// Numbas version: exam_results_page_options
{"name": "Jacobi iterative method", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Jacobi iterative method", "rulesets": {}, "functions": {}, "statement": "<p>Perform&nbsp;three iterations of the Jacobi method, taking&nbsp; \\(x_0=\\var{x0}\\) &nbsp;and &nbsp;\\(y_0=\\var{y0}\\) &nbsp;as your initial estimates, to partially solve the following system of equations:</p>\n<p style=\"padding-left: 60px;\">\\(\\var{a}x+\\var{b}y=\\var{r}\\)</p>\n<p style=\"padding-left: 60px;\">\\(\\var{c}x+\\var{d}y=\\var{s}\\)</p>\n<p>Give all your answers correct to three decimal places.</p>", "extensions": [], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "preamble": {"css": "", "js": ""}, "advice": "<p>(i) &nbsp; &nbsp; &nbsp;\\(\\var{a}x+\\var{b}y=\\var{r}\\)</p>\n<p>(ii) &nbsp; &nbsp; &nbsp;\\(\\var{c}x+\\var{d}y=\\var{s}\\)</p>\n<p>Rearrange equation (i) to make \\(x\\) the subject and rearrange equation (ii) to make \\(y\\) the subject</p>\n<p>(i) &nbsp; &nbsp;&nbsp;\\(x_{n+1}=\\frac{1}{\\var{a}}[\\var{r}-\\var{b}y_{n}]\\)</p>\n<p>(ii) &nbsp; &nbsp;\\(y_{n+1}=\\frac{1}{\\var{d}}[\\var{s}-\\var{c}x_n]\\)</p>\n<p></p>\n<p style=\"padding-left: 60px;\">\\((x_0,y_0)=(\\var{x0},\\,\\var{y0})\\) &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</p>\n<p>\\(x_{1}=\\frac{1}{\\var{a}}[\\var{r}-\\var{b}*(\\var{y0})]=\\var{x1}\\)</p>\n<p>\\(y_{1}=\\frac{1}{\\var{d}}[\\var{s}-\\var{c}*(\\var{x0})]=\\var{y1}\\)</p>\n<p></p>\n<p style=\"padding-left: 60px;\">\\((x_1,y_1)=(\\var{x1},\\,\\var{y1})\\)</p>\n<p>\\(x_{2}=\\frac{1}{\\var{a}}[\\var{r}-\\var{b}*(\\var{y1})]=\\var{x2}\\)</p>\n<p>\\(y_{2}=\\frac{1}{\\var{d}}[\\var{s}-\\var{c}*(\\var{x1})]=\\var{y2}\\)</p>\n<p></p>\n<p style=\"padding-left: 60px;\">\\((x_2,y_2)=(\\var{x2},\\,\\var{y2})\\)</p>\n<p>\\(x_{3}=\\frac{1}{\\var{a}}[\\var{r}-\\var{b}*(\\var{y2})]=\\var{x3}\\)</p>\n<p>\\(y_{3}=\\frac{1}{\\var{d}}[\\var{s}-\\var{c}*(\\var{x2})]=\\var{y3}\\)</p>\n<p></p>\n<p style=\"padding-left: 60px;\">\\((x_3,y_3)=(\\var{x3},\\,\\var{y3})\\)</p>\n<p></p>", "variable_groups": [], "variables": {"x2": {"name": "x2", "definition": "(r-b*y1)/a", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "y0": {"name": "y0", "definition": "y+e2", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "y": {"name": "y", "definition": "random(2..16#1)", "description": "", "templateType": "randrange", "group": "Ungrouped variables"}, "a": {"name": "a", "definition": "random(9..19#1)", "description": "", "templateType": "randrange", "group": "Ungrouped variables"}, "x": {"name": "x", "definition": "random(1..12#1)", "description": "", "templateType": "randrange", "group": "Ungrouped variables"}, "y2": {"name": "y2", "definition": "(s-c*x1)/d", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "y1": {"name": "y1", "definition": "(s-c*x0)/d", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "x4": {"name": "x4", "definition": "(r-b*y3)/a", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "s": {"name": "s", "definition": "c*x+d*y", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "d": {"name": "d", "definition": "random(7..16#1)", "description": "", "templateType": "randrange", "group": "Ungrouped variables"}, "y4": {"name": "y4", "definition": "(s-c*x3)/d", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "e2": {"name": "e2", "definition": "random(0.5..2#0.5)", "description": "", "templateType": "randrange", "group": "Ungrouped variables"}, "b": {"name": "b", "definition": "random(1..8#1)", "description": "", "templateType": "randrange", "group": "Ungrouped variables"}, "c": {"name": "c", "definition": "random(1..6#1)", "description": "", "templateType": "randrange", "group": "Ungrouped variables"}, "x0": {"name": "x0", "definition": "x+e1", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "r": {"name": "r", "definition": "a*x+b*y", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "x3": {"name": "x3", "definition": "(r-b*y2)/a", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "y3": {"name": "y3", "definition": "(s-c*x2)/d", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "x1": {"name": "x1", "definition": "(r-b*y0)/a", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "e1": {"name": "e1", "definition": "random(0.5..2.5#0.5)", "description": "<p>e1</p>", "templateType": "randrange", "group": "Ungrouped variables"}}, "variablesTest": {"maxRuns": 100, "condition": ""}, "tags": [], "ungrouped_variables": ["a", "b", "c", "x", "d", "y", "r", "s", "e1", "e2", "x0", "y0", "x1", "y1", "x2", "y2", "x3", "y3", "x4", "y4"], "parts": [{"showFeedbackIcon": true, "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "marks": 0, "gaps": [{"showFeedbackIcon": true, "allowResize": false, "precisionPartialCredit": 0, "allowFractions": false, "correctAnswer": "matrix([\n    [x1,y1]\n])    \n", "correctAnswerFractions": false, "showCorrectAnswer": true, "marks": 1, "precisionType": "dp", "numRows": 1, "numColumns": "2", "precision": "3", "markPerCell": false, "type": "matrix", "tolerance": 0, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "scripts": {}, "strictPrecision": false, "precisionMessage": "You have not given your answer to the correct precision."}, {"showFeedbackIcon": true, "allowResize": false, "precisionPartialCredit": 0, "allowFractions": false, "correctAnswer": "matrix([\n    [x2,y2]\n]) ", "correctAnswerFractions": false, "showCorrectAnswer": true, "marks": 1, "precisionType": "dp", "numRows": 1, "numColumns": "2", "precision": "3", "markPerCell": false, "type": "matrix", "tolerance": 0, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "scripts": {}, "strictPrecision": false, "precisionMessage": "You have not given your answer to the correct precision."}, {"showFeedbackIcon": true, "allowResize": false, "precisionPartialCredit": 0, "allowFractions": false, "correctAnswer": "matrix([\n    [x3,y3]\n]) ", "correctAnswerFractions": false, "showCorrectAnswer": true, "marks": 1, "precisionType": "dp", "numRows": 1, "numColumns": "2", "precision": "3", "markPerCell": false, "type": "matrix", "tolerance": 0, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "scripts": {}, "strictPrecision": false, "precisionMessage": "You have not given your answer to the correct precision."}], "scripts": {}, "prompt": "<p>Evaluate your answers to at least four decimal places but input your answers correct to two decimal places.</p>\n<p>The initial estimates are:</p>\n<p style=\"padding-left: 60px;\">\\((x_0,y_0)=(\\var{x0},\\,\\var{y0})\\)</p>\n<p>The values generated by the first iteration are:</p>\n<p style=\"padding-left: 60px;\">\\((x_1,y_1)=\\)&nbsp;[[0]]</p>\n<p>The values generated by the second iteration are:</p>\n<p style=\"padding-left: 60px;\">\\((x_2,y_2)=\\)&nbsp;[[1]]</p>\n<p>The values generated by the third iteration are:</p>\n<p style=\"padding-left: 60px;\">\\((x_3,y_3)=\\)&nbsp;[[2]]</p>\n<p></p>", "variableReplacementStrategy": "originalfirst"}], "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}