Exponential and integrating factor integrators

Given: The finite matrix, interval, curve, and semilinear differential equation in the Statement.

First-order systems, initial value problems, and solutions on intervals: a solution is differentiable and satisfies its differential equation at every time in its interval.

A bounded linear operator between normed spaces, The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||, and Every finite-dimensional normed space is Banach: a finite matrix is a bounded operator on the complete finite-dimensional space ${\mathbb{C}}^d$, so The exponential series of a bounded operator applies and gives $(e^{rA}{)}^{\mathrm{\prime }}=A{e}^{rA}=e^{rA}A$ and $e^{rA}{e}^{qA}=e^{(r+q)A}$.

The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral: a curve in ${\mathbb{C}}^d$ and its integral are interpreted coordinatewise in the real vector space ${\mathbb{R}}^{2d}$; an integral with equal limits is zero.

A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion: every continuous real coordinate of a finite-dimensional vector curve on a closed bounded interval is integrable.

If $f:[a,b]\to {\mathbb{R}}^m$ is differentiable with integrable $f^{\mathrm{\prime }}$ then ${\int }_a^b{f}^{\mathrm{\prime }}=f(b)-f(a)$; and a bounded derivative makes $f$ Lipschitz: for $s<t$ and a differentiable finite-dimensional real vector curve with integrable derivative, $z(t)-z(s)={\int }_s^t{z}^{\mathrm{\prime }}(r)dr$.

Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes and Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$: matrix multiplication is a finite sum of scalar products, so its derivative obeys the product rule.

Integrable functions on $[a,b]$ form a set closed under sums and scalar multiples, and ${\int }_a^b(\lambda f+\mu g)=\lambda {\int }_a^{b}f+\mu {\int }_a^{b}g$: a fixed finite-dimensional linear map commutes with componentwise vector integration.

$\mathbb{C}$ is the real coordinate plane, with coordinate arithmetic, The standard formulas $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ and ${\sum }_{k<n}{x}_k\overline{y_k}$ on ${\mathbb{C}}^n$ are inner products, and The norm $\parallel v\parallel =\sqrt{⟨v,v⟩}$ induced by a real or complex inner product: for $y=(y_0,\dots ,y_{d-1})$, set ${\mathrm{\Phi }}_d(y):=(\mathrm{Re}{y}_0,\mathrm{Im}{y}_0,\dots ,\mathrm{Re}{y}_{d-1},\mathrm{Im}{y}_{d-1})$. Applying the scalar coordinate map componentwise identifies ${\mathbb{C}}^d$ real-linearly with ${\mathbb{R}}^{2d}$, and $\mathrm{\parallel }{\mathrm{\Phi }}_d(y){\mathrm{\parallel }}_{{\mathbb{R}}^{2d}}^2={\sum }_{j<d}((\mathrm{Re}{y}_j{)}^2+(\mathrm{Im}{y}_j{)}^2)={\sum }_{j<d}\mathrm{\mid }{y}_j{\mathrm{\mid }}^2=\mathrm{\parallel }y{\mathrm{\parallel }}_2^2$. It carries each real-time difference quotient to the corresponding realified quotient and preserves its limit, while matrix-vector multiplication becomes the corresponding real coordinate arithmetic. Thus the real differentiation and integration suppliers apply without changing the norm.

Given: A finite complex matrix $A$ with $A^{\ast }=-A$ and the matrix exponential $U(t):=e^{tA}$.

The exponential series of a bounded operator: $U(0)=I_d$ and $U^{\mathrm{\prime }}(t)=AU(t)$.

The adjoint $T^{\ast }:W\to V$ is characterised by $⟨Tv,w{⟩}_W=⟨v,T^{\ast }w{⟩}_V$ and In orthonormal bases, the matrix of the adjoint is the conjugate transpose of the matrix: the adjoint of a matrix is its conjugate transpose and $(BC{)}^{\ast }=C^{\ast }{B}^{\ast }$.

Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$: entrywise conjugation commutes with differentiation in real time, and finite matrix products along a real parameter obey the product rule.

The mean value theorem, as the case $g(x)=x$ of Cauchy's: for $f$ continuous on $[a,b]$ with $a<b$ and differentiable on $(a,b)$ there is $c\in (a,b)$ with $f(b)-f(a)=f^{\mathrm{\prime }}(c)(b-a)$: a real-valued function with zero derivative on an interval is constant there.

The norm $\parallel v\parallel =\sqrt{⟨v,v⟩}$ induced by a real or complex inner product and The standard formulas $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ and ${\sum }_{k<n}{x}_k\overline{y_k}$ on ${\mathbb{C}}^n$ are inner products: the complex Euclidean norm satisfies $\mathrm{\parallel }v{\mathrm{\parallel }}_2^2=v^{\ast }v$.

Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes: matrix multiplication and $I_d$ have their standard coordinate meanings.

$\mathbb{C}$ is the real coordinate plane, with coordinate arithmetic, The standard formulas $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ and ${\sum }_{k<n}{x}_k\overline{y_k}$ on ${\mathbb{C}}^n$ are inner products, and The norm $\parallel v\parallel =\sqrt{⟨v,v⟩}$ induced by a real or complex inner product: for $v=(v_0,\dots ,v_{d-1})$, the map ${\mathrm{\Phi }}_d(v):=(\mathrm{Re}{v}_0,\mathrm{Im}{v}_0,\dots ,\mathrm{Re}{v}_{d-1},\mathrm{Im}{v}_{d-1})$ is real-linear and satisfies $\mathrm{\parallel }{\mathrm{\Phi }}_d(v){\mathrm{\parallel }}_{{\mathbb{R}}^{2d}}^2={\sum }_{j<d}\mathrm{\mid }{v}_j{\mathrm{\mid }}^2=\mathrm{\parallel }v{\mathrm{\parallel }}_2^2$. It commutes with real-time difference quotients and preserves limits, so entrywise real differentiation and the zero-derivative argument apply.

Given: $d\ge 1$ and $X\in M_d(\mathbb{C})$.

The phi-function family by everywhere-convergent series: ${\phi }_1(X)={\sum }_{j\ge 0}{X}^j\mathrm{/}(j+1)!$ and ${\phi }_2(X)={\sum }_{j\ge 0}{X}^j\mathrm{/}(j+2)!$.

The scalar and matrix phi series converge absolutely: these series converge absolutely in operator norm.

A bounded linear operator between normed spaces, The operator norm as the least bound and as the unit-sphere or unit-ball supremum, Every finite-dimensional normed space is Banach, and The exponential series of a bounded operator: a finite matrix acts as a bounded operator on the complete finite-dimensional space, and its exponential series converges locally uniformly in operator norm.

The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, The standard formulas $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ and ${\sum }_{k<n}{x}_k\overline{y_k}$ on ${\mathbb{C}}^n$ are inner products, Integrable functions on $[a,b]$ form a set closed under sums and scalar multiples, and ${\int }_a^b(\lambda f+\mu g)=\lambda {\int }_a^{b}f+\mu {\int }_a^{b}g$, and For $a\le b$ and $f:[a,b]\to {\mathbb{R}}^m$ integrable when $a<b$, $\parallel {\int }_a^{b}f{\parallel }_2\le {\int }_a^b\parallel f{\parallel }_2$; for $a<b$, $\parallel f{\parallel }_2$ is integrable: componentwise complex-vector integrals agree with real integrals in ${\mathbb{R}}^{2d}$, finite sums may be integrated termwise, and $\mathrm{\parallel }{\int }_0^{1}v(\theta )d\theta {\mathrm{\parallel }}_2\le {\int }_0^1\mathrm{\parallel }v(\theta ){\mathrm{\parallel }}_{2}d\theta$.

The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv|| and The operator norm is submultiplicative and satisfies ||T^*T|| = ||T||^2: the matrix operator norm is the maximum over unit vectors and $\mathrm{\parallel }{X}^{j}v{\mathrm{\parallel }}_2\le \mathrm{\parallel }X{\mathrm{\parallel }}_2^j\mathrm{\parallel }v{\mathrm{\parallel }}_2$.

$\mathbb{C}$ is the real coordinate plane, with coordinate arithmetic, The standard formulas $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ and ${\sum }_{k<n}{x}_k\overline{y_k}$ on ${\mathbb{C}}^n$ are inner products, and The norm $\parallel v\parallel =\sqrt{⟨v,v⟩}$ induced by a real or complex inner product: for $v=(v_0,\dots ,v_{d-1})$, the componentwise map ${\mathrm{\Phi }}_d(v):=(\mathrm{Re}{v}_0,\mathrm{Im}{v}_0,\dots ,\mathrm{Re}{v}_{d-1},\mathrm{Im}{v}_{d-1})$ is real-linear and satisfies $\mathrm{\parallel }{\mathrm{\Phi }}_d(v){\mathrm{\parallel }}_{{\mathbb{R}}^{2d}}^2={\sum }_{j<d}\mathrm{\mid }{v}_j{\mathrm{\mid }}^2=\mathrm{\parallel }v{\mathrm{\parallel }}_2^2$. Thus [F4] applies to these complex matrix-valued integrands with the stated Euclidean norm.

Given: The finite matrix, interval, methods, Lipschitz constant, and tube in the Statement.

Exponential Euler and the two-stage ETD2 step maps: ${\mathrm{\Psi }}_h^E$ and ${\mathrm{\Psi }}_h^2$ have the displayed exponential Euler and ETD2 formulas, with the predictor $a_h(t,y)={\mathrm{\Psi }}_h^E(t,y)$.

Integral representations of the first two phi functions: ${\phi }_1(X)={\int }_0^1{e}^{(1-\theta )X}d\theta$ and ${\phi }_2(X)={\int }_0^1\theta e^{(1-\theta )X}d\theta$.

A bounded linear operator between normed spaces, The operator norm as the least bound and as the unit-sphere or unit-ball supremum, Every finite-dimensional normed space is Banach, and The exponential series of a bounded operator: finite-matrix exponentials are continuous, differentiable and satisfy $e^{(r+q)A}=e^{rA}{e}^{qA}$.

The norm $\parallel v\parallel =\sqrt{⟨v,v⟩}$ induced by a real or complex inner product, The standard formulas $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ and ${\sum }_{k<n}{x}_k\overline{y_k}$ on ${\mathbb{C}}^n$ are inner products, and The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||: $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_2$ is the Euclidean vector norm and the induced operator norm.

The operator norm is submultiplicative and satisfies ||T^*T|| = ||T||^2 and For $a\le b$ and $f:[a,b]\to {\mathbb{R}}^m$ integrable when $a<b$, $\parallel {\int }_a^{b}f{\parallel }_2\le {\int }_a^b\parallel f{\parallel }_2$; for $a<b$, $\parallel f{\parallel }_2$ is integrable: products obey $\mathrm{\parallel }ST{\mathrm{\parallel }}_2\le \mathrm{\parallel }S{\mathrm{\parallel }}_2\mathrm{\parallel }T{\mathrm{\parallel }}_2$, and vector integrals obey $\mathrm{\parallel }\int v{\mathrm{\parallel }}_2\le \int \mathrm{\parallel }v{\mathrm{\parallel }}_2$.

The exponential series of a bounded operator: for finite $A$ on the finite-dimensional Banach space ${\mathbb{C}}^d$, $\mathrm{\parallel }{e}^{\pm rA}{\mathrm{\parallel }}_2\le e^{r\mathrm{\parallel }A{\mathrm{\parallel }}_2}$ for $0\le r\le T$. For $0\le r\le T$, nonnegativity of $\mathrm{\parallel }A{\mathrm{\parallel }}_2$ gives $r\mathrm{\parallel }A{\mathrm{\parallel }}_2\le T\mathrm{\parallel }A{\mathrm{\parallel }}_2$, so scalar exponential monotonicity (The exponential function is strictly increasing) gives $e^{r\mathrm{\parallel }A{\mathrm{\parallel }}_2}\le e^{T\mathrm{\parallel }A{\mathrm{\parallel }}_2}$. Both sets contain $\mathrm{\parallel }{I}_d{\mathrm{\parallel }}_2$ at $r=0$ and are therefore bounded above by $e^{T\mathrm{\parallel }A{\mathrm{\parallel }}_2}$; Dedekind completeness: the least-upper-bound property supplies finite suprema.

$\mathbb{C}$ is the real coordinate plane, with coordinate arithmetic, The standard formulas $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ and ${\sum }_{k<n}{x}_k\overline{y_k}$ on ${\mathbb{C}}^n$ are inner products, and The norm $\parallel v\parallel =\sqrt{⟨v,v⟩}$ induced by a real or complex inner product: for $v=(v_0,\dots ,v_{d-1})$, ${\mathrm{\Phi }}_d(v):=(\mathrm{Re}{v}_0,\mathrm{Im}{v}_0,\dots ,\mathrm{Re}{v}_{d-1},\mathrm{Im}{v}_{d-1})$ is real-linear and satisfies $\mathrm{\parallel }{\mathrm{\Phi }}_d(v){\mathrm{\parallel }}_{{\mathbb{R}}^{2d}}^2={\sum }_{j<d}\mathrm{\mid }{v}_j{\mathrm{\mid }}^2=\mathrm{\parallel }v{\mathrm{\parallel }}_2^2$. Thus the real vector-integral estimate [F5] applies to the complex-valued integrands with the same norms.

On the stated tube, $\mathrm{\parallel }g(r,u)-g(r,v){\mathrm{\parallel }}_2\le L\mathrm{\parallel }u-v{\mathrm{\parallel }}_2$, and all transformed inputs and ETD2 predictor stages considered in the Statement remain admissible.

Given: The exact semilinear solution, its forcing curve $q$, the finite matrix, and the bounds in the Statement.

Variation of constants for a finite-matrix semilinear system: $y(t+h)=e^{hA}y(t)+{\int }_0^h{e}^{(h-s)A}q(t+s)ds$.

Exponential Euler and the two-stage ETD2 step maps: ${\mathrm{\Psi }}_h^E(t,y(t))=e^{hA}y(t)+h{\phi }_1(hA)q(t)$.

Integral representations of the first two phi functions: $h{\phi }_1(hA)q(t)={\int }_0^h{e}^{(h-s)A}q(t)ds$.

If $f:[a,b]\to {\mathbb{R}}^m$ is differentiable with integrable $f^{\mathrm{\prime }}$ then ${\int }_a^b{f}^{\mathrm{\prime }}=f(b)-f(a)$; and a bounded derivative makes $f$ Lipschitz: a continuously differentiable vector curve with derivative norm bounded by $Q_1$ is $Q_1$-Lipschitz on its closed interval, including its endpoints.

The norm $\parallel v\parallel =\sqrt{⟨v,v⟩}$ induced by a real or complex inner product, The standard formulas $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ and ${\sum }_{k<n}{x}_k\overline{y_k}$ on ${\mathbb{C}}^n$ are inner products, The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||, and For $a\le b$ and $f:[a,b]\to {\mathbb{R}}^m$ integrable when $a<b$, $\parallel {\int }_a^{b}f{\parallel }_2\le {\int }_a^b\parallel f{\parallel }_2$; for $a<b$, $\parallel f{\parallel }_2$ is integrable: the induced Euclidean matrix norm bounds its action on vectors, and $\mathrm{\parallel }\int v{\mathrm{\parallel }}_2\le \int \mathrm{\parallel }v{\mathrm{\parallel }}_2$.

The forcing derivative obeys the stated bound $Q_1$, and $\mathrm{\parallel }{e}^{sA}{\mathrm{\parallel }}_2\le K_{+}$ for $0\le s\le h$.

A bounded linear operator between normed spaces, The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||, Every finite-dimensional normed space is Banach, and The exponential series of a bounded operator: for finite $A$ on ${\mathbb{C}}^d$, $\mathrm{\parallel }{e}^{sA}{\mathrm{\parallel }}_2\le e^{\mathrm{\mid }s\mathrm{\mid }\mathrm{\parallel }A{\mathrm{\parallel }}_2}$. For $0\le s\le T$, nonnegativity of $\mathrm{\parallel }A{\mathrm{\parallel }}_2$ gives $s\mathrm{\parallel }A{\mathrm{\parallel }}_2\le T\mathrm{\parallel }A{\mathrm{\parallel }}_2$, so scalar exponential monotonicity (The exponential function is strictly increasing) gives $e^{s\mathrm{\parallel }A{\mathrm{\parallel }}_2}\le e^{T\mathrm{\parallel }A{\mathrm{\parallel }}_2}$. The set defining $K_{+}$ contains $\mathrm{\parallel }{I}_d{\mathrm{\parallel }}_2$ at $s=0$ and is therefore bounded above by $e^{T\mathrm{\parallel }A{\mathrm{\parallel }}_2}$; completeness of $\mathbb{R}$ (Dedekind completeness: the least-upper-bound property) supplies its finite supremum.

$\mathbb{C}$ is the real coordinate plane, with coordinate arithmetic, The standard formulas $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ and ${\sum }_{k<n}{x}_k\overline{y_k}$ on ${\mathbb{C}}^n$ are inner products, and The norm $\parallel v\parallel =\sqrt{⟨v,v⟩}$ induced by a real or complex inner product: for $v=(v_0,\dots ,v_{d-1})$, ${\mathrm{\Phi }}_d(v):=(\mathrm{Re}{v}_0,\mathrm{Im}{v}_0,\dots ,\mathrm{Re}{v}_{d-1},\mathrm{Im}{v}_{d-1})$ is real-linear and satisfies $\mathrm{\parallel }{\mathrm{\Phi }}_d(v){\mathrm{\parallel }}_{{\mathbb{R}}^{2d}}^2={\sum }_{j<d}\mathrm{\mid }{v}_j{\mathrm{\mid }}^2=\mathrm{\parallel }v{\mathrm{\parallel }}_2^2$. It maps real-time difference quotients to realified difference quotients and preserves their limits, so $q$ and its derivative are legitimate real vector data for [F4] with the same norm bound.

Given: The exact solution, forcing curve, Lipschitz tube, and bounds in the Statement.

Variation of constants for a finite-matrix semilinear system: the exact endpoint is the exponential-linear term plus ${\int }_0^h{e}^{(h-s)A}q(t+s)ds$.

Exponential Euler and the two-stage ETD2 step maps: the ETD2 predictor is $a_h(t,y)={\mathrm{\Psi }}_h^E(t,y)$ and its correction uses ${\phi }_1-{\phi }_2$ at $q(t)$ and ${\phi }_2$ at $g(t+h,a_h)$.

Integral representations of the first two phi functions: ${\phi }_1(X)={\int }_0^1{e}^{(1-\theta )X}d\theta$ and ${\phi }_2(X)={\int }_0^1\theta e^{(1-\theta )X}d\theta$.

Exponential Euler has a second-order local defect: the exact-start predictor obeys $\mathrm{\parallel }{a}_h(t,y(t))-y(t+h){\mathrm{\parallel }}_2\le C_E{h}^2$.

If $f:[a,b]\to {\mathbb{R}}^m$ is differentiable with integrable $f^{\mathrm{\prime }}$ then ${\int }_a^b{f}^{\mathrm{\prime }}=f(b)-f(a)$; and a bounded derivative makes $f$ Lipschitz and For $a\le b$ and $f:[a,b]\to {\mathbb{R}}^m$ integrable when $a<b$, $\parallel {\int }_a^{b}f{\parallel }_2\le {\int }_a^b\parallel f{\parallel }_2$; for $a<b$, $\parallel f{\parallel }_2$ is integrable: Taylor integral remainders and their norms are bounded componentwise.

The norm $\parallel v\parallel =\sqrt{⟨v,v⟩}$ induced by a real or complex inner product, The standard formulas $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ and ${\sum }_{k<n}{x}_k\overline{y_k}$ on ${\mathbb{C}}^n$ are inner products, and The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||: vector and induced matrix norms are the Euclidean norms used in the bounds.

The second derivative bound $Q_2$, state Lipschitz bound $L$, and semigroup bound $K_{+}$ hold on the complete path and predictor tube.

A bounded linear operator between normed spaces, The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||, Every finite-dimensional normed space is Banach, and The exponential series of a bounded operator: for finite $A$ on ${\mathbb{C}}^d$, $\mathrm{\parallel }{e}^{sA}{\mathrm{\parallel }}_2\le e^{\mathrm{\mid }s\mathrm{\mid }\mathrm{\parallel }A{\mathrm{\parallel }}_2}$. For $0\le s\le T$, nonnegativity of $\mathrm{\parallel }A{\mathrm{\parallel }}_2$ gives $s\mathrm{\parallel }A{\mathrm{\parallel }}_2\le T\mathrm{\parallel }A{\mathrm{\parallel }}_2$, so scalar exponential monotonicity (The exponential function is strictly increasing) gives $e^{s\mathrm{\parallel }A{\mathrm{\parallel }}_2}\le e^{T\mathrm{\parallel }A{\mathrm{\parallel }}_2}$. The set defining $K_{+}$ contains $\mathrm{\parallel }{I}_d{\mathrm{\parallel }}_2$ at $s=0$ and is therefore bounded above by $e^{T\mathrm{\parallel }A{\mathrm{\parallel }}_2}$; completeness of $\mathbb{R}$ (Dedekind completeness: the least-upper-bound property) supplies its finite supremum.

$\mathbb{C}$ is the real coordinate plane, with coordinate arithmetic, The standard formulas $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ and ${\sum }_{k<n}{x}_k\overline{y_k}$ on ${\mathbb{C}}^n$ are inner products, and The norm $\parallel v\parallel =\sqrt{⟨v,v⟩}$ induced by a real or complex inner product: for $v=(v_0,\dots ,v_{d-1})$, ${\mathrm{\Phi }}_d(v):=(\mathrm{Re}{v}_0,\mathrm{Im}{v}_0,\dots ,\mathrm{Re}{v}_{d-1},\mathrm{Im}{v}_{d-1})$ is real-linear and satisfies $\mathrm{\parallel }{\mathrm{\Phi }}_d(v){\mathrm{\parallel }}_{{\mathbb{R}}^{2d}}^2={\sum }_{j<d}\mathrm{\mid }{v}_j{\mathrm{\mid }}^2=\mathrm{\parallel }v{\mathrm{\parallel }}_2^2$. It preserves real-time difference quotients and their limits, so the real vector-valued FTC and integral norm bound in [F4] apply to the complex forcing curve with the same norms.