MCA 247 Numerical and Scientific Computing Notes cover all 4 units of the GGSIPU MCA Semester 3 Core Elective III paper in 68 pages, with 72 solved examples, 50 short-answer questions with answers and 40 long-answer practice questions. They follow the MCA syllabus for affiliated institutes that applies from academic session 2025-26 onwards.
Written by Prateek Talwar, Founder, Unnati Educations · Reviewed by Ritik Rawat, B.Tech, Senior Full Stack Developer · Checked against syllmcaaff050825900p.pdf (AC Sub Committee, 01/08/2025) · Updated 13 October 2026 · Editorial policy
Scheme: MCA (2 years), affiliated institutes, A.S. 2025-26 onwards | Applies to: batches from academic session 2025-26 onwards | Verified against: syllmcaaff050825900p.pdf, approved by the AC Sub Committee on 01/08/2025
MCA 247 Numerical and Scientific Computing at a glance: GGSIPU MCA Sem 3 credits, units and exam pattern
MCA 247 is one of 11 Core Elective III papers, so you study it only if you pick it; it is assessed with the other GGSIPU MCA Semester 3 papers. Classes run only if at least 20 students choose it; otherwise the scheme sends students to an approved MOOC.
| Code and title | MCA-247 Numerical and Scientific Computing |
|---|---|
| Course and semester | MCA (2 years), Semester 3 |
| Category and credits | Core Elective III · 3 credits · L 3, T/P nil |
| Marks | 100: 40 internal + 60 end-term |
| Pass mark | 40 out of 100 (Important Note 9 of the MCA scheme); re-appear rules: GGSIPU Ordinance 11 |
| Applies to | Affiliated institutes, academic session 2025-26 onwards |
| End-term pattern | Q1 compulsory, objective or short answers from the whole syllabus, 12 marks; then one of two questions from each of the 4 units, 12 marks each (12 + 48 = 60 marks) |
| Units and hours | 4 units: 10 + 10 + 10 + 10 = 40 hours (the official PDF numbers them UNIT I to UNIT IV, without unit titles) |
| Textbooks | Gerald and Wheatley; Kincaid and Cheney; Kothari and Garg (Unit IV) |
| Paired lab | MCA-267 Lab based on Core Elective III (Numerical and Scientific Computing Lab) · 1 credit · 2 practical hours; marks not printed in the syllabus PDF |
MCA 247 Numerical and Scientific Computing Notes unit-wise: 72 solved examples and 90 exam questions across 4 units
These MCA 247 Numerical and Scientific Computing IPU notes follow the official unit order; each unit ends with short and long questions and a Quick Revision Table. The official PDF gives units no titles, so the headings use the notes' titles. One content page from each unit is in the sample PDF above. Notes for every other paper sit under GGSIPU MCA notes.
MCA 247 Unit 1: Floating Point Arithmetic and Roots of Equations
Unit 1 (pages 5 to 22) covers how a computer stores real numbers (normalisation, IEEE 754, −13.625 converted bit by bit), the pitfalls and error types, then root finding: bisection, regula falsi, secant, Newton-Raphson, Ramanujan's method, convergence proofs and complex roots by Muller and Lin-Bairstow. It has 17 solved examples, 12 short-answer and 10 long-answer questions.
| Method | Order p | Starting values | Derivative needed? |
|---|---|---|---|
| Bisection | 1 (C = ½) | 2 (bracket) | No |
| Secant | 1.618 | 2 | No |
| Newton-Raphson | 2 | 1 | Yes, f′(x) |
| Muller | ≈ 1.84 | 3 | No |
Quick look: order of convergence of four Unit 1 methods (from page 19)
- Method
- Bisection
- Order p
- 1 (C = ½)
- Starting values
- 2 (bracket)
- Derivative needed?
- No
- Method
- Secant
- Order p
- 1.618
- Starting values
- 2
- Derivative needed?
- No
- Method
- Newton-Raphson
- Order p
- 2
- Starting values
- 1
- Derivative needed?
- Yes, f′(x)
- Method
- Muller
- Order p
- ≈ 1.84
- Starting values
- 3
- Derivative needed?
- No
MCA 247 Unit 2: Linear Systems and Interpolation
Unit 2 (pages 23 to 40) covers Gauss elimination with pivoting, the Gauss-Jordan inverse, LU (Doolittle, Crout, Cholesky), Jacobi and Gauss-Seidel, condition number and refinement; then Lagrange, divided differences, Newton forward and backward on 1891 to 1931 population data, Gauss, Stirling, Bessel and Everett on one ex table, Hermite and Chebyshev. It has 18 solved examples, 13 short-answer and 10 long-answer questions.
| Measure | Gauss-Jacobi | Gauss-Seidel |
|---|---|---|
| Iterations to reach x = 1, y = −1, z = 1 | 6 | 4 |
| Spectral radius ρ(G) | ≈ 0.139 | ≈ 0.017 |
| Rate of convergence R = −log10 ρ | ≈ 0.86 | ≈ 1.76 |
Quick look: Jacobi vs Gauss-Seidel on the 3 × 3 system of Example 5.1 (from page 28)
- Measure
- Iterations to reach x = 1, y = −1, z = 1
- Gauss-Jacobi
- 6
- Gauss-Seidel
- 4
- Measure
- Spectral radius ρ(G)
- Gauss-Jacobi
- ≈ 0.139
- Gauss-Seidel
- ≈ 0.017
- Measure
- Rate of convergence R = −log10 ρ
- Gauss-Jacobi
- ≈ 0.86
- Gauss-Seidel
- ≈ 1.76
MCA 247 Unit 3: Numerical Integration and Differential Equations
Unit 3 (pages 41 to 54) covers numerical differentiation and why small h magnifies round-off, quadrature with error terms, Gauss-Legendre and a Romberg table reaching ln 2 = 0.693147; then one ODE, dy/dx = x + y with y(0) = 1, solved by Picard, Taylor, Euler, Modified Euler, RK2 and RK4, plus RKF45, stability and finite difference BVPs. It has 15 solved examples, 12 short-answer and 10 long-answer questions.
| Method | y(0.2) | Error at 0.2 | Order |
|---|---|---|---|
| Euler | 1.220000 | 0.0228 | 1 |
| Modified Euler / RK2 | 1.242050 | 0.00076 | 2 |
| RK4 | 1.242805 | 0.0000004 | 4 |
| Exact (2ex − x − 1) | 1.242806 | nil | nil |
Quick look: value of y(0.2) by each Unit 3 method (from page 49)
- Method
- Euler
- y(0.2)
- 1.220000
- Error at 0.2
- 0.0228
- Order
- 1
- Method
- Modified Euler / RK2
- y(0.2)
- 1.242050
- Error at 0.2
- 0.00076
- Order
- 2
- Method
- RK4
- y(0.2)
- 1.242805
- Error at 0.2
- 0.0000004
- Order
- 4
- Method
- Exact (2ex − x − 1)
- y(0.2)
- 1.242806
- Error at 0.2
- nil
- Order
- nil
MCA 247 Unit 4: Curve Fitting, Regression and Hypothesis Testing
Unit 4 (pages 55 to 68) covers least squares for lines, parabolas and exponential curves, correlation and both regression lines (father and son heights), multiple regression and moving averages; then sampling, standard errors, moments, maximum likelihood, confidence intervals and z, t, F and chi-square tests. It has 22 solved examples, 13 short-answer and 10 long-answer questions.
Quick look: the five-step testing procedure the notes use for every test (from page 63)
- Set up H0 and H1.
- Choose α (5% when the question gives none).
- Compute the statistic, for example z = (statistic − parameter) / S.E.
- Read the table value for α and the degrees of freedom.
- If |calculated| is below the table value, accept H0; otherwise reject it.
MCA 247 exam corner: 50 short-answer solved questions and 40 long-answer practice questions by unit
The 50 short answers prepare the compulsory 12-mark Q1; the 40 long-answer practice questions, most tied to a worked example or ending with the final answer, mirror the 12-mark unit questions. The notes label them "2.5 Marks" and "12.5 Marks"; the current scheme sets 12 marks per question.
| Unit | Short-answer (with answers) | Long-answer practice | Solved examples |
|---|---|---|---|
| Unit 1 | 12 | 10 | 17 |
| Unit 2 | 13 | 10 | 18 |
| Unit 3 | 12 | 10 | 15 |
| Unit 4 | 13 | 10 | 22 |
| Total | 50 | 40 | 72 |
MCA 247 questions and solved examples in the notes, by unit
- Unit
- Unit 1
- Short-answer (with answers)
- 12
- Long-answer practice
- 10
- Solved examples
- 17
- Unit
- Unit 2
- Short-answer (with answers)
- 13
- Long-answer practice
- 10
- Solved examples
- 18
- Unit
- Unit 3
- Short-answer (with answers)
- 12
- Long-answer practice
- 10
- Solved examples
- 15
- Unit
- Unit 4
- Short-answer (with answers)
- 13
- Long-answer practice
- 10
- Solved examples
- 22
- Unit
- Total
- Short-answer (with answers)
- 50
- Long-answer practice
- 40
- Solved examples
- 72
The MCA 247 notes solve 50 short-answer questions and set 40 long-answer practice questions across 4 units, plus 72 worked examples.
One question per unit, with answer pointers:
- Unit 1: How many bisection iterations give a root in [0, 1] with error below 10−4? Pointer: n ≥ log(1/10−4) / log 2 = 13.29, so 14 iterations.
- Unit 2: From x = 4, 5, 7, 10, 11, 13 and f = 48, 100, 294, 900, 1210, 2028, find f(8) by divided differences. Pointer: third differences are constant, so a cubic fits; f(8) = 448.
- Unit 3: State the advantage of Gauss quadrature. Pointer: an n-point formula is exact up to degree 2n − 1, with unequally spaced nodes.
- Unit 4: When is the t-test used instead of the z-test? Pointer: n < 30, σ unknown, normal population; n − 1 degrees of freedom.
More practice sits in the MCA 247 important questions.
Boole's and Weddle's rules on one integral in numerical computing: excerpt from page 44 of the MCA 247 notes
Excerpt from page 44
Boole's rule (n must be a multiple of 4)
Over four strips from x0 to x4, Boole's rule gives ∫ y dx ≈ (2h/45)[7y0 + 32y1 + 12y2 + 32y3 + 7y4]. For a longer range, repeat it for every group of 4 strips, which is why the total number of strips must be a multiple of 4.
Weddle's rule (n must be a multiple of 6)
Over six strips, Weddle's rule gives ∫ y dx ≈ (3h/10)[y0 + 5y1 + y2 + 6y3 + y4 + 5y5 + y6], repeated for every group of 6 strips.
Example 2.1: four rules on ∫ dx/(1 + x²) from 0 to 6, h = 1
The ordinates are y0 = 1.0000, y1 = 0.5000, y2 = 0.2000, y3 = 0.1000, y4 = 0.0588, y5 = 0.0385 and y6 = 0.0270. With n = 6, the strips are even, a multiple of 3 and a multiple of 6, so four rules can be applied to the same table.
| Rule | Working | Result |
|---|---|---|
| Trapezoidal | (1/2)[(1 + 0.0270) + 2(0.5 + 0.2 + 0.1 + 0.0588 + 0.0385)] | 1.4108 |
| Simpson ⅓ | (1/3)[(1 + 0.0270) + 4(0.5 + 0.1 + 0.0385) + 2(0.2 + 0.0588)] | 1.3662 |
| Simpson ⅜ | (3/8)[(1 + 0.0270) + 3(0.5 + 0.2 + 0.0588 + 0.0385) + 2(0.1)] | 1.3571 |
| Weddle | (3/10)[1 + 5(0.5) + 0.2 + 6(0.1) + 0.0588 + 5(0.0385) + 0.0270] | 1.3735 |
Results for ∫ dx/(1 + x²), 0 to 6, h = 1 (exact tan−16 = 1.4056)
- Rule
- Trapezoidal
- Working
- (1/2)[(1 + 0.0270) + 2(0.5 + 0.2 + 0.1 + 0.0588 + 0.0385)]
- Result
- 1.4108
- Rule
- Simpson ⅓
- Working
- (1/3)[(1 + 0.0270) + 4(0.5 + 0.1 + 0.0385) + 2(0.2 + 0.0588)]
- Result
- 1.3662
- Rule
- Simpson ⅜
- Working
- (3/8)[(1 + 0.0270) + 3(0.5 + 0.2 + 0.0588 + 0.0385) + 2(0.1)]
- Result
- 1.3571
- Rule
- Weddle
- Working
- (3/10)[1 + 5(0.5) + 0.2 + 6(0.1) + 0.0588 + 5(0.0385) + 0.0270]
- Result
- 1.3735
Here h = 1 is very coarse for a function that curves this sharply near zero, so no rule lands close to 1.4056. With a smaller h, every rule improves.
Example 2.2: Boole's rule on ∫ dx/(1 + x) from 0 to 1, h = 0.25
The ordinates are 1, 0.8, 0.666667, 0.571429 and 0.5. So I = (2 × 0.25/45)[7(1) + 32(0.8) + 12(0.666667) + 32(0.571429) + 7(0.5)] = (0.5/45)(62.3857) = 0.693175. The exact value is ln 2 = 0.693147, an error of only 0.000028.
Which rule needs what
| Rule | Condition on n | Exact for degree up to | Global error order |
|---|---|---|---|
| Trapezoidal | Any n | 1 | O(h²) |
| Simpson ⅓ | n even | 3 | O(h⁴) |
| Simpson ⅜ | n multiple of 3 | 3 | O(h⁴) |
| Boole | n multiple of 4 | 5 | O(h⁶) |
| Weddle | n multiple of 6 | 5 (approx.) | O(h⁶) |
Conditions and error order of the Newton-Cotes rules (excerpt, page 44)
- Rule
- Trapezoidal
- Condition on n
- Any n
- Exact for degree up to
- 1
- Global error order
- O(h²)
- Rule
- Simpson ⅓
- Condition on n
- n even
- Exact for degree up to
- 3
- Global error order
- O(h⁴)
- Rule
- Simpson ⅜
- Condition on n
- n multiple of 3
- Exact for degree up to
- 3
- Global error order
- O(h⁴)
- Rule
- Boole
- Condition on n
- n multiple of 4
- Exact for degree up to
- 5
- Global error order
- O(h⁶)
- Rule
- Weddle
- Condition on n
- n multiple of 6
- Exact for degree up to
- 5 (approx.)
- Global error order
- O(h⁶)
Do the MCA 247 notes cover the full GGSIPU syllabus?
Yes. Every topic printed for Units I to IV of MCA-247 has its own section in the notes, plus a few extras the syllabus does not name (IEEE 754, Muller, Lin-Bairstow, Cholesky, RKF45).
| Official unit | Official topics | Where in the notes | Status |
|---|---|---|---|
| Unit I | Floating point representation, operations, normalisation, pitfalls; absolute, relative, truncation, rounding off errors; Bisection, Secant, False Position, Newton-Raphson, Ramanujan's methods; rates of convergence; complex roots | Sections 1 to 10, pages 6 to 20 | Covered |
| Unit II | Gauss elimination, pivoting, LU, Gauss Seidel, Gauss Jacobi, Gauss Jordan inverse, ill conditioned systems, refinement; Lagrange, Newton divided difference, finite differences, Newton forward and backward, Gauss, Bessel, Stirling, Everett, Hermite, Taylor, Chebyshev | Sections 1 to 14, pages 24 to 38 | Covered |
| Unit III | Numerical differentiation; Trapezoidal, Simpson's, Boole's, Weddle's rules, Gauss quadrature, Romberg, rule errors; Picard, Taylor, Euler, Modified Euler, RK2 and RK4, error monitoring, stability; finite difference BVPs | Sections 1 to 7, pages 42 to 52 | Covered |
| Unit IV | Least squares, linear, non-linear and multiple regression, moving averages, smoothening; random sampling, sampling distributions; moments, maximum likelihood, interval estimation; hypothesis testing, Type I and II errors, chi-square, t and F tests | Sections 1 to 9, pages 56 to 66 | Covered |
MCA 247 official syllabus topics mapped to the notes (section and page)
- Official unit
- Unit I
- Official topics
- Floating point representation, operations, normalisation, pitfalls; absolute, relative, truncation, rounding off errors; Bisection, Secant, False Position, Newton-Raphson, Ramanujan's methods; rates of convergence; complex roots
- Where in the notes
- Sections 1 to 10, pages 6 to 20
- Status
- Covered
- Official unit
- Unit II
- Official topics
- Gauss elimination, pivoting, LU, Gauss Seidel, Gauss Jacobi, Gauss Jordan inverse, ill conditioned systems, refinement; Lagrange, Newton divided difference, finite differences, Newton forward and backward, Gauss, Bessel, Stirling, Everett, Hermite, Taylor, Chebyshev
- Where in the notes
- Sections 1 to 14, pages 24 to 38
- Status
- Covered
- Official unit
- Unit III
- Official topics
- Numerical differentiation; Trapezoidal, Simpson's, Boole's, Weddle's rules, Gauss quadrature, Romberg, rule errors; Picard, Taylor, Euler, Modified Euler, RK2 and RK4, error monitoring, stability; finite difference BVPs
- Where in the notes
- Sections 1 to 7, pages 42 to 52
- Status
- Covered
- Official unit
- Unit IV
- Official topics
- Least squares, linear, non-linear and multiple regression, moving averages, smoothening; random sampling, sampling distributions; moments, maximum likelihood, interval estimation; hypothesis testing, Type I and II errors, chi-square, t and F tests
- Where in the notes
- Sections 1 to 9, pages 56 to 66
- Status
- Covered
Verify the topic list yourself in the official MCA scheme and syllabus for affiliated institutes, A.S. 2025-26 onwards (MCA-247 starts on page 160 of that PDF).
Unit IV follows TB3, Kothari and Garg's Research Methodology, so its tests use that book's layout. Every course in GGSIPU notes is mapped to its official PDF the same way.
Who writes and checks the MCA 247 numerical computing notes?
- Prateek Talwar, author and editor: Founder, Unnati Educations; Associate Company Secretary (ACS), ICSI, membership no. 58704, since 15 May 2019; B.Com, University of Delhi (School of Open Learning), 2012. Author bio
- Ritik Rawat, subject reviewer: Senior Full Stack Developer, Unnati Educations; B.Tech. Checks method steps, iteration tables and worked values.
Mapped to the MCA scheme for affiliated institutes, A.S. 2025-26 onwards (AC Sub Committee, 01/08/2025). The 2026-27 edition has 72 worked examples, 19 figures and four Quick Revision Tables. They are Numerical and Scientific Computing notes in English, with each method as numbered steps. Last updated 13 October 2026. See our editorial policy.
The same team prepares GGSIPU important questions for every course, built from the same official PDFs.
MCA 247 and the MCA 267 numerical computing lab (1 credit), plus other GGSIPU MCA Sem 3 papers
Paired lab: MCA 267 numerical computing lab notes. MCA-267 is the "Lab based on Core Elective III"; with MCA 247 as your elective, it becomes the Numerical and Scientific Computing Lab (2 practical hours a week, 1 credit).
The official MCA-267 entry prints course outcomes but no practical list, so this mapping is ours. These are the methods you would code first:
- Unit 1: bisection and Newton-Raphson.
- Unit 2: Gauss elimination and Gauss-Seidel.
- Unit 3: Simpson's rules and RK4.
- Unit 4: least squares.
Other MCA Semester 3 papers: MCA 201 Design and Analysis of Algorithms and MCA 203 Artificial Intelligence and Machine Learning (core); one Core Elective II paper, such as MCA 205 Statistics and Data Analytics, which overlaps with Unit 4; and one Open Elective II paper, such as MCA 249 Research Methodology, whose first textbook is also by Kothari. Labs MCA 261 to 267, MCA 269 Minor Project III and NUES papers MCA 271 and MCA 273 complete the 27 credits.
How to write MCA 247 answers: iteration tables, difference tables and twelve-mark numericals
A twelve-mark MCA 247 numerical is marked on visible steps, not just the last value. The notes repeat one habit: write f(x), f′(x) and the iteration formula before the table, because those lines carry marks on their own. For Newton-Raphson, start where f(x) and f″(x) share a sign (the Fourier condition).
Worked example: find √12 correct to 4 decimal places (page 15)
- Take f(x) = x² − 12, so xn+1 = ½(xn + 12/xn).
- Start at x0 = 3.5: x1 = ½(3.5 + 3.428571) = 3.464286.
- x2 = ½(3.464286 + 3.463918) = 3.464102, and x3 = 3.464102 again.
- Two equal iterates: √12 = 3.4641.
For interpolation, write the difference table with subscripts (y−2, y−1, y0 and so on) before using Stirling, Bessel or Everett. Most mistakes come from picking the wrong row for Δ³ or Δ⁴. State the condition on n before any Newton-Cotes rule.
Three traps the notes flag:
- Truncation error (cutting a series) is not the same as chopping (dropping digits), and examiners ask the difference directly.
- Numerical differentiation magnifies round-off, so a smaller h can make the answer worse.
- In Unit 4, accepting H0 does not prove it true, and z is only for large samples or known σ.
For the 12-mark Q1, keep each short answer to a formula plus one line. For "methods for complex roots", name Newton-Raphson in complex arithmetic, Muller and Lin-Bairstow with a line on each. Practise this format on GGSIPU MCA important questions.
MCA 247 numerical computing notes FAQs
How do I get the full MCA 247 notes?
Message the Unnati Educations desk on WhatsApp at 9899436384 using the "Get Full Notes" button. The full MCA 247 notes come only as a printed book, not as a PDF. Delivery takes 5 to 7 days at most, depending on your location. The team confirms your order and delivery details on WhatsApp before dispatch.
What is the passing mark for MCA 247?
MCA 247 is marked out of 100: 40 internal and 60 end-term. The MCA scheme of examination states pass marks as 40%, which is 40 out of 100, for every paper. For end-term minimums, re-appear and other examination rules, read Ordinance 11 of the university on ipu.ac.in, which the MCA scheme says governs the programme.
Is MCA 247 compulsory in MCA Semester 3?
No. MCA 247 is one of 11 Core Elective III options, alongside papers such as MCA 237 Soft Computing and MCA 243 Compiler Design, and you choose one. The scheme runs classes for an elective only if at least 20 students choose it. Otherwise students complete it through an approved MOOC and still sit the institute's evaluation for the credits.
Which unit of the MCA 247 notes has the most solved examples?
Unit 4 has the most: 22 of the 72 solved examples. Half of them, 11, are hypothesis tests: four z-tests, two t-tests, an F-test for variances, an F-test for a regression fit and three chi-square tests. Unit 2 follows with 18, Unit 1 with 17 and Unit 3 with 15, where one ODE is solved by six methods.
Do the MCA 247 notes include programs for the MCA 267 lab?
No. The MCA 247 theory notes contain no program code. They give each method as numbered steps, such as the bisection algorithm, Gauss elimination steps and the finite difference procedure, which you can turn into lab programs. The MCA 267 lab has its own notes page, and the official syllabus prints no fixed practical list for it.
Why do the MCA 247 notes print 2.5 and 12.5 marks?
Those labels sit on the short-answer and long-answer question sets in the MCA 247 notes. The current MCA scheme sets Q1 at 12 marks of short answers and each unit question at 12 marks, for 60 end-term marks. Use the short answers for Q1 and the long-answer practice set for the unit questions; the depth still matches.
Get the full MCA 247 Numerical and Scientific Computing Notes on WhatsApp
The full GGSIPU MCA Sem 3 Numerical and Scientific Computing notes come as a 68-page printed book, not a PDF, with 72 solved examples and 90 unit-end questions.
| Part | Pages |
|---|---|
| Cover and contents | 1-4 (4) |
| Unit 1: Floating Point Arithmetic and Roots of Equations | 5-22 (18) |
| Unit 2: Linear Systems and Interpolation | 23-40 (18) |
| Unit 3: Numerical Integration and Differential Equations | 41-54 (14) |
| Unit 4: Curve Fitting, Regression and Hypothesis Testing | 55-68 (14) |
Pages per unit in the MCA 247 printed notes
- Part
- Cover and contents
- Pages
- 1-4 (4)
- Pages
- 5-22 (18)
- Pages
- 23-40 (18)
- Pages
- 41-54 (14)
- Pages
- 55-68 (14)
Delivery: within 5 to 7 days at most; the exact time may vary with your location.
Get the full MCA 247 notes on WhatsApp (4 units, 68-page printed book)
Unnati Educations is an independent education provider based in Vikaspuri, Delhi. We are not affiliated with, authorised by, or endorsed by Guru Gobind Singh Indraprastha University (GGSIPU). 'GGSIPU' and 'IPU' are used only to describe the syllabus our notes follow. For official information, visit ipu.ac.in.
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