 = (*f)(x2); + double y1 = (*f)(x1); + + return (y2 - y1) / (x2 - x1); +} +#+END_SRC + +*** ~backward_derivative_at~ ++ Author: Elizabeth Hunt ++ Name: ~backward_derivative_at~ ++ Location: ~src/approx_derivative.c~ ++ Input: + - ~f~ is a pointer to a one-ary function that takes a double as input and produces + a double as output + - ~a~ is the domain value at which we approximate ~f'~ + - ~h~ is the step size ++ Output: a ~double~ of the approximate value of ~f'(a)~ via the backward difference + method. + +#+BEGIN_SRC c +double backward_derivative_at(double (*f)(double), double a, double h) { + assert(h > 0); + + double x2 = a; + double x1 = a - h; + + double y2 = (*f)(x2); + double y1 = (*f)(x1); + + return (y2 - y1) / (x2 - x1); +} +#+END_SRC + +** Vector Routines +*** Vector Arithmetic: ~add_v, minus_v~ ++ Author: Elizabeth Hunt ++ Name(s): ~add_v~, ~minus_v~ ++ Location: ~src/vector.c~ ++ Input: two pointers to locations in memory wherein ~Array_double~'s lie ++ Output: a pointer to a new ~Array_double~ as the result of addition or subtraction + of the two input ~Array_double~'s + +#+BEGIN_SRC c +Array_double *add_v(Array_double *v1, Array_double *v2) { + assert(v1->size == v2->size); + + Array_double *sum = copy_vector(v1); + for (size_t i = 0; i < v1->size; i++) + sum->data[i] += v2->data[i]; + return sum; +} + +Array_double *minus_v(Array_double *v1, Array_double *v2) { + assert(v1->size == v2->size); + + Array_double *sub = InitArrayWithSize(double, v1->size, 0); + for (size_t i = 0; i < v1->size; i++) + sub->data[i] = v1->data[i] - v2->data[i]; + return sub; +} +#+END_SRC + +*** Norms: ~l1_norm~, ~l2_norm~, ~linf_norm~ ++ Author: Elizabeth Hunt ++ Name(s): ~l1_norm~, ~l2_norm~, ~linf_norm~ ++ Location: ~src/vector.c~ ++ Input: a pointer to a location in memory wherein an ~Array_double~ lies ++ Output: a ~double~ representing the value of the norm the function applies + +#+BEGIN_SRC c +double l1_norm(Array_double *v) { + double sum = 0; + for (size_t i = 0; i < v->size; ++i) + sum += fabs(v->data[i]); + return sum; +} + +double l2_norm(Array_double *v) { + double norm = 0; + for (size_t i = 0; i < v->size; ++i) + norm += v->data[i] * v->data[i]; + return sqrt(norm); +} + +double linf_norm(Array_double *v) { + assert(v->size > 0); + double max = v->data[0]; + for (size_t i = 0; i < v->size; ++i) + max = c_max(v->data[i], max); + return max; +} +#+END_SRC + +*** ~vector_distance~ ++ Author: Elizabeth Hunt ++ Name: ~vector_distance~ ++ Location: ~src/vector.c~ ++ Input: two pointers to locations in memory wherein ~Array_double~'s lie, and a pointer to a + one-ary function ~norm~ taking as input a pointer to an ~Array_double~ and returning a double + representing the norm of that ~Array_double~ + +#+BEGIN_SRC c +double vector_distance(Array_double *v1, Array_double *v2, + double (*norm)(Array_double *)) { + Array_double *minus = minus_v(v1, v2); + double dist = (*norm)(minus); + free(minus); + return dist; +} +#+END_SRC + +*** Distances: ~l1_distance~, ~l2_distance~, ~linf_distance~ ++ Author: Elizabeth Hunt ++ Name(s): ~l1_distance~, ~l2_distance~, ~linf_distance~ ++ Location: ~src/vector.c~ ++ Input: two pointers to locations in memory wherein ~Array_double~'s lie, and the distance + via the corresponding ~l1~, ~l2~, or ~linf~ norms ++ Output: A ~double~ representing the distance between the two ~Array_doubles~'s by the given + norm. + +#+BEGIN_SRC c +double l1_distance(Array_double *v1, Array_double *v2) { + return vector_distance(v1, v2, &l1_norm); +} + +double l2_distance(Array_double *v1, Array_double *v2) { + return vector_distance(v1, v2, &l2_norm); +} + +double linf_distance(Array_double *v1, Array_double *v2) { + return vector_distance(v1, v2, &linf_norm); +} +#+END_SRC + +*** ~sum_v~ ++ Author: Elizabeth Hunt ++ Name: ~sum_v~ ++ Location: ~src/vector.c~ ++ Input: a pointer to an ~Array_double~ ++ Output: a ~double~ representing the sum of all the elements of an ~Array_double~ + +#+BEGIN_SRC c +double sum_v(Array_double *v) { + double sum = 0; + for (size_t i = 0; i < v->size; i++) + sum += v->data[i]; + return sum; +} +#+END_SRC + + +*** ~scale_v~ ++ Author: Elizabeth Hunt ++ Name: ~scale_v~ ++ Location: ~src/vector.c~ ++ Input: a pointer to an ~Array_double~ and a scalar ~double~ to scale the vector ++ Output: a pointer to a new ~Array_double~ of the scaled input ~Array_double~ + +#+BEGIN_SRC c +Array_double *scale_v(Array_double *v, double m) { + Array_double *copy = copy_vector(v); + for (size_t i = 0; i < v->size; i++) + copy->data[i] *= m; + return copy; +} +#+END_SRC + +*** ~free_vector~ ++ Author: Elizabeth Hunt ++ Name: ~free_vector~ ++ Location: ~src/vector.c~ ++ Input: a pointer to an ~Array_double~ ++ Output: nothing. ++ Side effect: free the memory of the reserved ~Array_double~ on the heap + +#+BEGIN_SRC c +void free_vector(Array_double *v) { + free(v->data); + free(v); +} +#+END_SRC + +*** ~copy_vector~ ++ Author: Elizabeth Hunt ++ Name: ~copy_vector~ ++ Location: ~src/vector.c~ ++ Input: a pointer to an ~Array_double~ ++ Output: a pointer to a new ~Array_double~ whose ~data~ and ~size~ are copied from the input + ~Array_double~ + +#+BEGIN_SRC c +Array_double *copy_vector(Array_double *v) { + Array_double *copy = InitArrayWithSize(double, v->size, 0.0); + for (size_t i = 0; i < copy->size; ++i) + copy->data[i] = v->data[i]; + return copy; +} +#+END_SRC + +*** ~format_vector_into~ ++ Author: Elizabeth Hunt ++ Name: ~format_vector_into~ ++ Location: ~src/vector.c~ ++ Input: a pointer to an ~Array_double~ and a pointer to a c-string ~s~ to "print" the vector out + into ++ Output: nothing. ++ Side effect: overwritten memory into ~s~ + +#+BEGIN_SRC c +void format_vector_into(Array_double *v, char *s) { + if (v->size == 0) { + strcat(s, "empty"); + return; + } + + for (size_t i = 0; i < v->size; ++i) { + char num[64]; + strcpy(num, ""); + + sprintf(num, "%f,", v->data[i]); + strcat(s, num); + } + strcat(s, "\n"); +} +#+END_SRC + +** Matrix Routines +*** ~lu_decomp~ ++ Author: Elizabeth Hunt ++ Name: ~lu_decomp~ ++ Location: ~src/matrix.c~ ++ Input: a pointer to a ~Matrix_double~ $m$ to decompose into a lower triangular and upper triangular + matrix $L$, $U$, respectively such that $LU = m$. ++ Output: a pointer to the location in memory in which two ~Matrix_double~'s reside: the first + representing $L$, the second, $U$. ++ Errors: Exits and throws a status code of ~-1~ when encountering a matrix that cannot be + decomposed + +#+BEGIN_SRC c +Matrix_double **lu_decomp(Matrix_double *m) { + assert(m->cols == m->rows); + + Matrix_double *u = copy_matrix(m); + Matrix_double *l = InitMatrixWithSize(double, m->rows, m->cols, 0.0); + put_identity_diagonal(l); + + Matrix_double **u_l = malloc(sizeof(Matrix_double *) * 2); + + for (size_t y = 0; y < m->rows; y++) { + if (u->data[y]->data[y] == 0) { + printf("ERROR: a pivot is zero in given matrix\n"); + exit(-1); + } + } + + if (u && l) { + for (size_t x = 0; x < m->cols; x++) { + for (size_t y = x + 1; y < m->rows; y++) { + double denom = u->data[x]->data[x]; + + if (denom == 0) { + printf("ERROR: non-factorable matrix\n"); + exit(-1); + } + + double factor = -(u->data[y]->data[x] / denom); + + Array_double *scaled = scale_v(u->data[x], factor); + Array_double *added = add_v(scaled, u->data[y]); + free_vector(scaled); + free_vector(u->data[y]); + + u->data[y] = added; + l->data[y]->data[x] = -factor; + } + } + } + + u_l[0] = u; + u_l[1] = l; + return u_l; +} +#+END_SRC +*** ~bsubst~ ++ Author: Elizabeth Hunt ++ Name: ~bsubst~ ++ Location: ~src/matrix.c~ ++ Input: a pointer to an upper-triangular ~Matrix_double~ $u$ and a ~Array_double~ + $b$ ++ Output: a pointer to a new ~Array_double~ whose entries are given by performing + back substitution + +#+BEGIN_SRC c +Array_double *bsubst(Matrix_double *u, Array_double *b) { + assert(u->rows == b->size && u->cols == u->rows); + + Array_double *x = copy_vector(b); + for (int64_t row = b->size - 1; row >= 0; row--) { + for (size_t col = b->size - 1; col > row; col--) + x->data[row] -= x->data[col] * u->data[row]->data[col]; + x->data[row] /= u->data[row]->data[row]; + } + return x; +} +#+END_SRC +*** ~fsubst~ ++ Author: Elizabeth Hunt ++ Name: ~fsubst~ ++ Location: ~src/matrix.c~ ++ Input: a pointer to a lower-triangular ~Matrix_double~ $l$ and a ~Array_double~ + $b$ ++ Output: a pointer to a new ~Array_double~ whose entries are given by performing + forward substitution + +#+BEGIN_SRC c +Array_double *fsubst(Matrix_double *l, Array_double *b) { + assert(l->rows == b->size && l->cols == l->rows); + + Array_double *x = copy_vector(b); + + for (size_t row = 0; row < b->size; row++) { + for (size_t col = 0; col < row; col++) + x->data[row] -= x->data[col] * l->data[row]->data[col]; + x->data[row] /= l->data[row]->data[row]; + } + + return x; +} +#+END_SRC + +*** ~solve_matrix~ ++ Author: Elizabeth Hunt ++ Location: ~src/matrix.c~ ++ Input: a pointer to a ~Matrix_double~ $m$ and a pointer to an ~Array_double~ $b$ ++ Output: $x$ such that $mx = b$ if such a solution exists (else it's non LU-factorable as discussed + above) + +Here we make use of forward substitution to first solve $Ly = b$ given $L$ as the $L$ factor in +~lu_decomp~. Then we use back substitution to solve $Ux = y$ for $x$ similarly given $U$. + +Then, $LUx = b$, thus $x$ is a solution. + +#+BEGIN_SRC c +Array_double *solve_matrix(Matrix_double *m, Array_double *b) { + assert(b->size == m->rows); + assert(m->rows == m->cols); + + Array_double *x = copy_vector(b); + Matrix_double **u_l = lu_decomp(m); + Matrix_double *u = u_l[0]; + Matrix_double *l = u_l[1]; + + Array_double *b_fsub = fsubst(l, b); + x = bsubst(u, b_fsub); + free_vector(b_fsub); + + free_matrix(u); + free_matrix(l); + + return x; +} +#+END_SRC + +*** ~m_dot_v~ ++ Author: Elizabeth Hunt ++ Location: ~src/matrix.c~ ++ Input: a pointer to a ~Matrix_double~ $m$ and ~Array_double~ $v$ ++ Output: the dot product $mv$ as an ~Array_double~ + +#+BEGIN_SRC c +Array_double *m_dot_v(Matrix_double *m, Array_double *v) { + assert(v->size == m->cols); + + Array_double *product = copy_vector(v); + + for (size_t row = 0; row < v->size; ++row) + product->data[row] = v_dot_v(m->data[row], v); + + return product; +} +#+END_SRC + +*** ~put_identity_diagonal~ ++ Author: Elizabeth Hunt ++ Location: ~src/matrix.c~ ++ Input: a pointer to a ~Matrix_double~ ++ Output: a pointer to a copy to ~Matrix_double~ whose diagonal is full of 1's + +#+BEGIN_SRC c +Matrix_double *put_identity_diagonal(Matrix_double *m) { + assert(m->rows == m->cols); + Matrix_double *copy = copy_matrix(m); + for (size_t y = 0; y < m->rows; ++y) + copy->data[y]->data[y] = 1.0; + return copy; +} +#+END_SRC + +*** ~copy_matrix~ ++ Author: Elizabeth Hunt ++ Location: ~src/matrix.c~ ++ Input: a pointer to a ~Matrix_double~ ++ Output: a pointer to a copy of the given ~Matrix_double~ + +#+BEGIN_SRC c +Matrix_double *copy_matrix(Matrix_double *m) { + Matrix_double *copy = InitMatrixWithSize(double, m->rows, m->cols, 0.0); + for (size_t y = 0; y < copy->rows; y++) { + free_vector(copy->data[y]); + copy->data[y] = copy_vector(m->data[y]); + } + return copy; +} +#+END_SRC + +*** ~free_matrix~ ++ Author: Elizabeth Hunt ++ Location: ~src/matrix.c~ ++ Input: a pointer to a ~Matrix_double~ ++ Output: none. ++ Side Effects: frees memory reserved by a given ~Matrix_double~ and its member + ~Array_double~ vectors describing its rows. + +#+BEGIN_SRC c +void free_matrix(Matrix_double *m) { + for (size_t y = 0; y < m->rows; ++y) + free_vector(m->data[y]); + free(m); +} +#+END_SRC + +*** ~format_matrix_into~ ++ Author: Elizabeth Hunt ++ Name: ~format_matrix_into~ ++ Location: ~src/matrix.c~ ++ Input: a pointer to a ~Matrix_double~ and a pointer to a c-string ~s~ to "print" the vector out + into ++ Output: nothing. ++ Side effect: overwritten memory into ~s~ + +#+BEGIN_SRC c +void format_matrix_into(Matrix_double *m, char *s) { + if (m->rows == 0) + strcpy(s, "empty"); + + for (size_t y = 0; y < m->rows; ++y) { + char row_s[256]; + strcpy(row_s, ""); + + format_vector_into(m->data[y], row_s); + strcat(s, row_s); + } + strcat(s, "\n"); +} +#+END_SRC +** Linear Routines +*** ~least_squares_lin_reg~ ++ Author: Elizabeth Hunt ++ Name: ~least_squares_lin_reg~ ++ Location: ~src/lin.c~ ++ Input: two pointers to ~Array_double~'s whose entries correspond two ordered + pairs in R^2 ++ Output: a linear model best representing the ordered pairs via least squares + regression + +#+BEGIN_SRC c +Line *least_squares_lin_reg(Array_double *x, Array_double *y) { + assert(x->size == y->size); + + uint64_t n = x->size; + double sum_x = sum_v(x); + double sum_y = sum_v(y); + double sum_xy = v_dot_v(x, y); + double sum_xx = v_dot_v(x, x); + double denom = ((n * sum_xx) - (sum_x * sum_x)); + + Line *line = malloc(sizeof(Line)); + line->m = ((sum_xy * n) - (sum_x * sum_y)) / denom; + line->a = ((sum_y * sum_xx) - (sum_x * sum_xy)) / denom; + + return line; +} +#+END_SRC +** Appendix / Miscellaneous +*** Data Types +**** ~Line~ ++ Author: Elizabeth Hunt ++ Location: ~inc/types.h~ + +#+BEGIN_SRC c +typedef struct Line { + double m; + double a; +} Line; +#+END_SRC +**** The ~Array_~ and ~Matrix_~ ++ Author: Elizabeth Hunt ++ Location: ~inc/types.h~ + +We define two Pre processor Macros ~DEFINE_ARRAY~ and ~DEFINE_MATRIX~ that take +as input a type, and construct a struct definition for the given type for +convenient access to the vector or matrices dimensions. + +Such that ~DEFINE_ARRAY(int)~ would expand to: + +#+BEGIN_SRC c + typedef struct { + int* data; + size_t size; + } Array_int +#+END_SRC + +And ~DEFINE_MATRIX(int)~ would expand a to ~Matrix_int~; containing a pointer to +a collection of pointers of ~Array_int~'s and its dimensions. + +#+BEGIN_SRC c + typedef struct { + Array_int **data; + size_t cols; + size_t rows; + } Matrix_int +#+END_SRC + +*** Macros +**** ~c_max~ and ~c_min~ ++ Author: Elizabeth Hunt ++ Location: ~inc/macros.h~ ++ Input: two structures that define an order measure ++ Output: either the larger or smaller of the two depending on the measure + +#+BEGIN_SRC c +#define c_max(x, y) (((x) >= (y)) ? (x) : (y)) +#define c_min(x, y) (((x) <= (y)) ? (x) : (y)) +#+END_SRC + +**** ~InitArray~ ++ Author: Elizabeth Hunt ++ Location: ~inc/macros.h~ ++ Input: a type and array of values to initialze an array with such type ++ Output: a new ~Array_type~ with the size of the given array and its data + +#+BEGIN_SRC c +#define InitArray(TYPE, ...) \ + ({ \ + TYPE temp[] = __VA_ARGS__; \ + Array_##TYPE *arr = malloc(sizeof(Array_##TYPE)); \ + arr->size = sizeof(temp) / sizeof(temp[0]); \ + arr->data = malloc(arr->size * sizeof(TYPE)); \ + memcpy(arr->data, temp, arr->size * sizeof(TYPE)); \ + arr; \ + }) +#+END_SRC + +**** ~InitArrayWithSize~ ++ Author: Elizabeth Hunt ++ Location: ~inc/macros.h~ ++ Input: a type, a size, and initial value ++ Output: a new ~Array_type~ with the given size filled with the initial value + +#+BEGIN_SRC c +#define InitArrayWithSize(TYPE, SIZE, INIT_VALUE) \ + ({ \ + Array_##TYPE *arr = malloc(sizeof(Array_##TYPE)); \ + arr->size = SIZE; \ + arr->data = malloc(arr->size * sizeof(TYPE)); \ + for (size_t i = 0; i < arr->size; i++) \ + arr->data[i] = INIT_VALUE; \ + arr; \ + }) +#+END_SRC + +**** ~InitMatrixWithSize~ ++ Author: Elizabeth Hunt ++ Location: ~inc/macros.h~ ++ Input: a type, number of rows, columns, and initial value ++ Output: a new ~Matrix_type~ of size ~rows x columns~ filled with the initial + value + +#+BEGIN_SRC c +#define InitMatrixWithSize(TYPE, ROWS, COLS, INIT_VALUE) \ + ({ \ + Matrix_##TYPE *matrix = malloc(sizeof(Matrix_##TYPE)); \ + matrix->rows = ROWS; \ + matrix->cols = COLS; \ + matrix->data = malloc(matrix->rows * sizeof(Array_##TYPE *)); \ + for (size_t y = 0; y < matrix->rows; y++) \ + matrix->data[y] = InitArrayWithSize(TYPE, COLS, INIT_VALUE); \ + matrix; \ + }) +#+END_SRc + diff --git a/doc/software_manual.pdf b/doc/software_manual.pdf new file mode 100644 index 0000000000000000000000000000000000000000..7dffa58df3c3b1bbe6dd4356862969c56104bfc4 GIT binary patch literal 129228 zcmb5UQ;=xgwyl}AGgsQSZQHhO+qP}nwr$&5X`6SSTjy4tsQpLOuK6?`=EI0NqkpZp 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+\usepackage{amssymb} +\usepackage{capt-of} +\usepackage{hyperref} +\notindent \notag \usepackage{amsmath} \usepackage[a4paper,margin=1in,portrait]{geometry} +\author{Elizabeth Hunt} +\date{\today} +\title{LIZFCM Software Manual (v0.1)} +\hypersetup{ + pdfauthor={Elizabeth Hunt}, + pdftitle={LIZFCM Software Manual (v0.1)}, + pdfkeywords={}, + pdfsubject={}, + pdfcreator={Emacs 28.2 (Org mode 9.7-pre)}, + pdflang={English}} +\begin{document} + +\maketitle +\tableofcontents + +\setlength\parindent{0pt} + +\section{Design} +\label{sec:org23cc15b} +The LIZFCM static library is a successor to my attempt at writing codes for the +Fundamentals of Computational Mathematics course in Common Lisp, but the effort required +to meet the requirement of creating a static library became too difficult to integrate +outside of the \texttt{ASDF} solution that Common Lisp already brings to the table. + +All of the work established in \texttt{deprecated-cl} has been painstakingly translated into +the C programming language. I have a couple tenets for its design: + +\begin{itemize} +\item Implemntations of routines should all be done immutably in respect to arguments. +\item Functional programming is good (it's\ldots{} rough in this language though). +\end{itemize} + +\section{Compilation} +\label{sec:orgc704fb9} +A provided \texttt{Makefile} is added for convencience. It has been tested on an M1 machine running MacOS as +well as Arch Linux. + +\begin{enumerate} +\item \texttt{cd} into the root of the repo +\item \texttt{make} +\end{enumerate} + +Then, as of homework 4, the testing routine in \texttt{test/main.c} can be run via +\texttt{./dist/lizfcm.test}. + +Execution of the Makefile will perform compilation of individual routines. + +But, in the requirement of manual intervention (should the little alien workers +inside the computer fail to do their job), one can use the following command to +produce an object file: + +\begin{verbatim} + gcc -Iinc/ -lm -Wall -c src/.c -o build/.o +\end{verbatim} + +Which is then bundled into a static library in \texttt{lib/lizfcm.a} which can be linked +in the standard method. + +\section{The LIZFCM API} +\label{sec:org01b31c2} +\subsection{Simple Routines} +\label{sec:org5355145} +\subsubsection{\texttt{smaceps}} +\label{sec:org4ed063f} +\begin{itemize} +\item Author: Elizabeth Hunt +\item Name: \texttt{smaceps} +\item Location: \texttt{src/maceps.c} +\item Input: none +\item Output: a \texttt{float} returning the specific "Machine Epsilon" of a machine on a +single precision floating point number at which it becomes "indistinguishable". +\end{itemize} + +\begin{verbatim} +float smaceps() { + float one = 1.0; + float machine_epsilon = 1.0; + float one_approx = one + machine_epsilon; + + while (fabsf(one_approx - one) > 0) { + machine_epsilon /= 2; + one_approx = one + machine_epsilon; + } + + return machine_epsilon; +} +\end{verbatim} + +\subsubsection{\texttt{dmaceps}} +\label{sec:orgcd0dfff} +\begin{itemize} +\item Author: Elizabeth Hunt +\item Name: \texttt{dmaceps} +\item Location: \texttt{src/maceps.c} +\item Input: none +\item Output: a \texttt{double} returning the specific "Machine Epsilon" of a machine on a +double precision floating point number at which it becomes "indistinguishable". +\end{itemize} + +\begin{verbatim} +double dmaceps() { + double one = 1.0; + double machine_epsilon = 1.0; + double one_approx = one + machine_epsilon; + + while (fabs(one_approx - one) > 0) { + machine_epsilon /= 2; + one_approx = one + machine_epsilon; + } + + return machine_epsilon; +} +\end{verbatim} + +\subsection{Derivative Routines} +\label{sec:org8f54012} +\subsubsection{\texttt{central\_derivative\_at}} +\label{sec:org2c81fc1} +\begin{itemize} +\item Author: Elizabeth Hunt +\item Name: \texttt{central\_derivative\_at} +\item Location: \texttt{src/approx\_derivative.c} +\item Input: +\begin{itemize} +\item \texttt{f} is a pointer to a one-ary function that takes a double as input and produces +a double as output +\item \texttt{a} is the domain value at which we approximate \texttt{f'} +\item \texttt{h} is the step size +\end{itemize} +\item Output: a \texttt{double} of the approximate value of \texttt{f'(a)} via the central difference +method. +\end{itemize} + +\begin{verbatim} +double central_derivative_at(double (*f)(double), double a, double h) { + assert(h > 0); + + double x2 = a + h; + double x1 = a - h; + + double y2 = (*f)(x2); + double y1 = (*f)(x1); + + return (y2 - y1) / (x2 - x1); +} +\end{verbatim} + +\subsubsection{\texttt{forward\_derivative\_at}} +\label{sec:org149b09e} +\begin{itemize} +\item Author: Elizabeth Hunt +\item Name: \texttt{forward\_derivative\_at} +\item Location: \texttt{src/approx\_derivative.c} +\item Input: +\begin{itemize} +\item \texttt{f} is a pointer to a one-ary function that takes a double as input and produces +a double as output +\item \texttt{a} is the domain value at which we approximate \texttt{f'} +\item \texttt{h} is the step size +\end{itemize} +\item Output: a \texttt{double} of the approximate value of \texttt{f'(a)} via the forward difference +method. +\end{itemize} + +\begin{verbatim} +double forward_derivative_at(double (*f)(double), double a, double h) { + assert(h > 0); + + double x2 = a + h; + double x1 = a; + + double y2 = (*f)(x2); + double y1 = (*f)(x1); + + return (y2 - y1) / (x2 - x1); +} +\end{verbatim} + +\subsubsection{\texttt{backward\_derivative\_at}} +\label{sec:orgbaa2238} +\begin{itemize} +\item Author: Elizabeth Hunt +\item Name: \texttt{backward\_derivative\_at} +\item Location: \texttt{src/approx\_derivative.c} +\item Input: +\begin{itemize} +\item \texttt{f} is a pointer to a one-ary function that takes a double as input and produces +a double as output +\item \texttt{a} is the domain value at which we approximate \texttt{f'} +\item \texttt{h} is the step size +\end{itemize} +\item Output: a \texttt{double} of the approximate value of \texttt{f'(a)} via the backward difference +method. +\end{itemize} + +\begin{verbatim} +double backward_derivative_at(double (*f)(double), double a, double h) { + assert(h > 0); + + double x2 = a; + double x1 = a - h; + + double y2 = (*f)(x2); + double y1 = (*f)(x1); + + return (y2 - y1) / (x2 - x1); +} +\end{verbatim} + +\subsection{Vector Routines} +\label{sec:orgf9dd708} +\subsubsection{Vector Arithmetic: \texttt{add\_v, minus\_v}} +\label{sec:orgbb91d9d} +\begin{itemize} +\item Author: Elizabeth Hunt +\item Name(s): \texttt{add\_v}, \texttt{minus\_v} +\item Location: \texttt{src/vector.c} +\item Input: two pointers to locations in memory wherein \texttt{Array\_double}'s lie +\item Output: a pointer to a new \texttt{Array\_double} as the result of addition or subtraction +of the two input \texttt{Array\_double}'s +\end{itemize} + +\begin{verbatim} +Array_double *add_v(Array_double *v1, Array_double *v2) { + assert(v1->size == v2->size); + + Array_double *sum = copy_vector(v1); + for (size_t i = 0; i < v1->size; i++) + sum->data[i] += v2->data[i]; + return sum; +} + +Array_double *minus_v(Array_double *v1, Array_double *v2) { + assert(v1->size == v2->size); + + Array_double *sub = InitArrayWithSize(double, v1->size, 0); + for (size_t i = 0; i < v1->size; i++) + sub->data[i] = v1->data[i] - v2->data[i]; + return sub; +} +\end{verbatim} + +\subsubsection{Norms: \texttt{l1\_norm}, \texttt{l2\_norm}, \texttt{linf\_norm}} +\label{sec:org88bb4c5} +\begin{itemize} +\item Author: Elizabeth Hunt +\item Name(s): \texttt{l1\_norm}, \texttt{l2\_norm}, \texttt{linf\_norm} +\item Location: \texttt{src/vector.c} +\item Input: a pointer to a location in memory wherein an \texttt{Array\_double} lies +\item Output: a \texttt{double} representing the value of the norm the function applies +\end{itemize} + +\begin{verbatim} +double l1_norm(Array_double *v) { + double sum = 0; + for (size_t i = 0; i < v->size; ++i) + sum += fabs(v->data[i]); + return sum; +} + +double l2_norm(Array_double *v) { + double norm = 0; + for (size_t i = 0; i < v->size; ++i) + norm += v->data[i] * v->data[i]; + return sqrt(norm); +} + +double linf_norm(Array_double *v) { + assert(v->size > 0); + double max = v->data[0]; + for (size_t i = 0; i < v->size; ++i) + max = c_max(v->data[i], max); + return max; +} +\end{verbatim} + +\subsubsection{\texttt{vector\_distance}} +\label{sec:org4499de6} +\begin{itemize} +\item Author: Elizabeth Hunt +\item Name: \texttt{vector\_distance} +\item Location: \texttt{src/vector.c} +\item Input: two pointers to locations in memory wherein \texttt{Array\_double}'s lie, and a pointer to a +one-ary function \texttt{norm} taking as input a pointer to an \texttt{Array\_double} and returning a double +representing the norm of that \texttt{Array\_double} +\end{itemize} + +\begin{verbatim} +double vector_distance(Array_double *v1, Array_double *v2, + double (*norm)(Array_double *)) { + Array_double *minus = minus_v(v1, v2); + double dist = (*norm)(minus); + free(minus); + return dist; +} +\end{verbatim} + +\subsubsection{Distances: \texttt{l1\_distance}, \texttt{l2\_distance}, \texttt{linf\_distance}} +\label{sec:org1e61aea} +\begin{itemize} +\item Author: Elizabeth Hunt +\item Name(s): \texttt{l1\_distance}, \texttt{l2\_distance}, \texttt{linf\_distance} +\item Location: \texttt{src/vector.c} +\item Input: two pointers to locations in memory wherein \texttt{Array\_double}'s lie, and the distance +via the corresponding \texttt{l1}, \texttt{l2}, or \texttt{linf} norms +\item Output: A \texttt{double} representing the distance between the two \texttt{Array\_doubles}'s by the given +norm. +\end{itemize} + +\begin{verbatim} +double l1_distance(Array_double *v1, Array_double *v2) { + return vector_distance(v1, v2, &l1_norm); +} + +double l2_distance(Array_double *v1, Array_double *v2) { + return vector_distance(v1, v2, &l2_norm); +} + +double linf_distance(Array_double *v1, Array_double *v2) { + return vector_distance(v1, v2, &linf_norm); +} +\end{verbatim} + +\subsubsection{\texttt{sum\_v}} +\label{sec:org57e2591} +\begin{itemize} +\item Author: Elizabeth Hunt +\item Name: \texttt{sum\_v} +\item Location: \texttt{src/vector.c} +\item Input: a pointer to an \texttt{Array\_double} +\item Output: a \texttt{double} representing the sum of all the elements of an \texttt{Array\_double} +\end{itemize} + +\begin{verbatim} +double sum_v(Array_double *v) { + double sum = 0; + for (size_t i = 0; i < v->size; i++) + sum += v->data[i]; + return sum; +} +\end{verbatim} + + +\subsubsection{\texttt{scale\_v}} +\label{sec:org61b466a} +\begin{itemize} +\item Author: Elizabeth Hunt +\item Name: \texttt{scale\_v} +\item Location: \texttt{src/vector.c} +\item Input: a pointer to an \texttt{Array\_double} and a scalar \texttt{double} to scale the vector +\item Output: a pointer to a new \texttt{Array\_double} of the scaled input \texttt{Array\_double} +\end{itemize} + +\begin{verbatim} +Array_double *scale_v(Array_double *v, double m) { + Array_double *copy = copy_vector(v); + for (size_t i = 0; i < v->size; i++) + copy->data[i] *= m; + return copy; +} +\end{verbatim} + +\subsubsection{\texttt{free\_vector}} +\label{sec:org398d778} +\begin{itemize} +\item Author: Elizabeth Hunt +\item Name: \texttt{free\_vector} +\item Location: \texttt{src/vector.c} +\item Input: a pointer to an \texttt{Array\_double} +\item Output: nothing. +\item Side effect: free the memory of the reserved \texttt{Array\_double} on the heap +\end{itemize} + +\begin{verbatim} +void free_vector(Array_double *v) { + free(v->data); + free(v); +} +\end{verbatim} + +\subsubsection{\texttt{copy\_vector}} +\label{sec:orgf6b116b} +\begin{itemize} +\item Author: Elizabeth Hunt +\item Name: \texttt{copy\_vector} +\item Location: \texttt{src/vector.c} +\item Input: a pointer to an \texttt{Array\_double} +\item Output: a pointer to a new \texttt{Array\_double} whose \texttt{data} and \texttt{size} are copied from the input +\texttt{Array\_double} +\end{itemize} + +\begin{verbatim} +Array_double *copy_vector(Array_double *v) { + Array_double *copy = InitArrayWithSize(double, v->size, 0.0); + for (size_t i = 0; i < copy->size; ++i) + copy->data[i] = v->data[i]; + return copy; +} +\end{verbatim} + +\subsubsection{\texttt{format\_vector\_into}} +\label{sec:org595519d} +\begin{itemize} +\item Author: Elizabeth Hunt +\item Name: \texttt{format\_vector\_into} +\item Location: \texttt{src/vector.c} +\item Input: a pointer to an \texttt{Array\_double} and a pointer to a c-string \texttt{s} to "print" the vector out +into +\item Output: nothing. +\item Side effect: overwritten memory into \texttt{s} +\end{itemize} + +\begin{verbatim} +void format_vector_into(Array_double *v, char *s) { + if (v->size == 0) { + strcat(s, "empty"); + return; + } + + for (size_t i = 0; i < v->size; ++i) { + char num[64]; + strcpy(num, ""); + + sprintf(num, "%f,", v->data[i]); + strcat(s, num); + } + strcat(s, "\n"); +} +\end{verbatim} + +\subsection{Matrix Routines} +\label{sec:org53505d6} +\subsubsection{\texttt{lu\_decomp}} +\label{sec:org22ad28d} +\begin{itemize} +\item Author: Elizabeth Hunt +\item Name: \texttt{lu\_decomp} +\item Location: \texttt{src/matrix.c} +\item Input: a pointer to a \texttt{Matrix\_double} \(m\) to decompose into a lower triangular and upper triangular +matrix \(L\), \(U\), respectively such that \(LU = m\). +\item Output: a pointer to the location in memory in which two \texttt{Matrix\_double}'s reside: the first +representing \(L\), the second, \(U\). +\item Errors: Exits and throws a status code of \texttt{-1} when encountering a matrix that cannot be +decomposed +\end{itemize} + +\begin{verbatim} +Matrix_double **lu_decomp(Matrix_double *m) { + assert(m->cols == m->rows); + + Matrix_double *u = copy_matrix(m); + Matrix_double *l = InitMatrixWithSize(double, m->rows, m->cols, 0.0); + put_identity_diagonal(l); + + Matrix_double **u_l = malloc(sizeof(Matrix_double *) * 2); + + for (size_t y = 0; y < m->rows; y++) { + if (u->data[y]->data[y] == 0) { + printf("ERROR: a pivot is zero in given matrix\n"); + exit(-1); + } + } + + if (u && l) { + for (size_t x = 0; x < m->cols; x++) { + for (size_t y = x + 1; y < m->rows; y++) { + double denom = u->data[x]->data[x]; + + if (denom == 0) { + printf("ERROR: non-factorable matrix\n"); + exit(-1); + } + + double factor = -(u->data[y]->data[x] / denom); + + Array_double *scaled = scale_v(u->data[x], factor); + Array_double *added = add_v(scaled, u->data[y]); + free_vector(scaled); + free_vector(u->data[y]); + + u->data[y] = added; + l->data[y]->data[x] = -factor; + } + } + } + + u_l[0] = u; + u_l[1] = l; + return u_l; +} +\end{verbatim} +\subsubsection{\texttt{bsubst}} +\label{sec:org15fec98} +\begin{itemize} +\item Author: Elizabeth Hunt +\item Name: \texttt{bsubst} +\item Location: \texttt{src/matrix.c} +\item Input: a pointer to an upper-triangular \texttt{Matrix\_double} \(u\) and a \texttt{Array\_double} +\(b\) +\item Output: a pointer to a new \texttt{Array\_double} whose entries are given by performing +back substitution +\end{itemize} + +\begin{verbatim} +Array_double *bsubst(Matrix_double *u, Array_double *b) { + assert(u->rows == b->size && u->cols == u->rows); + + Array_double *x = copy_vector(b); + for (int64_t row = b->size - 1; row >= 0; row--) { + for (size_t col = b->size - 1; col > row; col--) + x->data[row] -= x->data[col] * u->data[row]->data[col]; + x->data[row] /= u->data[row]->data[row]; + } + return x; +} +\end{verbatim} +\subsubsection{\texttt{fsubst}} +\label{sec:orgdeab27c} +\begin{itemize} +\item Author: Elizabeth Hunt +\item Name: \texttt{fsubst} +\item Location: \texttt{src/matrix.c} +\item Input: a pointer to a lower-triangular \texttt{Matrix\_double} \(l\) and a \texttt{Array\_double} +\(b\) +\item Output: a pointer to a new \texttt{Array\_double} whose entries are given by performing +forward substitution +\end{itemize} + +\begin{verbatim} +Array_double *fsubst(Matrix_double *l, Array_double *b) { + assert(l->rows == b->size && l->cols == l->rows); + + Array_double *x = copy_vector(b); + + for (size_t row = 0; row < b->size; row++) { + for (size_t col = 0; col < row; col++) + x->data[row] -= x->data[col] * l->data[row]->data[col]; + x->data[row] /= l->data[row]->data[row]; + } + + return x; +} +\end{verbatim} + +\subsubsection{\texttt{solve\_matrix}} +\label{sec:orge57c26b} +\begin{itemize} +\item Author: Elizabeth Hunt +\item Location: \texttt{src/matrix.c} +\item Input: a pointer to a \texttt{Matrix\_double} \(m\) and a pointer to an \texttt{Array\_double} \(b\) +\item Output: \(x\) such that \(mx = b\) if such a solution exists (else it's non LU-factorable as discussed +above) +\end{itemize} + +Here we make use of forward substitution to first solve \(Ly = b\) given \(L\) as the \(L\) factor in +\texttt{lu\_decomp}. Then we use back substitution to solve \(Ux = y\) for \(x\) similarly given \(U\). + +Then, \(LUx = b\), thus \(x\) is a solution. + +\begin{verbatim} +Array_double *solve_matrix(Matrix_double *m, Array_double *b) { + assert(b->size == m->rows); + assert(m->rows == m->cols); + + Array_double *x = copy_vector(b); + Matrix_double **u_l = lu_decomp(m); + Matrix_double *u = u_l[0]; + Matrix_double *l = u_l[1]; + + Array_double *b_fsub = fsubst(l, b); + x = bsubst(u, b_fsub); + free_vector(b_fsub); + + free_matrix(u); + free_matrix(l); + + return x; +} +\end{verbatim} + +\subsubsection{\texttt{m\_dot\_v}} +\label{sec:org6afa7d5} +\begin{itemize} +\item Author: Elizabeth Hunt +\item Location: \texttt{src/matrix.c} +\item Input: a pointer to a \texttt{Matrix\_double} \(m\) and \texttt{Array\_double} \(v\) +\item Output: the dot product \(mv\) as an \texttt{Array\_double} +\end{itemize} + +\begin{verbatim} +Array_double *m_dot_v(Matrix_double *m, Array_double *v) { + assert(v->size == m->cols); + + Array_double *product = copy_vector(v); + + for (size_t row = 0; row < v->size; ++row) + product->data[row] = v_dot_v(m->data[row], v); + + return product; +} +\end{verbatim} + +\subsubsection{\texttt{put\_identity\_diagonal}} +\label{sec:orgdd1c373} +\begin{itemize} +\item Author: Elizabeth Hunt +\item Location: \texttt{src/matrix.c} +\item Input: a pointer to a \texttt{Matrix\_double} +\item Output: a pointer to a copy to \texttt{Matrix\_double} whose diagonal is full of 1's +\end{itemize} + +\begin{verbatim} +Matrix_double *put_identity_diagonal(Matrix_double *m) { + assert(m->rows == m->cols); + Matrix_double *copy = copy_matrix(m); + for (size_t y = 0; y < m->rows; ++y) + copy->data[y]->data[y] = 1.0; + return copy; +} +\end{verbatim} + +\subsubsection{\texttt{copy\_matrix}} +\label{sec:org3d1b7b0} +\begin{itemize} +\item Author: Elizabeth Hunt +\item Location: \texttt{src/matrix.c} +\item Input: a pointer to a \texttt{Matrix\_double} +\item Output: a pointer to a copy of the given \texttt{Matrix\_double} +\end{itemize} + +\begin{verbatim} +Matrix_double *copy_matrix(Matrix_double *m) { + Matrix_double *copy = InitMatrixWithSize(double, m->rows, m->cols, 0.0); + for (size_t y = 0; y < copy->rows; y++) { + free_vector(copy->data[y]); + copy->data[y] = copy_vector(m->data[y]); + } + return copy; +} +\end{verbatim} + +\subsubsection{\texttt{free\_matrix}} +\label{sec:org697f6cc} +\begin{itemize} +\item Author: Elizabeth Hunt +\item Location: \texttt{src/matrix.c} +\item Input: a pointer to a \texttt{Matrix\_double} +\item Output: none. +\item Side Effects: frees memory reserved by a given \texttt{Matrix\_double} and its member +\texttt{Array\_double} vectors describing its rows. +\end{itemize} + +\begin{verbatim} +void free_matrix(Matrix_double *m) { + for (size_t y = 0; y < m->rows; ++y) + free_vector(m->data[y]); + free(m); +} +\end{verbatim} + +\subsubsection{\texttt{format\_matrix\_into}} +\label{sec:orgc43bda3} +\begin{itemize} +\item Author: Elizabeth Hunt +\item Name: \texttt{format\_matrix\_into} +\item Location: \texttt{src/matrix.c} +\item Input: a pointer to a \texttt{Matrix\_double} and a pointer to a c-string \texttt{s} to "print" the vector out +into +\item Output: nothing. +\item Side effect: overwritten memory into \texttt{s} +\end{itemize} + +\begin{verbatim} +void format_matrix_into(Matrix_double *m, char *s) { + if (m->rows == 0) + strcpy(s, "empty"); + + for (size_t y = 0; y < m->rows; ++y) { + char row_s[256]; + strcpy(row_s, ""); + + format_vector_into(m->data[y], row_s); + strcat(s, row_s); + } + strcat(s, "\n"); +} +\end{verbatim} +\subsection{Linear Routines} +\label{sec:org1e850f2} +\subsubsection{\texttt{least\_squares\_lin\_reg}} +\label{sec:org02e6d37} +\begin{itemize} +\item Author: Elizabeth Hunt +\item Name: \texttt{least\_squares\_lin\_reg} +\item Location: \texttt{src/lin.c} +\item Input: two pointers to \texttt{Array\_double}'s whose entries correspond two ordered +pairs in R\textsuperscript{2} +\item Output: a linear model best representing the ordered pairs via least squares +regression +\end{itemize} + +\begin{verbatim} +Line *least_squares_lin_reg(Array_double *x, Array_double *y) { + assert(x->size == y->size); + + uint64_t n = x->size; + double sum_x = sum_v(x); + double sum_y = sum_v(y); + double sum_xy = v_dot_v(x, y); + double sum_xx = v_dot_v(x, x); + double denom = ((n * sum_xx) - (sum_x * sum_x)); + + Line *line = malloc(sizeof(Line)); + line->m = ((sum_xy * n) - (sum_x * sum_y)) / denom; + line->a = ((sum_y * sum_xx) - (sum_x * sum_xy)) / denom; + + return line; +} +\end{verbatim} +\subsection{Appendix / Miscellaneous} +\label{sec:org83c0f8d} +\subsubsection{Data Types} +\label{sec:org22f30f4} +\begin{enumerate} +\item \texttt{Line} +\label{sec:orgd014841} +\begin{itemize} +\item Author: Elizabeth Hunt +\item Location: \texttt{inc/types.h} +\end{itemize} + +\begin{verbatim} +typedef struct Line { + double m; + double a; +} Line; +\end{verbatim} +\item The \texttt{Array\_} and \texttt{Matrix\_} +\label{sec:org3f90e03} +\begin{itemize} +\item Author: Elizabeth Hunt +\item Location: \texttt{inc/types.h} +\end{itemize} + +We define two Pre processor Macros \texttt{DEFINE\_ARRAY} and \texttt{DEFINE\_MATRIX} that take +as input a type, and construct a struct definition for the given type for +convenient access to the vector or matrices dimensions. + +Such that \texttt{DEFINE\_ARRAY(int)} would expand to: + +\begin{verbatim} +typedef struct { + int* data; + size_t size; +} Array_int +\end{verbatim} + +And \texttt{DEFINE\_MATRIX(int)} would expand a to \texttt{Matrix\_int}; containing a pointer to +a collection of pointers of \texttt{Array\_int}'s and its dimensions. + +\begin{verbatim} +typedef struct { + Array_int **data; + size_t cols; + size_t rows; +} Matrix_int +\end{verbatim} +\end{enumerate} + +\subsubsection{Macros} +\label{sec:org60b549e} +\begin{enumerate} +\item \texttt{c\_max} and \texttt{c\_min} +\label{sec:org04ff2db} +\begin{itemize} +\item Author: Elizabeth Hunt +\item Location: \texttt{inc/macros.h} +\item Input: two structures that define an order measure +\item Output: either the larger or smaller of the two depending on the measure +\end{itemize} + +\begin{verbatim} +#define c_max(x, y) (((x) >= (y)) ? (x) : (y)) +#define c_min(x, y) (((x) <= (y)) ? (x) : (y)) +\end{verbatim} + +\item \texttt{InitArray} +\label{sec:orgf67f153} +\begin{itemize} +\item Author: Elizabeth Hunt +\item Location: \texttt{inc/macros.h} +\item Input: a type and array of values to initialze an array with such type +\item Output: a new \texttt{Array\_type} with the size of the given array and its data +\end{itemize} + +\begin{verbatim} +#define InitArray(TYPE, ...) \ + ({ \ + TYPE temp[] = __VA_ARGS__; \ + Array_##TYPE *arr = malloc(sizeof(Array_##TYPE)); \ + arr->size = sizeof(temp) / sizeof(temp[0]); \ + arr->data = malloc(arr->size * sizeof(TYPE)); \ + memcpy(arr->data, temp, arr->size * sizeof(TYPE)); \ + arr; \ + }) +\end{verbatim} + +\item \texttt{InitArrayWithSize} +\label{sec:org47e5e66} +\begin{itemize} +\item Author: Elizabeth Hunt +\item Location: \texttt{inc/macros.h} +\item Input: a type, a size, and initial value +\item Output: a new \texttt{Array\_type} with the given size filled with the initial value +\end{itemize} + +\begin{verbatim} +#define InitArrayWithSize(TYPE, SIZE, INIT_VALUE) \ + ({ \ + Array_##TYPE *arr = malloc(sizeof(Array_##TYPE)); \ + arr->size = SIZE; \ + arr->data = malloc(arr->size * sizeof(TYPE)); \ + for (size_t i = 0; i < arr->size; i++) \ + arr->data[i] = INIT_VALUE; \ + arr; \ + }) +\end{verbatim} + +\item \texttt{InitMatrixWithSize} +\label{sec:org3b96b75} +\begin{itemize} +\item Author: Elizabeth Hunt +\item Location: \texttt{inc/macros.h} +\item Input: a type, number of rows, columns, and initial value +\item Output: a new \texttt{Matrix\_type} of size \texttt{rows x columns} filled with the initial +value +\end{itemize} + +\begin{verbatim} +#define InitMatrixWithSize(TYPE, ROWS, COLS, INIT_VALUE) \ + ({ \ + Matrix_##TYPE *matrix = malloc(sizeof(Matrix_##TYPE)); \ + matrix->rows = ROWS; \ + matrix->cols = COLS; \ + matrix->data = malloc(matrix->rows * sizeof(Array_##TYPE *)); \ + for (size_t y = 0; y < matrix->rows; y++) \ + matrix->data[y] = InitArrayWithSize(TYPE, COLS, INIT_VALUE); \ + matrix; \ + }) +\end{verbatim} +\end{enumerate} +\end{document} \ No newline at end of file diff --git a/homeworks/hw-4.org b/homeworks/hw-4.org index ba82878..8884f63 100644 --- a/homeworks/hw-4.org +++ b/homeworks/hw-4.org @@ -1,10 +1,34 @@ -* My Basic Routines -* Linear Solution Routines -** Gaussian Elimination with Backwards Substitution -** LU factorization -* Matrix Routines -** dotproduct -** crossproduct -** matrixvector -** matrixmatrix +#+TITLE: Homework 4 +#+AUTHOR: Elizabeth Hunt +#+LATEX_HEADER: \notindent \notag \usepackage{amsmath} \usepackage[a4paper,margin=1in,portrait]{geometry} +#+LATEX: \setlength\parindent{0pt} +#+OPTIONS: toc:nil +* Question 1 +See the attached LIZFCM Software Manual. + +* Question 2, 3, 4 +#+attr_latex: :width 350px +[[./img/make_run.png]] + +* Question 5 +#+attr_latex: :width 350px +[[./img/test_routine_1.png]] + +#+attr_latex: :width 350px +[[./img/test_routine_2.png]] + +* Question 6 +See the LIZFCM Software Manual. + +* Question 7 +See ~src/matrix.c -> lu_decomp, fsubst, bsubst, solve_matrix~ + +* Question 8 +See ~test/main.c -> lines 109 - 113~ in correspondence to the run in Question 5 + +* Question 9 +See ~test/main.c -> lines 118 - 121~ in correspondence to the run in Question 5 + +* Question 10 +See the TOC on the first page of the LIZFCM Software Manual. diff --git a/homeworks/hw-4.pdf b/homeworks/hw-4.pdf new file mode 100644 index 0000000000000000000000000000000000000000..8d515c71fb2a9d5a5b268df5eb0ea21deedca0ec GIT binary patch literal 215081 zcmd43c|4Tw-#)BGeIhD}6pWmgg+4B59*)@+ewY?F|& z4B4}9!w5sRVXV(N`h36l_r9O|d;jj|^*nz(m)9%0%ynMZbuRDYeH`!OBydkzRrm%( zoI&76cf(HxvFkk7c^u4b7-VG`pc)n^E9WOXqT)Ar{u~TYRckwE3p5W@)y~Y>LfPW6 z1JZ&)UY^0p8Es)^&+u%(KC%||?fUUoi{7s};nv4KCx0YIGac^w@p8txi+&5oAo z?dYoqBX90pPK_rV_JbtVeU-L9&Y<$D#{-T2x%VJ7`r=cTEP4uz1#KT){@_qrieJ@!9c=YJL` z;#DM@RTTYM_52_61Wc(jnvP!Y}+Rk;Tlz1^$Dd+ zRB`7Aq!eS_%zIV3N4?u+!Y{RXU9g%o?mEjhCVFS5veFNE|I*j0Q#^D!o;4@ryT5yU zmcKL_OzD*#bs2g&j4AFGh>L$OnZSKZR=g=P_0|KXidofXLpu_Cl)!)pa$CJHWrVa8K7_%bLagtckIoqEEu5r z)<`gzH?B+GWPqw$SX(`D<`EaW&Hz=ic6PdFfxhiv@92QCKsobBGC*%T*g2r#j%JTR zgUS}J){iajt0{pdHO-uCdBh~aP4EA6dWcBf6qDMYAOGj+F+aTJd#2BOW(PO%h2)Yz zT(usM&lHrq1JAThS#64ny7cDpbryBROMd?>-X?^7e8|g}*I7=&c=%r){>v`n&;;t` zi_;MnhmV{#_sTXV)xRu1bXp`q@Skb#pk_3UxV5cO=&#kSAyxL1!rP3lH$3rZ~W z#BF=s!YwJdoy}<9Lv;VYI}{&A%l~#uD=U_&msOM4?h&krKjK$O$Ma!x?>Fvc%{_(R)GhwgP4P!=^(K=yCuUx?$9aF+ zDe&D{D90gds=cj4nkktIl26>{I^j--*si7)>5hEcJ*Ag5?~>%*F;Y~OW~W-0_S)+h zQTPr|(?lr3MDFuO^K*JZqdg@-^qr5x<_iJNst`S{k~8Po$Q-{)`DL7bS-Tsb#C{Y; 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sum_y = sum_v(y); - double sum_xy = dot_v(x, y); - double sum_xx = dot_v(x, x); + double sum_xy = v_dot_v(x, y); + double sum_xx = v_dot_v(x, x); double denom = ((n * sum_xx) - (sum_x * sum_x)); Line *line = malloc(sizeof(Line)); diff --git a/src/matrix.c b/src/matrix.c index 438a468..51de22c 100644 --- a/src/matrix.c +++ b/src/matrix.c @@ -3,11 +3,23 @@ #include #include -void put_identity_diagonal(Matrix_double *m) { - assert(m->rows == m->cols); +Array_double *m_dot_v(Matrix_double *m, Array_double *v) { + assert(v->size == m->cols); + Array_double *product = copy_vector(v); + + for (size_t row = 0; row < v->size; ++row) + product->data[row] = v_dot_v(m->data[row], v); + + return product; +} + +Matrix_double *put_identity_diagonal(Matrix_double *m) { + assert(m->rows == m->cols); + Matrix_double *copy = copy_matrix(m); for (size_t y = 0; y < m->rows; ++y) - m->data[y]->data[y] = 1.0; + copy->data[y]->data[y] = 1.0; + return copy; } Matrix_double *copy_matrix(Matrix_double *m) { @@ -89,6 +101,25 @@ Array_double *fsubst(Matrix_double *l, Array_double *b) { return x; } +Array_double *solve_matrix(Matrix_double *m, Array_double *b) { + assert(b->size == m->rows); + assert(m->rows == m->cols); + + Array_double *x = copy_vector(b); + Matrix_double **u_l = lu_decomp(m); + Matrix_double *u = u_l[0]; + Matrix_double *l = u_l[1]; + + Array_double *b_fsub = fsubst(l, b); + x = bsubst(u, b_fsub); + free_vector(b_fsub); + + free_matrix(u); + free_matrix(l); + + return x; +} + void free_matrix(Matrix_double *m) { for (size_t y = 0; y < m->rows; ++y) free_vector(m->data[y]); diff --git a/src/vector.c b/src/vector.c index a7ff783..4a6250c 100644 --- a/src/vector.c +++ b/src/vector.c @@ -2,30 +2,18 @@ #include #include #include -#include #include +#include -double l2_norm(Array_double *v) { - double norm = 0; - for (size_t i = 0; i < v->size; ++i) - norm += v->data[i] * v->data[i]; - return sqrt(norm); -} +Array_double *add_v(Array_double *v1, Array_double *v2) { + assert(v1->size == v2->size); -double l1_norm(Array_double *v) { - double sum = 0; - for (size_t i = 0; i < v->size; ++i) - sum += fabs(v->data[i]); + Array_double *sum = copy_vector(v1); + for (size_t i = 0; i < v1->size; i++) + sum->data[i] += v2->data[i]; return sum; } -double linf_norm(Array_double *v) { - double max = -DBL_MAX; - for (size_t i = 0; i < v->size; ++i) - max = c_max(v->data[i], max); - return max; -} - Array_double *minus_v(Array_double *v1, Array_double *v2) { assert(v1->size == v2->size); @@ -35,13 +23,6 @@ Array_double *minus_v(Array_double *v1, Array_double *v2) { return sub; } -double sum_v(Array_double *v) { - double sum = 0; - for (size_t i = 0; i < v->size; i++) - sum += v->data[i]; - return sum; -} - Array_double *scale_v(Array_double *v, double m) { Array_double *copy = copy_vector(v); for (size_t i = 0; i < v->size; i++) @@ -49,16 +30,29 @@ Array_double *scale_v(Array_double *v, double m) { return copy; } -Array_double *add_v(Array_double *v1, Array_double *v2) { - assert(v1->size == v2->size); - - Array_double *sum = copy_vector(v1); - for (size_t i = 0; i < v1->size; i++) - sum->data[i] += v2->data[i]; +double l1_norm(Array_double *v) { + double sum = 0; + for (size_t i = 0; i < v->size; ++i) + sum += fabs(v->data[i]); return sum; } -double dot_v(Array_double *v1, Array_double *v2) { +double l2_norm(Array_double *v) { + double norm = 0; + for (size_t i = 0; i < v->size; ++i) + norm += v->data[i] * v->data[i]; + return sqrt(norm); +} + +double linf_norm(Array_double *v) { + assert(v->size > 0); + double max = v->data[0]; + for (size_t i = 0; i < v->size; ++i) + max = c_max(v->data[i], max); + return max; +} + +double v_dot_v(Array_double *v1, Array_double *v2) { assert(v1->size == v2->size); double dot = 0; @@ -67,25 +61,24 @@ double dot_v(Array_double *v1, Array_double *v2) { return dot; } -double l2_distance(Array_double *v1, Array_double *v2) { +double vector_distance(Array_double *v1, Array_double *v2, + double (*norm)(Array_double *)) { Array_double *minus = minus_v(v1, v2); - double dist = l2_norm(minus); + double dist = (*norm)(minus); free(minus); return dist; } double l1_distance(Array_double *v1, Array_double *v2) { - Array_double *minus = minus_v(v1, v2); - double dist = l1_norm(minus); - free(minus); - return dist; + return vector_distance(v1, v2, &l1_norm); +} + +double l2_distance(Array_double *v1, Array_double *v2) { + return vector_distance(v1, v2, &l2_norm); } double linf_distance(Array_double *v1, Array_double *v2) { - Array_double *minus = minus_v(v1, v2); - double dist = linf_norm(minus); - free(minus); - return dist; + return vector_distance(v1, v2, &linf_norm); } Array_double *copy_vector(Array_double *v) { @@ -115,3 +108,10 @@ void format_vector_into(Array_double *v, char *s) { } strcat(s, "\n"); } + +double sum_v(Array_double *v) { + double sum = 0; + for (size_t i = 0; i < v->size; i++) + sum += v->data[i]; + return sum; +} diff --git a/test/main.c b/test/main.c index 0cf3cfd..d31c92b 100644 --- a/test/main.c +++ b/test/main.c @@ -14,12 +14,12 @@ int main() { printf("========\n"); printf("Norm, Distance\n"); - Array_double *v = InitArray(double, {3, 1, -4, 1, 5, -9, 3}); + Array_double *v = InitArray(double, {3, 1, -4}); strcpy(s, ""); format_vector_into(v, s); printf("v: %s", s); - Array_double *w = InitArray(double, {-2, 7, 1, -8, -2, 8, 5}); + Array_double *w = InitArray(double, {-2, 7, 1}); strcpy(s, ""); format_vector_into(w, s); printf("w: %s", s); @@ -51,7 +51,6 @@ int main() { printf("w: %s", s); Line *line = least_squares_lin_reg(v, w); - printf("least_squares_lin_reg(v, w): (%f)x + %f\n", line->m, line->a); v = InitArray(double, {1, 2, 3, 4, 5, 6, 7}); strcpy(s, ""); @@ -64,6 +63,7 @@ int main() { line = least_squares_lin_reg(v, w); printf("least_squares_lin_reg(v, w): (%f)x + %f\n", line->m, line->a); + free(line); printf("========\n"); printf("LU Decomp\n"); @@ -77,8 +77,9 @@ int main() { format_matrix_into(a, s); printf("a = %s", s); - uint32_t solution = 100; - Array_double *b = InitArrayWithSize(double, n, (double)solution); + uint32_t solution = 1; + Array_double *y = InitArrayWithSize(double, n, (double)solution); + Array_double *b = m_dot_v(a, y); // q8 steps Matrix_double **u_l = lu_decomp(a); Matrix_double *u = u_l[0]; Matrix_double *l = u_l[1]; @@ -91,12 +92,11 @@ int main() { printf("l = %s", s); strcpy(s, ""); format_vector_into(b, s); - printf("b = %s", s); + printf("(after following q8) b = %s", s); printf("========\n"); printf("Forward / Backward Substitution Solution to ax=b\n"); Array_double *b_fsub = fsubst(l, b); - free_vector(b); strcpy(s, ""); format_vector_into(b_fsub, s); printf("b_fsub: %s", s); @@ -106,16 +106,19 @@ int main() { format_vector_into(x_bsub, s); printf("x_bsub: %s", s); + Array_double *x_solve_matrix = solve_matrix(a, b); + free_vector(b); + strcpy(s, ""); + format_vector_into(x_solve_matrix, s); + printf("\\--> == x_solve_matrix: %s", s); + free_vector(b_fsub); + free_vector(x_solve_matrix); printf("Verifications\n"); - for (size_t row = 0; row < a->rows; row++) { - double curr = 0; - for (size_t col = 0; col < a->cols; col++) - curr += a->data[row]->data[col] * x_bsub->data[col]; - printf("Substituions for values in row %zu = %f, true value err=%.10e\n", - row, curr, fabs(curr - solution)); - } + for (size_t i = 0; i < x_bsub->size; i++) + printf("in row %zu, solution = %f, true value err=%.10e\n", i, + x_bsub->data[i], fabs(x_bsub->data[i] - solution)); return 0; }