Nonuniform trapezoids and signed error

Goal and scope

For f(x)=x^2 and nodes (0,1/2,2), compute T, I-T and absolute error.

Nonuniform partitions are allowed; the full quadratic gives an independent reference integral, not a general error bound or convergence order.

Givens and symbols

• algebra
• polynomial
• trap_def
• integral_poly

• G1 — f(x)=x^2 on [0,2], nodes (0,1/2,2), values (0,1/4,4).

• T — trapezoid value
• E=I-T — fixed signed error
• h_i — individual interval width

Rules and sources

• mko_definition · cited — Source semantic paraphrase: on a qualified nonuniform partition, sum each width times its endpoint-value average; a finite table does not determine a general integral. · mko-trapezoidal-quadrature
• algebra · prerequisite — Use real-field identities; division requires a nonzero denominator and multiplication by a positive number preserves inequalities.
• polynomial · prerequisite — Real polynomials are continuous and differentiate termwise: the derivative of x^k is k*x^(k-1), and constants differentiate to zero.
• trap_def · prerequisite — The composite trapezoid value sums (x_i-x_(i-1))*(y_(i-1)+y_i)/2; nodes are strictly increasing with aligned endpoint values, and n intervals have n+1 nodes.
• integral_poly · prerequisite — Integrate real polynomials on finite intervals termwise using antiderivative x^(k+1)/(k+1).

Main derivation

0<1/2<2,\quad\#\text{nodes}=3

n=2,\quad\#\text{values}=3

(f(0),f(1/2),f(2))

(0,1/4,4)

h_1=x_1-x_0,\quad h_2=x_2-x_1

h_1=1/2,\quad h_2=3/2

T_1=\tfrac12\tfrac{0+1/4}{2},\quad T_2=\tfrac32\tfrac{1/4+4}{2}

T_1=1/16,\quad T_2=51/16

T=T_1+T_2

T=52/16=13/4

I=\int_0^2x^2\,dx

I=[x^3/3]_0^2=8/3

E=I-T=8/3-13/4

E=-7/12,\quad|E|=7/12

\ell(x)=mx+b,\quad u<v

\int_u^v\ell(x)\,dx=(v-u)\left(m\frac{u+v}{2}+b\right)=(v-u)\frac{\ell(u)+\ell(v)}2

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · S03 — Affine exactness is not a general second-derivative error bound or convergence rate.
• S07 · inverse_equivalence · S05, S06 — The negative sign means overestimation; an error magnitude is not negative.
• S06 · premise_audit · G1 — A node table alone would not supply this full-integrand certificate.
• S05 · inverse_equivalence · S04 — T is the quadrature value, not yet called the exact integral.
• S04 · premise_audit · S02, S03 — The interior node participates once in each adjacent interval as intended.
• S03 · inverse_equivalence · G1, S01 — Their sum is 2=b-a as a consistency check.
• S02 · inverse_equivalence · G1, S01 — The table is bound to a specified full function.
• S01 · premise_audit · G1 — Three nodes do not make three subintervals.

Error controls and negative controls

Each width

Compute each h_i before multiplying its endpoint average.

Error sign

Keep I-T fixed and report its absolute value separately.

• Three nodes give three intervals. → reject — There are only two adjacent endpoint pairs.
• One h applies to this nonuniform partition. → reject — The widths are 1/2 and 3/2.
• Three sampled values uniquely determine the integral. → reject — Adding a positive multiple of x^2*(x-1/2)^2*(x-2)^2 preserves nodes but increases the integral.

Result and limitations

T=13/4, I=8/3, I-T=-7/12 and absolute error 7/12; affine functions are exact on each interval.

Verification and source binding

3 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 0523238679e2cdf17d8dbeaaa405cec37b031589a6c3c5e271550f3786ebdf06
• original source SHA-256: 0523238679e2cdf17d8dbeaaa405cec37b031589a6c3c5e271550f3786ebdf06
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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