Study guide
6.1 Numerical Root-Finding & Solution of Linear Systems
PRC Registered Electrical Engineer REE
Track your progress as you read
Finished sections get a check mark.
1 Chapter 1: Overview, RA 7920 & Electrical Engineering Ethics
1.1 PRC Board of Electrical Engineering & REE Examination Overview 1.2 Republic Act No. 7920 (New Electrical Engineering Law) & License Classes 1.3 Code of Ethics & Professional Practice for Electrical Engineers
2 Chapter 2: Math — Algebra, Trig & Geometry
2.1 Algebraic Equations, Polynomials & Complex Numbers 2.2 Trigonometry — Identities, Triangles & Spherical Trig 2.3 Analytic & Solid Geometry
3 Chapter 3: Engineering Mathematics — Calculus & Differential Equations
3.1 Differential Calculus, Limits, Derivatives & Applications 3.2 Integral Calculus, Integration Techniques & Area/Volume Applications 3.3 Ordinary Differential Equations & Initial-Value Problems
4 Chapter 4: Engineering Mathematics — Advanced Math & Transform Methods
4.1 Laplace Transforms, Inverse Transforms & Circuit Applications 4.2 Fourier Series, Fourier Transforms & Frequency-Domain Analysis 4.3 Vector Analysis, Linear Algebra, Matrices & Complex Analysis
5 Chapter 5: Engineering Mathematics — Probability, Statistics & Engineering Data Analysis
5.1 Probability Theory, Counting Techniques & Bayes' Theorem 5.2 Engineering Data Analysis I — Descriptive Statistics & Probability Distributions 5.3 Engineering Data Analysis II — Confidence Intervals, Hypothesis Testing & Regression
6 Chapter 6: Engineering Mathematics — Numerical Methods & Analysis
6.1 Numerical Root-Finding & Solution of Linear Systems 6.2 Interpolation, Numerical Integration & Numerical Solution of ODEs
7 Chapter 7: ESAS — Physics, Chemistry & Computer Fundamentals
7.1 Physics for Engineers 7.2 General Chemistry & Materials Science 7.3 Computer Fundamentals, Programming & Data Logic 7.4 Logic Circuits, Switching Theory & Microprocessor Systems
8 Chapter 8: ESAS — Mechanics & Thermal-Fluid Engineering
8.1 Engineering Mechanics — Statics & Dynamics 8.2 Strength of Materials & Mechanics of Deformable Bodies 8.3 Thermal-Fluid Engineering — Thermodynamics & Fluid Mechanics
9 Chapter 9: ESAS — Engineering Economics, Management & Contracts
9.1 Engineering Economics & Depreciation 9.2 Engineering Management & Ethics 9.3 Philippine Electrical Engineering Law, Code & Contracts 9.4 Technopreneurship 101 & Management of Engineering Projects
10 Chapter 10: PEC 1 — Safety, Sizing & Protection
10.1 PEC Fundamentals, Safety Standards & Grounding/Bonding 10.2 Conductor Sizing, Ampacity Adjustment & Voltage Drop 10.3 Overcurrent Protection, Branch Circuit & Feeder Sizing 10.4 Basic Occupational Safety & Health (BOSH) and Electrical Safety Standards
11 Chapter 11: PEC 1 & 2 — Equipment & Utility Systems
11.1 PEC Equipment Installation & Wiring Methods 11.2 Utility Systems, Transformers & Switchgear 11.3 Special Occupancies & Hazardous Locations
12 Chapter 12: Electrical Professional — DC & AC Circuit Fundamentals
12.1 DC Circuit Analysis & Network Theorems 12.2 Single-Phase AC Circuits & Phasor Analysis 12.3 Balanced & Unbalanced Three-Phase AC Circuits
13 Chapter 13: Electrical Professional — EM Fields & Transients
13.1 Electrostatics, Magnetostatics & Maxwell's Equations 13.2 Electromagnetic Waves & Transmission Line Fields 13.3 First-Order & Second-Order Circuit Transients
14 Chapter 14: Electrical Professional — Electronics, Industrial Electronics & Communications
14.1 Electronics 1 & 2 — Semiconductor Devices & Analog Circuits 14.2 Electrical Apparatus & Devices — Industrial Electronics & Power Converters 14.3 Fundamentals of Electronic Communications
15 Chapter 15: Electrical Professional — Transformers & DC Machines
15.1 Single-Phase & Three-Phase Transformers 15.2 Transformer Connections & Special Transformers 15.3 DC Generators & DC Motors
16 Chapter 16: Electrical Professional — AC Synchronous & Induction Machinery
16.1 Synchronous Generators (Alternators) 16.2 Synchronous Motors & Special AC Machines 16.3 Three-Phase & Single-Phase Induction Motors
17 Chapter 17: Electrical Professional — Power Plants & Energy Conversion
17.1 Hydroelectric, Thermal & Combined Cycle Power Plants 17.2 Renewable Energy Systems & Geothermal Power Plants 17.3 Power Plant Economics, Load Curves & Energy Conversion
18 Chapter 18: Electrical Professional — Transmission & Distribution
18.1 Transmission Line Parameters & Performance Models 18.2 Mechanical Overhead Line Design, Underground Cables & Insulators 18.3 Distribution Systems, Substation Layouts & Power Flow Control
19 Chapter 19: Electrical Professional — Power System Analysis
19.1 The Per-Unit System, Base Conversion & Impedance Diagrams 19.2 Symmetrical Components & Unbalanced Fault Analysis 19.3 Load Flow, Economic Dispatch & Power System Stability
20 Chapter 20: Electrical Professional — Protection, Control, Instrumentation & Illumination
20.1 Power System Protection & Protective Relaying 20.2 Industrial Motor Control, Logic Circuits & PLC Automation 20.3 Electrical Instrumentation, Transducers & Measurement Systems 20.4 Illumination Engineering, Lighting Design & Photometrics 20.5 Feedback Control Systems & Research Methods
6.1 Numerical Root-Finding & Solution of Linear Systems
Key Takeaways Explain This Topic AI
Numerical Methods & Analysis carries 3.75% of the PRC REE examination and 15 of the 100 Mathematics items under the Enhanced TOS (PRBEE Res. 40, s. 2024).
Bisection always converges but only linearly, halving the bracket each pass; the required iterations are n ≥ log₂[(b − a)/ε].
Newton-Raphson uses x(k+1) = x(k) − f(x(k))/f′(x(k)) and converges quadratically near a simple root, but fails when f′ approaches zero.
The secant method replaces the analytic derivative with a finite difference and converges at order 1.618, needing two starting points instead of a derivative.
Gauss-Seidel updates each unknown immediately within the sweep and converges for diagonally dominant systems — the property that makes it the classical load-flow solver.
Last updated: August 2026
6.1 Numerical Root-Finding & Solution of Linear Systems
Numerical Methods & Analysis is topic I of the Mathematics subject in the PRC Enhanced Table of Specifications (PRBEE Resolution No. 40, s. 2024 ), weighted 3.75% of the whole examination and 15 of the 100 Mathematics items . It entered the REE syllabus with CHED Memorandum Order No. 88, s. 2017, which lists Numerical Methods and Analysis among the required BSEE professional courses. Every modern power-system tool — load flow, short-circuit study, transient stability, harmonic analysis — is a numerical method underneath, so this topic is applied as well as examinable.
1. Error, Precision & Termination
Absolute error: E a = ∣ x true − x approx ∣ E_a = |x_{\text{true}} - x_{\text{approx}}| E a = ∣ x true − x approx ∣
Relative error: E r = ∣ x true − x approx x true ∣ × 100 % E_r = \left|\dfrac{x_{\text{true}} - x_{\text{approx}}}{x_{\text{true}}}\right| \times 100\% E r = x true x true − x approx × 100%
Approximate relative error between successive iterates, which is what an algorithm can actually compute: ε a = ∣ x k + 1 − x k x k + 1 ∣ × 100 % \varepsilon_a = \left|\frac{x_{k+1} - x_{k}}{x_{k+1}}\right| \times 100\% ε a = x k + 1 x k + 1 − x k × 100%
A result is said to be correct to n n n significant figures when ε a < ( 0.5 × 10 2 − n ) % \varepsilon_a < (0.5 \times 10^{2-n})\% ε a < ( 0.5 × 1 0 2 − n ) % .
Two error sources are always in play: truncation error (from cutting an infinite process short) and round-off error (from finite machine precision). Refining a step size reduces truncation error but eventually amplifies round-off, so there is an optimum step, not a monotonic improvement.
- 2. Bracketing Methods
- Bisection Method
Requires a sign change: f ( a ) f ( b ) < 0 f(a)f(b) < 0 f ( a ) f ( b ) < 0 . Each pass replaces the bracket with its better half at c = ( a + b ) / 2 c = (a+b)/2 c = ( a + b ) /2 . Error after n iterations ≤ b − a 2 n ⟹ n ≥ ln ( b − a ε ) ln 2 = log 2 b − a ε \text{Error after } n \text{ iterations} \le \frac{b-a}{2^{n}} \qquad \Longrightarrow \qquad n \ge \frac{\ln\!\left(\frac{b-a}{\varepsilon}\right)}{\ln 2} = \log_{2}\!\frac{b-a}{\varepsilon} Error after n iterations ≤ 2 n b − a ⟹ n ≥ ln 2 ln ( ε b − a ) = lo g 2 ε b − a
Bisection is guaranteed to converge but only linearly . It is the safety net when faster methods diverge.
False Position (Regula Falsi)
Same bracketing guarantee, but the new estimate is the straight-line intercept rather than the midpoint: c = b − f ( b ) ( a − b ) f ( a ) − f ( b ) c = b - \frac{f(b)(a-b)}{f(a)-f(b)} c = b − f ( a ) − f ( b ) f ( b ) ( a − b )
Usually faster than bisection, but one endpoint can stagnate for strongly curved functions.
3. Open Methods
Newton-Raphson Method x k + 1 = x k − f ( x k ) f ′ ( x k ) x_{k+1} = x_{k} - \frac{f(x_{k})}{f'(x_{k})} x k + 1 = x k − f ′ ( x k ) f ( x k )
Quadratic convergence near a simple root — the number of correct digits roughly doubles each iteration. Its failure modes are examinable:
f ′ ( x k ) → 0 f'(x_k) \to 0 f ′ ( x k ) → 0 sends the tangent nearly horizontal and the next iterate to infinity;
a poor initial guess can converge to a different root or cycle indefinitely;
convergence degrades to linear at a repeated (multiple) root.
Secant Method
Replaces the analytic derivative with a backward difference: x k + 1 = x k − f ( x k ) ( x k − x k − 1 ) f ( x k ) − f ( x k − 1 ) x_{k+1} = x_{k} - \frac{f(x_{k})\,(x_{k} - x_{k-1})}{f(x_{k}) - f(x_{k-1})} x k + 1 = x k − f ( x k ) − f ( x k − 1 ) f ( x k ) ( x k − x k − 1 )
Needs two starting values and no derivative; converges at order ≈ 1.618 \approx 1.618 ≈ 1.618 (the golden ratio) — slower than Newton-Raphson but cheaper per step.
Comparison
Method Bracketing? Convergence order Derivative needed Guaranteed?
- Bisection Yes 1 (linear) No Yes
- False position Yes ~1 (superlinear in practice) No Yes
- Newton-Raphson No 2 (quadratic) Yes No
- Secant No 1.618 No No
- Fixed-point No 1 (linear) No Only if ∣ g ′ ( x ) ∣ < 1 \vert g'(x)\vert< 1 ∣ g ′ ( x ) ∣ < 1
- 4. Systems of Linear Equations
- Direct Methods
Gaussian elimination reduces [ A ∣ b ] [A|b] [ A ∣ b ] to upper-triangular form by row operations, then back-substitutes. Partial pivoting — swapping in the row with the largest available pivot magnitude — is what keeps the process numerically stable and is required whenever a pivot is small.
LU decomposition factors A = L U A = LU A = LU once, after which any number of right-hand sides is solved by one forward and one back substitution. This is why fault studies that reuse the same network matrix factor it a single time.
Cramer's rule , x i = det ( A i ) / det ( A ) x_i = \det(A_i)/\det(A) x i = det ( A i ) / det ( A ) , is examinable for 2 × 2 2\times2 2 × 2 and 3 × 3 3\times3 3 × 3 systems but is computationally hopeless beyond that.
Iterative Methods
Jacobi: all unknowns are updated from the previous sweep's values. x i ( k + 1 ) = 1 a i i ( b i − ∑ j ≠ i a i j x j ( k ) ) x_i^{(k+1)} = \frac{1}{a_{ii}}\left(b_i - \sum_{j \neq i} a_{ij} x_j^{(k)}\right) x i ( k + 1 ) = a ii 1 b i − j = i ∑ a ij x j ( k )
Gauss-Seidel: each new value is used immediately within the same sweep, typically halving the iteration count. x i ( k + 1 ) = 1 a i i ( b i − ∑ j < i a i j x j ( k + 1 ) − ∑ j > i a i j x j ( k ) ) x_i^{(k+1)} = \frac{1}{a_{ii}}\left(b_i - \sum_{j < i} a_{ij} x_j^{(k+1)} - \sum_{j > i} a_{ij} x_j^{(k)}\right) x i ( k + 1 ) = a ii 1 ( b i − j < i ∑ a ij x j ( k + 1 ) − j > i ∑ a ij x j ( k ) )
Both converge if A A A is strictly diagonally dominant : ∣ a i i ∣ > ∑ j ≠ i ∣ a i j ∣ for every row i |a_{ii}| > \sum_{j \neq i} |a_{ij}| \quad \text{for every row } i ∣ a ii ∣ > j = i ∑ ∣ a ij ∣ for every row i
A power network's bus admittance matrix is naturally diagonally dominant and extremely sparse, which is exactly why Gauss-Seidel was the first practical load-flow algorithm and why Newton-Raphson load flow later displaced it on convergence speed.
- Solved Board Exam Examples
- Example 1: Newton-Raphson on a Load Equation
Find the root of f ( x ) = x 3 − 2 x − 5 f(x) = x^{3} - 2x - 5 f ( x ) = x 3 − 2 x − 5 near x 0 = 2 x_0 = 2 x 0 = 2 , performing two Newton-Raphson iterations.
Solution. f ′ ( x ) = 3 x 2 − 2 f'(x) = 3x^{2} - 2 f ′ ( x ) = 3 x 2 − 2 .
Iteration 1: f ( 2 ) = 8 − 4 − 5 = − 1 f(2) = 8 - 4 - 5 = -1 f ( 2 ) = 8 − 4 − 5 = − 1 , f ′ ( 2 ) = 12 − 2 = 10 f'(2) = 12 - 2 = 10 f ′ ( 2 ) = 12 − 2 = 10 . x 1 = 2 − − 1 10 = 2.100 x_1 = 2 - \frac{-1}{10} = 2.100 x 1 = 2 − 10 − 1 = 2.100
Iteration 2: f ( 2.1 ) = 9.261 − 4.2 − 5 = 0.061 f(2.1) = 9.261 - 4.2 - 5 = 0.061 f ( 2.1 ) = 9.261 − 4.2 − 5 = 0.061 , f ′ ( 2.1 ) = 13.23 − 2 = 11.23 f'(2.1) = 13.23 - 2 = 11.23 f ′ ( 2.1 ) = 13.23 − 2 = 11.23 . x 2 = 2.1 − 0.061 11.23 = 2.09457 x_2 = 2.1 - \frac{0.061}{11.23} = \boxed{2.09457} x 2 = 2.1 − 11.23 0.061 = 2.09457
The true root is 2.094551, so two iterations already deliver six-figure accuracy — the signature of quadratic convergence.
Example 2: Bisection Iteration Count
How many bisection iterations are needed to locate a root in [ 1 , 2 ] [1, 2] [ 1 , 2 ] to within 10 − 4 10^{-4} 1 0 − 4 ?
Solution. n ≥ log 2 2 − 1 10 − 4 = log 2 ( 10 000 ) = ln 10 000 ln 2 = 9.2103 0.6931 = 13.29 n \ge \log_{2}\frac{2-1}{10^{-4}} = \log_{2}(10\,000) = \frac{\ln 10\,000}{\ln 2} = \frac{9.2103}{0.6931} = 13.29 n ≥ lo g 2 1 0 − 4 2 − 1 = lo g 2 ( 10 000 ) = ln 2 ln 10 000 = 0.6931 9.2103 = 13.29
Rounding up, 14 iterations \boxed{14 \text{ iterations}} 14 iterations are required. Newton-Raphson would reach the same tolerance in about four.
Example 3: One Gauss-Seidel Sweep
Solve the system below with one Gauss-Seidel sweep starting from x = y = 0 x = y = 0 x = y = 0 : 4 x + y = 9 , x + 3 y = 10 4x + y = 9, \qquad x + 3y = 10 4 x + y = 9 , x + 3 y = 10
Solution. Rearranged, x = ( 9 − y ) / 4 x = (9 - y)/4 x = ( 9 − y ) /4 and y = ( 10 − x ) / 3 y = (10 - x)/3 y = ( 10 − x ) /3 . x ( 1 ) = 9 − 0 4 = 2.25 , y ( 1 ) = 10 − 2.25 3 = 2.5833 x^{(1)} = \frac{9 - 0}{4} = 2.25, \qquad y^{(1)} = \frac{10 - 2.25}{3} = 2.5833 x ( 1 ) = 4 9 − 0 = 2.25 , y ( 1 ) = 3 10 − 2.25 = 2.5833
Note that y ( 1 ) y^{(1)} y ( 1 ) used the new x ( 1 ) x^{(1)} x ( 1 ) , not the old zero — that immediate substitution is precisely what distinguishes Gauss-Seidel from Jacobi (Jacobi would have given y ( 1 ) = 10 / 3 = 3.3333 y^{(1)} = 10/3 = 3.3333 y ( 1 ) = 10/3 = 3.3333 ). The exact solution is x = 1.5454 x = 1.5454 x = 1.5454 , y = 2.8182 y = 2.8182 y = 2.8182 ; each row satisfies ∣ a i i ∣ > ∑ j ≠ i ∣ a i j ∣ |a_{ii}| > \sum_{j \ne i}|a_{ij}| ∣ a ii ∣ > ∑ j = i ∣ a ij ∣ , so convergence is assured.