# Decision-oriented two-parameter Fisher information sensitivity using
symplectic decomposition

Jiannan Yang
Department of Engineering
University of Cambridge
Trumpington Street, Cambridge CB2 1PZ, UK
<EMAIL_ADDRESS>

(October 12, 2022)

###### Abstract

The eigenvalues and eigenvectors of the Fisher information matrix (FIM) can
reveal the most and least sensitive directions of a system and it has wide
application across science and engineering. We present a symplectic variant of
the eigenvalue decomposition for the FIM and extract the sensitivity
information with respect to two-parameter conjugate pairs. The symplectic
approach decomposes the FIM onto an even dimensional symplectic basis. This
symplectic structure can reveal additional sensitivity information between
two-parameter pairs, otherwise concealed in the orthogonal basis from the
standard eigenvalue decomposition. The proposed sensitivity approach can be
applied to naturally paired two-parameter distribution parameters, or a
decision-oriented pairing via re-grouping or re-parameterization of the FIM.
It can be utilised in tandem with the standard eigenvalue decomposition and
offer additional insight into the sensitivity analysis at negligible extra
cost.

_Keywords_ decision under uncertainty $\cdot$ probabilistic sensitivity
$\cdot$ Williamson’s theorem $\cdot$ symplectic eigenvalue $\cdot$ conjugate
parameters

##  1 Introduction

###  1.1 Background

Sensitivity analysis is an integral part of mathematical modelling, and in
particular a crucial element of decision making in presence of uncertainties.
The Fisher information, first introduced by Fisher [1] and is widely used for
parameter estimation and statistical inference, has found increasing
application in many area of science and engineering for probabilistic
sensitivity analysis. For example, the Fisher Information Matrix (FIM) has
been applied to the parametric sensitivity study of stochastic biological
systems [2]; the FIM is used to study sensitivity, robustness, and parameter
identifiability in stochastic chemical kinetics models [3]; through link with
relative entropy, the Fisher information is used to assess the most sensitive
directions for climate change given a model for the present climate [4]; used
in conjunction with the principle of Max Entropy, the FIM is used to identify
the pivotal voters that could perturb the collective voting outcomes in social
systems [5]; and more recently in [6] the Fisher information have been
proposed as one of the process-tailored sensitivity metrics for engineering
design. Despite the wide scope, the applications mentioned above all utilise
the spectral analysis of the FIM, i.e., the eigenvalues and eigenvectors of
the FIM reveal the most and least sensitive directions of the system.

In this paper, we apply a symplectic spectral analysis of the FIM, and
demonstrate that the resulted symplectic eigenvalues and eigenvectors are
oriented towards a better decision support by extracting sensitivity
information with respect to two-parameter pairs (e.g., the mean and standard
deviation of a Normal distribution). Consider a general function $\bf y=h(x)$,
the probabilistic sensitivity analysis characterise the uncertainties of the
output $\bf y$ that is induced by the random input $\bf x$ [7, 8]. When the
input can be described by parametric probability distributions, i.e., $x\sim
p(\mathbf{x|b})$, the FIM can then be estimated as the covariance matrix of
the random gradient vector ${\partial\ln
p(\mathbf{y|b})}/{\partial\mathbf{b}}$, with the $jk^{\text{th}}$ entry of the
Fisher Information Matrix (FIM) as (e.g., [6]):

$F_{jk}=\int\frac{\partial p(\mathbf{y|b})}{\partial b_{j}}\frac{\partial
p(\mathbf{y|b})}{\partial
b_{k}}\frac{1}{p}d\mathbf{y}=\mathbb{E}_{Y}\left[\frac{\partial\ln p}{\partial
b_{j}}\frac{\partial\ln p}{\partial b_{k}}\right]$ (1)

The eigenvalues of the FIM represent the magnitudes of the sensitivities with
respect to simultaneous variations of the parameters $\mathbf{b}$, and the
relative magnitudes and directions of the variations are given by the
corresponding eigenvectors.

The FIM depends on the parametrization used. Suppose
$b_{j}=g_{j}(\theta_{i}),\>i=1,2,\dots,s$, then the FIM with respect to the
parameter $\bm{\uptheta}$ is [9]:

$\mathbf{F}(\bm{\uptheta})=\mathbb{J}^{\mathsf{T}}\mathbf{F}(\mathbf{b})\mathbb{J}$
(2)

where $\mathbb{J}$ is the Jacobian matrix with $\mathbb{J}_{ji}={\partial
b_{j}}/{\partial\theta_{i}}$.

It should be noted that the sensitivity analysis based on FIM is fundamentally
different from the commonly used variance based analysis [10]. The Fisher
sensitivity examines the perturbation of the entire joint probability density
function (PDF) of the output, more specifically, the entropy of output
uncertainty. Moreover, the sensitivity measure from the Fisher analysis are
the eigenvectors, which can be regarded as the principal directions for a
simultaneous variation of the input parameters. This is in contrast to
variance based ranking where it is assumed that the uncertainty of the input
factors can be completely reduced to zero [7]. As pointed out in [11], using
principal sensitivity directions is based on a pragmatic view that given a
finite budget to change the parameters, maximizing the impact on the output
follows the principal sensitivity directions, which tend to be a simultaneous
variation of the parameters because their effect on the output are likely to
be correlated. The constrained maximization view also leads to the symplectic
eigenvectors in a symplectic basis as discussed in Section 3.1.

The Fisher sensitivity is based on partial derivatives, but it is different
from the derivative based global sensitivity measure [12] which is defined as
the integral of the squared derivatives of the function output. The Fisher
information, on the other hand, is defined as the variance of the partial
derivatives of the log probability of the uncertain function output, as seen
in Eq 1. And this differentiation is with respect to the distribution
parameters of the uncertain input, not with respect to the uncertain variables
themselves. Therefore, the Fisher sensitivity examines the impact of the
perturbation of the input probability distribution, and as the input
distributions are often estimated from data, it is equivalent to assess which
uncertain dataset to be focused on. Sensitivity index based on modification of
the input PDF has been proposed in [13] for reliability sensitivity analysis,
where the input perturbation is derived from minimising the probability
divergence under constraints. In contrast, we consider parametric uncertain
inputs in this paper to form the Fisher information matrix (FIM) and the
resulted eigenvectors provide the principal directions for the input
perturbation.

Many widely applied parametric distributions are in the two-parameter
families, e.g., the location-scale families including the Normal distribution
and Gamma distribution. Although the Fisher sensitivity is with respect to
these distribution parameters $\bf b$, the quantities of interest for
decision-making are ultimately the uncertain variables $\mathbf{x}$
themselves, e.g., to rank the relative importance of $\bf x$. We will
demonstrate in this paper that the symplectic decomposition of the FIM
identifies the influential two-parameter pairs, or equivalently the
corresponding variables, and can be used in tandem with the standard
eigenvalue decomposition for a better decision support.

###  1.2 A motivating example

As a motivating example, we consider an engineering design problem under
uncertainties. Consider a simple cantilever beam where the Young’s modulus $E$
and the length $L$ are uncertain, i.e., $\mathbf{x}=(E,L)$, and the
uncertainties can be described by Normal distributions with
$E\sim\mathcal{N}(\mu_{1}=69e9,\>\sigma_{1}^{2}=11.5e9^{2})$ and
$L\sim\mathcal{N}(\mu_{2}=0.45,\>\sigma_{2}^{2}=0.045^{2})$. To keep it
analytically tractable for this motivating example, we assume a trivial
function $\mathbf{y=x}$ (a random vibration problem considered in Section 5).
Assuming the two random variables are independent, the Fisher information
matrix in this case is a diagonal matrix [14]:

$\mathbf{F}(\mu_{1},\sigma_{1},\mu_{2},\sigma_{2})=\text{diag}\left(\sigma_{1}^{-2},2\sigma_{1}^{-2},\sigma_{2}^{-2},2\sigma_{2}^{-2}\right)$
(3)

The eigenvalues, the diagonal entries of the FIM in Eq 3 in this case, and the
corresponding eigenvectors then provide the sensitivity information of the
uncertain output $\bf y$ with respect to (w.r.t) the input parameter vector
$\mathbf{b}=(\mu_{1},\sigma_{1},\mu_{2},\sigma_{2})$. More specifically, the
sensitivity of the entropy of the random output as to be discussed in Section
3.

For practical utilisation of the Fisher sensitivity information, there are two
issues that need to be addressed and that motivates our research in this
paper. First, the FIM needs to be normalised. On one hand, the un-normalised
FIM given in Eq 3 tends to be ill conditioned. For example, the conditional
number is in the order of $10^{22}$ given that $\sigma_{1}=11.5e9$ and
$\sigma_{2}=0.045$. On the other hand, as the Young’s modulus $E$ and the
length $L$ are of different units, the FIM needs to be normalised so that the
sensitivities w.r.t the different parameters are comparable. One option is to
consider sensitivity w.r.t a percentage change of the parameter and this is
called proportional [6] or logarithmic [15] normalised FIM. Normalization is
equivalent to a re-parametrization. In the case of proportional normalization,
the change of parameter is $b_{j}=\bar{b}_{j}\theta_{j}$ with $\bar{b}_{j}$
the nominal value for normalization, and the Jacobian matrix in Eq 2 is just a
diagonal matrix with $\bar{b}_{j}$ on the diagonal.

However, the proportional normalisation might provide unrealistic sensitivity
information for practical applications. For example, unless the probability
distribution of the input variables is far from the real distribution, it is
most likely that the change of the mean should be within one or two standard
deviations. The FIM can instead be normalised by the standard deviations,
which implies that the allowable range of the mean value is limited to a local
region and it is quantified by the standard deviation. Normalising the FIM
from Eq 3 by the standard deviations, i.e., $\bar{b}_{j}$ equal to the
corresponding $\sigma$, we have the normalised FIM as:

$\mathbf{F}_{\text{nor}}=\text{diag}(\sigma_{1}^{2}\sigma_{1}^{-2},2\sigma_{1}^{2}\sigma_{1}^{-2},\sigma_{2}^{2}\sigma_{2}^{-2},2\sigma_{1}^{2}\sigma_{2}^{-2})=\text{diag}(1,2,1,2)$
(4)

where it is evident the condition number of the normalised FIM is much
smaller. However, the $\mathbf{F}_{\text{nor}}$ in Eq 4 has repeated
eigenvalues, and as a result, the corresponding eigenvectors are not unique.
Although the situation with repeated eigenvalues might seem extreme, as to be
seen with more examples, the eigenvalues of the normalised FIM tend to be of
similar magnitudes. In other words, the sensitivity information has been
compressed by normalization (in exchange for better conditioning). As we shall
see, the symplectic decomposition of the FIM has a unique symplectic
structure, and that tends to mitigate this issue by making the sensitivity
information for different variables more distinctive (by pairing the
parameters).

The second issue with the Fisher sensitivity with respect to the distribution
parameters is the gap to decision making. The purpose of the sensitivity
analysis is to identify the influential variables so that informed decisions
can be made. Although it is possible to make changes to the mean and standard
deviation independently, the quantities of interest are ultimately the
variables themselves, i.e. $E$ and $L$ in this case. As to be demonstrated,
the symplectic approach would naturally put parameters in pairs, e.g.,
$(\mu,\>\sigma)$ as a conjugate pair for random input with Normal
distribution, and provide more direct support for decision making. It is noted
in passing that even the true distribution of the uncertain input is not
Normal, a common practice is still to use mean and standard deviation as the
summary statistic for the dataset at hand. As a result, the two-parameter pair
sensitivity proposed in this paper still applies.

###  1.3 Summary and paper outline

In summary, the use of the Fisher information as a sensitivity measure has
wide application across science and engineering. Nevertheless, practical
issues can hinder the translation of the sensitivity information into
actionable decisions. In this paper, we propose a new approach using the
symplectic decomposition to extract the Fisher sensitivity information. The
symplectic decomposition utilises Williamson’s theorem [16, 17] which is a key
theorem in Gaussian quantum information theory [18]. Originated from
Hamiltonian mechanics, the symplectic transformations preserve Hamilton’s
equations in phase space [19]. In analogy to the conjugate coordinates for the
phase space, i.e. position and momentum, we regard the input parameters as
conjugate pairs and use symplectic matrix for the decomposition of the Fisher
information matrix (FIM). The resulted symplectic eigenvalues of large
magnitude, and the corresponding symplectic eigenvectors of the FIM, then
reveal the most sensitive two-parameter pairs.

It should be noted that the proposed symplectic decomposition is only
applicable for parameter space of even dimensions, i.e.,
$\mathbf{b}\in\mathbb{R}^{2n}$, and the requirement that the parameters can be
regarded as two-parameter pairs. For two-parameter family of probability
distributions, such as the widely used location-scale families, there is a
natural pairing of the parameters. For other cases, a decision-oriented
pairing might be needed. For example, a re-parametrization with respect to
(w.r.t) the mean and standard deviation, or two moments of the random
variables, using Eq 2 would transform the Fisher information matrix (FIM) into
even dimensions. Once the FIM is obtained w.r.t parameters of even dimensions,
it is envisaged that the proposed symplectic decomposition is best utilised in
tandem with the standard eigenvalue decomposition for sensitivity analysis
using the FIM. This offers additional insight into the sensitivity analysis,
and as the main computational burden is often at estimating the FIM, at
negligible extra cost.

In what follows, we will first review the approach of symplectic decomposition
using the Williamson’s theorem in Section 2. The details of finding the
symplectic eigenvalues and eigenvectors are given in the _Supplementary
Material_ together with the corresponding Matlab script. In Section 3, we give
a theoretical comparison between the symplectic decomposition and the standard
eigenvalue decomposition, in terms of the sensitivity of entropy and also from
optimization point of view using trace maximization. A benchmark study is
conducted in Section 4, where the similarity and difference between the Fisher
sensitivity and the main effects indices used in variance based analysis are
discussed. In Section 5, a numerical example using a simple cantilever beam is
used to demonstrate the effect of symplectic decomposition. Concluding remarks
are given in Section 6.

##  2 Symplectic decomposition

From elementary linear algebra, we know that a real symmetric matrix
$\mathbf{F}$ can be diagonalized by orthogonal matrices:

$\bf Q^{-1}FQ=\Lambda$ (5)

where $\mathbf{Q}$ is the orthogonal eigenvector matrix, i.e.
$\mathbf{Q}^{\operatorname{\mathsf{T}}}=\mathbf{Q}^{-1}$, and
$\mathbf{\Lambda}=\text{diag}(\lambda_{1},\lambda_{2},\dots)$ contains the
real eigenvalues. And the solution to Eq 5 can be solved using the standard
eigenvalue equation:

$\bf{FQ=Q\Lambda},\quad\text{with}\quad\det(\mathbf{F}-\lambda\mathbf{I})=0$
(6)

The Williamson’s theorem provides us with a symplectic variant of the results
above. Let $\mathbf{F\in\mathbb{R}^{2n\times 2n}}$ be a symmetric and positive
definite matrix, the Williamson’s theorem says that $\mathbf{F}$ can be
diagonalized using symplectic matrices [16, 20]:

$\bf S^{\mathsf{T}}FS=\hat{D}=\begin{bmatrix}\mathbf{D}&\\\
&\mathbf{D}\end{bmatrix}$ (7)

where $\mathbf{D}=\text{diag}(d_{1},d_{2},\dots,d_{n})$ is a diagonal matrix
with positive entries ($d_{j}$ maybe zero if $\mathbf{F}$ is semidefinite).
The $d_{j},\>j=1,2,\dots,n$ are said to be the symplectic eigenvalues of
matrix $\mathbf{F}$ [21] and are in general not equal to the eigenvalues given
in Eq 5. The matrix
$\mathbf{S}=[\mathbf{u}_{1},\dots,\mathbf{u}_{n},\mathbf{v}_{1},\dots,\mathbf{v}_{n}]$
is a real symplectic matrix that satisfies the condition:

$\bf
S^{\mathsf{T}}JS=J,\quad\text{where}\quad\mathbf{J}=\begin{bmatrix}\mathbf{0_{n}}&\mathbf{I_{n}}\\\
-\mathbf{I_{n}}&\mathbf{0_{n}}\end{bmatrix}$ (8)

The matrix $\mathbf{I}_{n}$ is the identity matrix of size $n$ and the matrix
$\mathbf{J}$ is itself a symplectic matrix. From Eq 8, we have the following
expression for $\mathbf{S}^{-\operatorname{\mathsf{T}}}$ , i.e. the inverse
transpose:

$\bf S^{\mathsf{T}}JS=J\rightarrow\bf JS=S^{-\mathsf{T}}J\rightarrow\bf
JSJ^{-1}=S^{-\mathsf{T}}$ (9)

Substitute the expression for $\mathbf{S}^{-\operatorname{\mathsf{T}}}$ into
Eq 7, we have:

$\bf FS=JSJ^{-1}\begin{bmatrix}\mathbf{D}&\\\
&\mathbf{D}\end{bmatrix}=JS\begin{bmatrix}&-\mathbf{D}\\\
\mathbf{D}&\end{bmatrix}$ (10)

where the identity $\mathbf{J}^{-1}=-\mathbf{J}$ is used. Taking the analogy
with Eq 6, the matrix $\mathbf{S}$ is called the symplectic eigenvector matrix
of $\mathbf{F}$. From Eq 10, we can see that each symplectic eigenvalue
$d_{j}$ corresponds to a pair of eigenvectors
$\mathbf{u}_{j},\>\mathbf{v}_{j}\in\mathbb{R}^{2n}$:

$\mathbf{Fu}_{j}=d_{j}\mathbf{Jv}_{j};\quad\mathbf{Fv}_{j}=-d_{j}\mathbf{Ju}_{j}$
(11)

These eigenvector pairs can be normalized so that they form an orthonormal
basis for the symplectic vector space:

$\mathbf{u}_{i}^{\mathsf{T}}\mathbf{J}\mathbf{v}_{j}=\delta_{ij},\quad\text{for}i,j=1,2,\dots,n$
(12)

From Eq 10, the symplectic decomposition of $\mathbf{F}$ can also be written
as:

$\bf F=JSJ^{-1}\begin{bmatrix}\mathbf{D}&\\\
&\mathbf{D}\end{bmatrix}S^{-1}=JSJ^{-1}\begin{bmatrix}\mathbf{D}&\\\
&\mathbf{D}\end{bmatrix}\left(JSJ^{-1}\right)^{\operatorname{\mathsf{T}}}=M\begin{bmatrix}\mathbf{D}&\\\
&\mathbf{D}\end{bmatrix}M^{\operatorname{\mathsf{T}}}$ (13)

where the expression in Eq 9 is substituted for $\bf S^{-1}$ in the second to
last step. From Eq 13, it is clear that the symmetric form of the matrix
$\mathbf{F}$ is preserved. In addition, as the determinant of symplectic
matrices are always one, from Eq 7, the determinant of the FIM can be
expressed as the product of its squared symplectic eigenvalues:

$\det(\mathbf{S^{\mathsf{T}}FS})=\det(\mathbf{S}^{\mathsf{T}})\det(\mathbf{F})\det(\mathbf{S})=\det(\mathbf{F})=\Pi_{j=1}^{n}{d_{j}^{2}}$
(14)

As the determinant of the FIM is also equal to the product of its standard
eigenvalues, i.e., $\det\mathbf{F}=\Pi_{j=1}^{2n}\lambda_{j}$, Eq 14 implies
the total sensitivity volume is conserved in the symplectic decomposition. The
volume is in terms of the relative entropy and more details will be given in
Section 3.2.
It should be noted that the symplectic eigenvectors from Eq 10 cannot be
solved using the standard eigenvalue algorithms directly. One approach uses
the Schur form of skew-symmetric matrices [22] and the details are given in
the _Supplementary Material_ together with the corresponding Matlab script.

##  3 Discussion

The procedure given in the previous section tells us that there exist
symplectic matrices that can decompose the Fisher information matrix (FIM).
Taking analogy with the standard eigenvalue problem, in this section, we first
show that the symplectic eigenvector matrix maximises a matrix trace subject
to a symplectic constraint, and then demonstrate that the symplectic
eigenvalues of the FIM indicate the sensitivity of the Kullback-Leibler (K-L)
divergence in a symplectic basis.

###  3.1 Constrained maximization

The standard eigenvalue equation can be obtained from a trace maximization
problem subject to an orthogonal constraint:

$\text{max}\>\mathrm{tr}\left(\bf{X^{\mathsf{T}}AX}\right)\quad\text{s.t.}\>\bf
X^{\mathsf{T}}X=I$ (15)

To solve the constrained optimization in Eq 15, the method of Lagrange
Multiplier can be used. In this case, the Lagrangian and its first derivative
are:

$L(\bf X,\Lambda)=\mathrm{tr}(\bf
X^{\mathsf{T}}AX)-\mathrm{tr}\left[\bf\Lambda(X^{\mathsf{T}}X)-I)\right]$ (16)
$\frac{\partial L(\bf X,\Lambda)}{\partial\mathbf{X}}=\bf
AX+A^{\operatorname{\mathsf{T}}}X-X(\Lambda+\Lambda^{\operatorname{\mathsf{T}}})=2\mathbf{AX}-2\mathbf{X\Lambda}$
(17)

where the matrix $\mathbf{\Lambda}$ is the Lagrange Multiplier and we have
assumed the matrix $\mathbf{A}$ is symmetric. In addition, since the
constraint $\bf X^{\operatorname{\mathsf{T}}}X$ is symmetric, the Lagrange
multiplier $\mathbf{\Lambda}$ is also symmetric [23]. Setting the first order
optimality condition for the Lagrangian, the orthogonality constraint leads to
the standard eigenvalue problem $\bf AX=X\Lambda$, as Eq (6) when the Fisher
information matrix $\mathbf{F}$ is the symmetric matrix of interest.
Taking the analogy, the maximization problem can be formulated subject to a
symplectic constraint [22] :

$\text{max}\>\mathrm{tr}\left(\bf{X^{\mathsf{T}}AX}\right)\quad\text{s.t.}\>\bf
X^{\mathsf{T}}X=J$ (18) $\frac{\partial L(\bf
X,\Lambda)}{\partial\mathbf{X}}=\bf
AX+A^{\operatorname{\mathsf{T}}}X-JX\Lambda-J^{\operatorname{\mathsf{T}}}X\Lambda^{\operatorname{\mathsf{T}}})=2\mathbf{AX}-2\mathbf{JX\Lambda}$
(19)

where both matrix $\mathbf{J}$ and $\mathbf{\Lambda}$ are skew symmetric, i.e.
$\bf J^{\operatorname{\mathsf{T}}}=-J$. The optimality condition then results
the following symplectic eigenvalue problem:

$\bf AX=JX\Lambda$ (20)

which has exactly the same form as Eq 10.

The standard and the symplectic eigenvectors thus provide the directions to
maximise the matrix trace in an orthogonal and a symplectic basis
respectively, and the corresponding eigenvalues indicate the sensitivities. A
special case with the Fisher information matrix (FIM) links its eigenvalues to
the sensitivities of the Kullback-Leibler (K-L) divergence, aka relative
entropy, and this is described in the next section below.

###  3.2 Sensitivity of entropy

As mentioned in the introduction, consider a general function $\bf y=h(x)$,
the probabilistic sensitivity analysis characterise the uncertainties of the
output $\bf y$ that is induced by the random input $\bf x$. When the joint
probability distribution of the output is known, the entropy of the
uncertainty can be estimated as [14]:

$H=-\int p(\mathbf{y|b})\ln p(\mathbf{y|b})d\mathbf{y}$ (21)

The perturbation of the entropy, defined as a relative entropy quantified
using the K-L divergence, can be approximated by a quadratic form in the
Fisher information matrix [6]:

$\Delta H\equiv
KL\left[p(\mathbf{y|b})||p(\mathbf{y|b}+\Delta\mathbf{b})\right]=\int
p(\mathbf{y|b})\ln\frac{p(\mathbf{y|b})}{p(\mathbf{y|b}+\Delta\mathbf{b})}d\mathbf{y}\approx\frac{1}{2}\Delta\mathbf{b}^{\operatorname{\mathsf{T}}}\mathbf{F}\Delta\mathbf{b}$
(22)

where the perturbed probability is approximated using its second order Taylor
expansion (e.g., see the appendix of [24]). It is noted in passing that, even
without the quadratic approximation to entropy, the Fisher information can be
used to quantify the distribution perturbation in its own right [25].
Consider the standard eigenvalue decomposition of the FIM and substitute Eq 5
into the expression for the relative entropy in Eq 22:

$2\Delta
H=\Delta\mathbf{b}^{\operatorname{\mathsf{T}}}\mathbf{F}\Delta\mathbf{b}=(\mathbf{Q}^{-1}\Delta\mathbf{b})^{\operatorname{\mathsf{T}}}\mathbf{\Lambda}(\mathbf{Q}^{-1}\Delta\mathbf{b})=\sum_{j}^{2n}\lambda_{j}\xi_{j}^{2}=\sum_{j}^{n}\lambda_{j}\xi_{j}^{2}+\lambda_{n+j}\xi_{n+j}^{2}$
(23)

where $\xi_{j}=(\mathbf{Q}^{-1}\Delta\mathbf{b})_{j}$ and it is clear that the
eigenvalues $\lambda_{j}$ indicate the magnitude of the entropy sensitivity.
It can be seen from Eq 23 that the relative entropy in this quadratic form can
be regarded as an ellipsoid geometrically, i.e.,
$\sum\lambda_{j}\xi_{j}^{2}=1$ . This is a consequence of the semi-positive
definiteness of the FIM and the ellipsoid is proper when the FIM is positive-
definite. The eigenvectors of the FIM define the principal axes and the
inverse of the square roots of the corresponding eigenvalues, i.e.,
$1/{\sqrt{\lambda_{j}}}$, are the principal radii of the ellipse. Since the
principal axes are orthogonal to each other, there is no direct relationship
between any pair of coordinates, say $(\xi_{j},\>\xi_{n+j})$, even they are
dominated by the two-parameter pairs for the same variable of interest as
discussed in the introduction.
Similarly, the relative entropy in the symplectic basis can be expressed as:

$2\Delta
H=\Delta\mathbf{b}^{\operatorname{\mathsf{T}}}\mathbf{F}\Delta\mathbf{b}=(\mathbf{S}^{-1}\Delta\mathbf{b})^{\operatorname{\mathsf{T}}}\hat{\mathbf{D}}(\mathbf{S}^{-1}\Delta\mathbf{b})=\begin{bmatrix}\bm{\upalpha}^{\operatorname{\mathsf{T}}}&\bm{\upbeta}^{\operatorname{\mathsf{T}}}\end{bmatrix}\begin{bmatrix}\mathbf{D}&\\\
&\mathbf{D}\end{bmatrix}\begin{bmatrix}\bm{\upalpha}\\\
\bm{\upbeta}\end{bmatrix}=\sum_{j}^{n}d_{j}(\alpha_{j}^{2}+\beta_{j}^{2})$
(24)

where $\alpha_{j}=(\mathbf{S}^{-1}\Delta\mathbf{b}))_{j}$ and
$\beta_{j}=(\mathbf{S}^{-1}\Delta\mathbf{b}))_{j+n}$. In contrast to Eq 23, it
can be seen that the coordinate pair $(\alpha_{j},\>\beta_{j})$ is now forced
to form a circle with radius $1/{\sqrt{d_{j}}}$. The consequence is that if
$(\alpha_{j},\>\beta_{j})$ corresponds to the two-parameter pairs of interest,
they are symplectically equivalent, in analogy to the conjugate pair, position
and momentum, in Hamiltonian mechanics.

##  4 Benchmark study

The Fisher information has been introduced in [6] for sensitivity analysis
with respect to distribution parameters. A benchmark study for Fisher
sensitivity, using a linear function with decreasing coefficients and a
product function with constant coefficients, has been conducted in [11]. In
this section, we apply the Fisher sensitivity analysis, using both standard
eigenvalue decomposition and the proposed symplectic decomposition, to a high
dimensional function:

$f(\mathbf{x})=\mathbf{a}_{1}^{\operatorname{\mathsf{T}}}\mathbf{x}+\mathbf{a}_{2}^{\operatorname{\mathsf{T}}}\sin(\mathbf{x})+\mathbf{a}_{3}^{\operatorname{\mathsf{T}}}\cos(\mathbf{x})+\mathbf{x}^{\operatorname{\mathsf{T}}}\mathbf{M}\mathbf{x}$
(25)

This function has a 15 dimensional input vector $\mathbf{x}$ and has been used
in [7] for variance based sensitivity analysis. This function’s coefficients,
$\mathbf{a}_{1},\mathbf{a}_{2}$ and $\mathbf{a}_{3}$, are chosen so that first
five of the input variables have almost no effect on the output variance,
$x_{6}$ to $x_{10}$ have a much larger effect, and the remaining five
contribute significantly to the output variance. All input variables are
assumed to be independent and from a standard Gaussian distribution, i.e.,
$x\sim\mathbb{N}(0,1)$.

The results from the standard eigenvalue analysis of the Fisher information
matrix are shown in Figure 1 and Figure 2. The eigenvalue spectrum has been
computed using different number of Monte Carlo samples for convergence check
and the eigenvector results in Figure 2 are from 20000 samples. Only the 1st
eigenvector is shown in Figure 2 as the eigenvalues corresponding to the rest
of the eigenvectors are of much smaller amplitudes as seen in Figure 1. The
sensitivity to the mean parameters of the input variables in Figure 2 clearly
indicates that there are there importance groups, $x_{11}$ to $x_{15}$ being
the most important and $x_{1}$ to $x_{5}$ being the least important. This is
in good agreement to [7] from a variance-based sensitivity analysis. The
sensitivity to the standard deviations, on the other hand, does not show a
clear clustered trend, although it is clear that the first few variables have
almost no effect. Different from the variance-based analysis where only the
amplitudes of the importance are measured, the Fisher sensitivity vectors also
provide the relative phases of the sensitivity to the distribution parameters.
For example, in Figure 2, it is clear that the effects of the input mean
parameters on the output PDF uncertainty are in opposite directions to the
effects due to the perturbation of the standard deviations. Note that the
absolute sign of the eigenvector is arbitrary.

Figure 1: Eigenvalue spectrum of the FIM, for the sensitivity of the benchmark
function in Eq 25. Results are given for different number of MC samples for
convergence check. Figure 2: The first eigenvector of the FIM for the
benchmark function Eq 25 with respect to distribution parameters, mean and
standard deviation (Std Dev) in this Gaussian input case, of the 15 input
variables. Only the dominant first eigenvector, as seen in Figure 1, is shown
here.

The symplectic analysis of the FIM, for the benchmark function in Eq 25, is
shown in Figure 3 and Figure 4, in similar format to the results presented
above for the standard analysis. Different from the standard eigenvalue
results, the symplectic eigenvalue spectrum has a dimension half of the
standard one but the symplectic eigenvectors always come in pairs. The
symplectic sensitivity results in Figure 4 present a similar picture as the
standard results, especially that the sensitivity to the standard deviations
for $\mathbf{u1}$ vector indicate clear group importance as discussed above.
However, in this case, the symplectic results do not provide any new insights.
This is because the FIM is dominated by its first eigenvector in this case.
For this benchmark function, there is no normalisation required for the
sensitivity analysis as all input variables are equivalent. As a result, there
is no compression of the sensitivity information as discussed in the
motivating example and the engineering example to be studied in the Section 5.

Figure 3: The symplectic eigenvalue spectrum of the FIM (S-Eig), for the
sensitivity of the benchmark function in Eq 25. Results are given for
different number of MC samples for convergence check. Note that the dimension
of symplectic spectrum is 15, which is half of the size of the standard
eigenvalue spectrum shown in Figure 1. Figure 4: The first pair of symplectic
eigenvectors (S-EigVector) of the FIM for the benchmark function Eq 25 with
respect to distribution parameters, mean and standard deviation (Std Dev) in
this Gaussian input case, of the 15 input variables. Only the dominant first
pair, as seen in Figure 3, is shown here

For the purpose of benchmarking, the sensitivity vectors from the Fisher
analysis can be further summarised for the individual variables and the
results are compared to the true main effect indices given in [7]. The
sensitivity vectors from the FIM provide the principal directions for a
simultaneous variation of the input parameters. To look at the effect of
individual parameters, Eq 23 and Eq 24 can be used. For example, assuming only
parameter $b_{k}$ is varied, from Eq 23:

$2\Delta H(\Delta b_{k})=\sum_{j}^{2n}\lambda_{j}(q_{kj}\Delta
b_{k})^{2}=\left[\sum_{j}^{2n}\lambda_{j}q_{kj}^{2}\right]\Delta b_{k}^{2}$
(26)

where $q_{jk}$ is the $j_{th}$ element of the eigenvector $\mathbf{q}_{k}$.
The term inside the square bracket can be regarded as the contribution to the
entropy change due to perturbation of $b_{k}$ alone. And the contributions
from the mean and the standard deviation parameter can be further aggregated
for the corresponding variables, assuming the the perturbations are
independent. The resulted relative importance of the variables from the Fisher
analysis, using the dominant first eigenvector (Fisher Eig) and the first pair
of symplectic eigenvector (Fisher S-Eig), can then be compared to the variance
based main effects and the comparison is shown in Figure 5. Although there are
small deviations, the relative importance of the three group of variables and
the order of difference are clearly identified from the Fisher sensitivity
analysis. Furthermore, the ratio of the first eigenvalue to the sum of all
eigenvalues, as seen in Figure 1, is about 0.75 in this case and that can be
regarded as the contribution of the first eigenvector to the entropy change.
Although not directly comparable, 0.75 is similar to the 72$\%$ main effects
contribution to the output variance as reported in [7]. This offers plausible
suggestion that, in this case, the dominant first eigenvector of the FIM
corresponds to the main effects from variance based sensitivity analysis.

Figure 5: Variable importance ranking using three different indices: the 1st
set of S-EigVector, the 1st Eigvector and the true main effect indices given
in [7]. The results are normalised by the largest value.

It should be noted although the contributions from perturbation of individual
parameters are useful for benchmarking against variance based main effects,
the purpose of the Fisher sensitivity analysis is to look at the simultaneous
variations of the input parameters. Contrast to the total effects from
variance based analysis, the eigenvectors and symplectic eigenvectors of FIM
provide principal sensitivity directions based on the impact on the output
PDF, or more specifically the entropy of the output uncertainty. Not only do
the eigenvectors indicate the relative amplitude, they also provide the
relative phase information of the input parameter variations as discussed
earlier.

In the next section, we will consider an engineering example where the input
variables are normally of different units. As their value tend to be of
different order of magnitude, normalisation is required for the Fisher
sensitivity analysis. In addition, different from the scalar output from this
benchmark function, engineering problems tend to have multiple outputs as the
example given below.

##  5 Demonstrating application to an engineering example

Figure 6: A cantilever beam subject to white noise excitation of unit
amplitude; the responses consists of peak r.m.s acceleration and strain
response. ‘peak’ indicates the maximum response along the beam for each sample
of the random input. The two type of responses are normalised by the maximum
values across the ensemble of the random samples.

In this section, we consider an engineering design example where the Fisher
information is used for parametric sensitivity analysis of a cantilever beam.
The beam is subject to a white noise excitation at the middle span position,
see Figure 6, where the excitation is bandlimited and only the first three
modes are excited. In this case, the quantities of interest are the peak r.m.s
responses, i.e., the maximum response along the beam, for both acceleration
and strain (output $\bf y$ in Eq 21 is 2-dimensional). The frequency response
functions for both acceleration and strain responses, at different positions
along the beam, are obtained via modal summation and the modal damping is
assumed to be 0.1 for all modes, see the _Supplementary Material_ for details.

It is assumed that five input variables are random,
$\mathbf{x}=(E,\rho,L,w,t)$, and can be described by Normal distributions,
$\mathbf{b}=(\mu_{m},\>\sigma_{m}),\>m=1,2,\dots,5$, as listed in Table 1. Two
cases with different parameter values for the standard deviations of the input
random variables are considered, case-1 with small variance and case-2 with
big variance, with the mean values same for both cases. To estimate the Fisher
information matrix (FIM) in Eq 1, an efficient numerical method based on Monte
Carlo sampling and the Likelihood Ratio method is use here. The Likelihood
Ratio (aka score function) method obtains a gradient estimation of a
performance measure w.r.t continuous parameters in a single simulation run.
More details of the method can be found in [6].

For the numerical results below, the FIM is normalised by the standard
deviations (for completeness, the results with proportionally normalised FIM
are given in the _Supplementary Material_):

$\mathbf{F}_{\text{nor}}=\sigma_{m}\sigma_{n}\mathbf{F}_{jk}$ (27)

where $j,k=1,2,\dots,10$ and $m=j/2$, $n=k/2$ when $j,k$ are even, and
$m=(j+1)/2$, $n=(k+1)/2$ when $j,\>k$ are odd numbers. As discussed in the
introduction, the normalisation is necessary for practical applications as the
input variables are of different units and often differ by orders of
magnitude. Moreover, it largely improves the condition number of the FIM. For
example, in this case study, the condition number of the FIM in the order of
$10^{27}$ for both case-1 and case-2, and it reduces to the order of $10^{2}$
for both cases after normalisation.

Table 1: Mean ($\mu$) and Coefficient of Variation (CoV) for the random
variables. CoV different for the two cases considered. 20000 Monte Carlo
samples are used for both Case-1 and Case-2.

Young’s Modulus Density Length Width Thickness $E[Pa]$ $\rho[kg/m^{3}]$ $L[m]$
$w[m]$ $t[m]$ $\mu$ (mean) 69e9 2700 0.45 2e-2 2e-3 Case-1
$\sigma\textfractionsolidus\mu$ (CoV) 1/200 1/80 1/100 1/60 1/80 Case-2
$\sigma\textfractionsolidus\mu$ (CoV) 1/5 1/5 1/30 1/6 1/8

Figure 7: Eigenvalues (Eig) and Symplectic eigenvalues (S-Eig) of the FIM for
(1) Case-1 ; (2) Case-2

Once the FIM is estimated and normalised, the standard approach is to compute
the eigenvalues and eigenvectors of the FIM for sensitivity analysis [6]. As
discussed, the eigenvalues of the FIM represent the magnitudes of the
sensitivities with respect to simultaneous parameter variations, and the most
sensitive directions are given by the eigenvectors corresponding to largest
eigenvalues. The standard eigenvalues and eigenvectors are denoted as
‘EigValue’ and ‘EigVector’ and are shown in Figure 7, Figure 8 (a), Figure 9
(a).

In Figure 7, the eigenvalues are ordered from large to small, with the first
four of similar magnitude but much larger than the rest. Note that the
spectrum here is quite different from the benchmark case shown in Figure 1
where one eigenvalue dominates. The corresponding first four eigenvectors are
displayed in Figure 8 and Figure 9. The results for case-1 in Figure 8 (a) are
relatively straightforward as it can be seen that the influential parameters
are $\sigma_{L}$, $\sigma_{t}$, $\mu_{L}$, $\mu_{t}$ from the first to the
fourth eigenvectors. As the first eigenvalue is almost two times of the second
one, this implies we can focus on the parameter $\sigma_{L}$ to change the
entropy the most, as discussed in Section 3.2. However, the sensitivity
information from the eigenvectors for case-2 in Figure 9 (a) are less clear.
This is because the four eigenvalues in this case are of very similar
magnitude. Furthermore, there are three different parameters that are
important for the 2nd eigenvector, $\mu_{E}$, $\mu_{\rho}$, $\mu_{t}$, and the
4th eigenvector, $\sigma_{E}$, $\sigma_{\rho}$, $\sigma_{t}$.

If we take a closer look at the eigenvector results in Figure 8 (a) and Figure
9 (a), it seems that there is a split phenomenon between the mean and standard
deviation of the same variables. In Figure 8 (a), the 1st and 2nd eigenvectors
point us to the standard deviations of the variables $L$ and $t$, while it is
the mean values of the two variables that are important for the 3rd and 4th
eigenvectors. Similarly, in Figure 9 (a) for case-2, $\sigma_{L}$ and
$\mu_{L}$, the mean and standard deviation of the variable $L$, dominates the
1st and the 3rd eigenvector respectively, while the mean and standard
deviation of $E,\>\rho,\>t$ are the influential parameters for the 2nd and 4th
eigenvector. In other words, the dominance of the sensitivity to the mean and
the standard deviation of the same variable splits into different
eigenvectors, e.g., $\sigma_{L}$ dominates the 1st eigenvector while $\mu_{L}$
dominates the 3rd.

(a) standard eigenvectors (EigVector)

(b) symplectic eigenvectors (S-EigVector)

Figure 8: Eigenvectors (EigVector) and symplectic eigenvectors (S-EigVector)
of the FIM for Case-1. The symplectic eigenvectors come in pairs, $\bf
u_{1},v_{1}$ and $\bf u_{2},v_{2}$ , and each pair corresponds to the same
symplectic eigenvalue

(a) standard eigenvectors (EigVector)

(b) symplectic eigenvectors (S-EigVector)

Figure 9: Case-2, same key as Figure 8

This split phenomenon can be understood as a consequence of the normalisation
as mentioned for the motivating example in Section 1.2. The normalisation
compresses the relative magnitudes between the eigenvalues so that the Fisher
information matrix is better conditioned. This makes the ellipsoid for the
relative entropy (c.f. Section 3.2) closer to a sphere. The orthogonality of
the principal axes could then result a split between the mean and standard
deviation parameters of the same variable as their influences are similar. As
a result, it is difficult to identify the most influential variables. On the
contrary, the symplectic decomposition enforces a symplectic structure that
tends to mitigate this issue by making the sensitivity information more
distinctive by pairing the parameters of the same variable.

As described in Section 2, the same Fisher information matrix (FIM) can also
be decomposed onto a symplectic basis. The symplectic eigenvalues and
eigenvectors are named as ‘S-EigValue’ and ‘S-EigVector’ and are shown in
Figure 7, (a)-2 and (b)-2, Figure 8 (b), Figure 9 (b). The dimension of the
symplectic eigenvalue spectrum, 5 in this case, is half of the standard
eigenvalues. The symplectic eigenvectors come in pairs, $(\bf u_{1},v_{1})$
and $(\bf u_{2},v_{2})$ , as shown in Figure 8 (b), Figure 9 (b), and each
pair corresponds to the same symplectic eigenvalue. As compared to the
standard eigenvectors, the split parameters are grouped together in the
symplectic eigenvectors. For example, for case-1 results in Figure 8 (b), the
1st symplectic eigenvector pairs identify $L$ as the influential variable,
with its mean and standard deviation dominates $\bf u_{1}$ and $\bf v_{1}$
respectively. The same can be found for the variable $t$ for 2nd pair of
symplectic eigenvectors $(\bf u_{2},v_{2})$ in Figure 8 (b). Similar
conclusion can be found for the case-2 results in Figure 9 (b). The grouping
of the parameters is a consequence of the symplectic structure, where
parameters are regarded as two-parameter pairs, e.g., $(\mu,\sigma)$ in this
case. This is really pertinent in the sensitivity analysis as it makes the
influential variables, or two-parameter pairs, very distinctive.

It is interesting to note that in both cases, the square of the 1st symplectic
eigenvalue is almost the same as the product of the two standard eigenvalues
that split. For example, for case-1, $d_{1}^{2}=1.96$ and that is about the
same as the product $\lambda_{1}\times\lambda_{3}=1.97$, which corresponds to
the 1st and 3rd eigenvectors that are dominated by $\sigma_{L}$ and $\mu_{L}$
respectively. This is a consequence of Eq 14, where when two of the standard
eigenvectors splits, the product of their eigenvalues tends to be conserved in
the corresponding symplectic decomposition. This also occurs for decision
oriented pairings to be presented in Figure 10. For case-1, the 1st and 2nd
eigenvectors are dominated by $\sigma_{L}$ and $\sigma_{t}$. When these two
parameters are paired together, as to be seen in Figure 10 (a),
$d_{1}^{2}=2.56$ is very similar to the product
$\lambda_{1}\times\lambda_{2}=2.6$ in Figure 8 (a).

Although in this simple example the parameter split found from the standard
eigenvalue analysis can be easily identified, it will get more difficult with
a larger number of parameters. On the contrary, the parameter pairing
structure is enforced by the symplectic decomposition. The same conclusion can
also be made for the proportionally normalised FIM as presented in the
_Supplementary Material_.

In addition, contrary to the standard eigenvalue decomposition where the
sensitivity information is fixed for a given FIM, the symplectic variant takes
account of user input for the pairing decisions. As an example, two different
pairing decisions are considered for the same FIM from case 1 presented in
Figure 8. The symplectic eigenvectors are shown in Figure 10, with the rows
and columns of the FIM rearranged as per the pairing requirement. It should be
noted that while the standard eigenvalue analysis is invariant with respect to
the row/column operation, the symplectic spectra are different as shown in
Figure 10.

Instead of using the mean and stand deviation as natural pairs for the same
variables, we consider pairing the mean and standard deviation parameters for
two different variables. In Figure 10 (a), we pair $L$ and $t$, i.e.,
($\mu_{L}$,$\mu_{t}$) and ($\sigma_{L}$,$\sigma_{t}$) in pairs, while in
Figure 10 (b), we pair $L$ and $w$, i.e., ($\mu_{L}$,$\mu_{w}$) and
($\sigma_{L}$,$\sigma_{w}$) in pairs. It is noted in passing that although
mainly for demonstrating purposes, these pairing decisions can arise in
practice where the actions to reduce the uncertainties of two independent
variables can impact both. For example, modifying of the production line can
have the same effect on the uncertainties of the length $L$ and the thickness
$t$, and this would prompt a decision-oriented sensitivity analysis with
respect to the parameter pairs. It is clear from Figure 10 that the
sensitivity to the paired parameters are grouped together as before. For
example, the sensitivity to the pair ($\sigma_{L}$, $\sigma_{t}$) dominates
the first group of the symplectic eigenvectors in Figure 10 (a), and
($\sigma_{L}$, $\sigma_{w}$) are grouped together in the 2nd symplectic
eigenvector pair in Figure 10 (b). It is interesting to note that the 1st
symplectic eigenvector pair in Figure 10 (b) is very similar to the 2nd
symplectic eigenvector pair in Figure 8 (b). This is mostly because the
pairing for $E$, $\rho$ and $t$ are the same for the two figures. However, for
$L$ and $w$ pairing in Figure 10 (b), the dominance of $L$ seen in Figure 8
disappears and $t$ becomes the most sensitive variable for the case described
in Figure 10 (b). This demonstrates that the symplectic decomposition is
decision oriented, as even for the same FIM, it extracts different sensitivity
information according to different pairing strategies.

(a) S-EigVector ($L$ and $t$ pairing)

(b) S-EigVector ($L$ and $w$ pairing)

Figure 10: Symplectic eigenvectors (S-EigVector) of the FIM for Case-1, for
two different pairing decisions. (a) ($\mu_{L}$,$\mu_{t}$) and
($\sigma_{L}$,$\sigma_{t}$) in pairs; (b) ($\mu_{L}$,$\mu_{w}$) and
($\sigma_{L}$,$\sigma_{w}$) in pairs. The rest of the pairs are same as Figure
8

While only an engineering design example is considered here, the benefits of
the symplectic decomposition are expected for general decision problems,
whenever the spectral analysis of the FIM is used for sensitivity analysis. As
the additional computation cost is negligible once the Fisher information
matrix is obtained, the symplectic decomposition can be used in tandem with
the standard eigenvalue decomposition to extract more useful sensitivity
information.

##  6 Conclusions

A new probabilistic sensitivity metric has been proposed based on the
symplectic spectral analysis of the Fisher information matrix (FIM).
Contrasting to the standard eigenvalue decomposition, the symplectic
decomposition of the FIM naturally identifies the sensitivity information with
respect to two-parameter pairs, e.g., mean and standard deviation of a random
input. The resulted symplectic eigenvalues of large magnitude, and the
corresponding symplectic eigenvectors of the FIM, then reveal the most
sensitive two-parameter pairs.

Through an engineering design example using a simple cantilever beam, it is
observed that the normalisation of the FIM tends to compress the relative
magnitudes between the eigenvalues. Geometrically the relative entropy
ellipsoid becomes near-spherical (c.f. Section 3.2) due to the normalisation,
and this results a split phenomenon of different distribution parameters of
the same variable. It is demonstrated that the proposed symplectic
decomposition can reveal the concealed sensitivity information between the
parameter pairs. Contrary to the standard eigen decomposition where the
sensitivity information is fixed for a given FIM, the symplectic variant takes
account of user input for the pairing decisions. As the additional computation
cost is negligible once the Fisher information matrix is obtained, the
symplectic decomposition can thus be used in tandem with the standard
eigenvalue decomposition to gain more insight into the sensitivity
information, and orient towards a better decision support under uncertainties.

The proposed symplectic decomposition is only applicable for parameter space
of even dimensions. For distribution parameters that belong to two-parameter
family of probability distributions, such as the widely used location-scale
families, there is a natural pairing of the parameters. For more general
cases, a decision-oriented two-parameter re-parametrization of the Fisher
information matrix is necessary and that is one of the future research to be
explored.

## Acknowledgment

This work has been funded by the Engineering and Physical Sciences Research
Council through the award of a Programme Grant “Digital Twins for Improved
Dynamic Design”, Grant No. EP/R006768. For the purpose of open access, the
author has applied a Creative Commons Attribution (CC BY) licence to any
Author Accepted Manuscript version arising. The author is grateful to
Professor Robin Langley, University of Cambridge, for comments on an early
draft of this paper and the support to publish this work.

## Data availability statement

The datasets generated during and/or analysed during the current study are
available in the GitHub repository: https://github.com/longitude-
jyang/SymplecticFisherSensitivity

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# On Mixed Concatenations of Fibonacci and Lucas Numbers Which are Fibonacci
Numbers

Alaa ALTASSAN1 and Murat ALAN2
1King Abdulaziz University, Department of Mathematics,
P.O. Box 80203, Jeddah 21589, Saudi Arabia
e-mail<EMAIL_ADDRESS>
2Yildiz Technical University
Mathematics Department, 34210, Istanbul, Turkey.
e-mail<EMAIL_ADDRESS>
###### Abstract

Let $(F_{n})_{n\geq 0}$ and $(L_{n})_{n\geq 0}$ be the Fibonacci and Lucas
sequences, respectively. In this paper we determine all Fibonacci numbers
which are mixed concatenations of a Fibonacci and a Lucas numbers. By mixed
concatenations of $a$ and $b$, we mean the both concatenations $\overline{ab}$
and $\overline{ba}$ together, where $a$ and $b$ are any two non negative
integers. So, the mathematical formulation of this problem leads us searching
the solutions of two Diophantine equations $F_{n}=10^{d}F_{m}+L_{k}$ and
$F_{n}=10^{d}L_{m}+F_{k}$ in non-negative integers $(n,m,k),$ where $d$
denotes the number of digits of $L_{k}$ and $F_{k}$, respectively. We use
lower bounds for linear forms in logarithms and reduction method in
Diophantine approximation to get the results.

##  1 Introduction

Let $(F_{n})_{n\geq 0}$ and $(L_{n})_{n\geq 0}$ be the Fibonacci and Lucas
sequences given by $F_{0}=0$, $F_{1}=1$, $L_{0}=2$, $L_{1}=1$,
$F_{n+2}=F_{n+1}+F_{n}$ and $L_{n+2}=L_{n+1}+L_{n}$ for $n\geq 0,$
respectively.

In recent years, the numbers in Fibonacci, Lucas or some similar sequences
which are concatenations of two or more repdigits are investigated by a series
of papers [2, 5, 6, 8, 13, 14]. In the case of the concatenation of binary
recurrent sequences, there is a general result due to Banks and Luca [4]. In
[4], they proved that if $u_{n}$ is any binary recurrent sequence of integers,
then only finitely many terms of the sequence $u_{n}$ can be written as
concatenations of two or more terms of the same sequence $u_{n}$ under some
mild hypotheses on $u_{n}.$ In particular, they proved that 13, 21 and 55 are
the only Fibonacci numbers which are non trivial concatenations of two terms
of Fibonacci numbers. In [1], Fibonacci and Lucas numbers which can be written
as concatenations of two terms of the other sequences also investigated and it
is shown that 13, 21 and 34 (1, 2, 3, 11, 18 and 521 ) are the only Fibonacci
(Lucas) numbers which are concatenations of two Lucas (Fibonacci) numbers.

In this paper, we study the mixed concatenations of these famous two sequences
which forms a Fibonacci number. By mixed concatenations of $a$ and $b$, for
any two non negative integers $a$ and $b,$ we mean the both concatenations
$\overline{ab}$ and $\overline{ba},$ together. So, we search all Fibonacci
numbers of the form $\overline{F_{m}L_{k}}$, that is the concatenations of
$F_{m}$ and $L_{k}$, as well as of the form $\overline{L_{m}F_{k}},$ that is
the concatenations of $L_{m}$ and $F_{k}.$ In other words, by the mathematical
expression of this problem, we solve the Diophantine equations

$F_{n}=10^{d}F_{m}+L_{k}$ (1)

and

$F_{n}=10^{d}L_{m}+F_{k}$ (2)

in non negative integers $(n,m,k)$ where $d$ denotes the number of digits of
$L_{k}$ and $F_{k}$, respectively and we get the following results.

###### Theorem 1

All Fibonacci numbers which are concatenations of a Fibonacci and a Lucas
numbers are only the numbers 1, 2, 3, 13, 21 and 34.

###### Theorem 2

All Fibonacci numbers which are concatenations of a Lucas and a Fibonacci
numbers are only the numbers 13 and 21.

In the next chapter, we give some details of the methods we used in this study
to prove the above theorems. In fact, we mainly use two powerful tools. The
first one is the theory of non zero linear forms in logarithms of algebraic
numbers which is due to Matveev [12] and the second one is reduction method
based on the theory of continued fractions given in [7] which is a version of
Baker-Davenport lemma [3]. In the third section, we give the proofs of the
above theorem. All calculations and computations are made with the help of the
software Maple.

##  2 Preliminaries

Let $F_{n}$ and $L_{n}$ be the Fibonacci and Lucas numbers, respectively. The
Binet formula for Fibonacci and Lucas numbers are given by

$F_{n}=\dfrac{\alpha^{n}-\beta^{n}}{\sqrt{5}},\qquad
L_{n}=\alpha^{n}+\beta^{n}\quad\quad n\geq 0,$

where

$\alpha=\frac{1+\sqrt{5}}{2}\quad\text{and}\quad\beta=\frac{1-\sqrt{5}}{2}$

are the roots of the equation $x^{2}-x-1=0.$ By using the Binet formula of
these sequences, one can see that, by induction,

$\alpha^{n-2}\leq F_{n}\leq\alpha^{n-1}$ (3)

and

$\alpha^{n-1}\leq L_{n}\leq 2\alpha^{n}$ (4)

hold for all $n\geq 1$ and $n\geq 0,$ respectively.

Let $\eta$ be an algebraic number of degree $d$ with minimal polynomial

$a_{0}x^{d}+a_{1}x^{d-1}+\cdots+a_{d}=a_{0}\prod_{i=1}^{d}(x-\eta^{(i)}),$

where the $a_{i}$’s are relatively prime integers with $a_{0}>0$ and the
$\eta^{(i)}$’s are conjugates of $\eta$. Recall that, the logarithmic height
of $\eta$ is defined by

$h(\eta)=\frac{1}{d}\left(\log
a_{0}+\sum_{i=1}^{d}\log\left(\max\\{|\eta^{(i)}|,1\\}\right)\right).$

In particular, for a rational number $p/q$ with $\gcd(p,q)=1$ and $q>0$,
$h(p/q)=\log\max\\{|p|,q\\}$. The logarithmic height $h(\eta)$ has the
following properties:

  * $\bullet$

$h(\eta\pm\gamma)\leq h(\eta)+h(\gamma)+\log 2$.

  * $\bullet$

$h(\eta\gamma^{\pm 1})\leq h(\eta)+h(\gamma)$.

  * $\bullet$

$h(\eta^{s})=|s|h(\eta),$ $s\in\mathbb{Z}$.

######  Theorem 3 (Matveev’s Theorem)

Assume that $\gamma_{1},\ldots,\gamma_{t}$ are positive real algebraic numbers
in a real algebraic number field $\mathbb{K}$ of degree $d_{\mathbb{K}}$,
$b_{1},\ldots,b_{t}$ are rational integers, and

$\Lambda:=\eta_{1}^{b_{1}}\cdots\eta_{t}^{b_{t}}-1,$

is not zero. Then

$|\Lambda|>\exp\left(-1.4\cdot 30^{t+3}\cdot t^{4.5}\cdot
d_{\mathbb{K}}^{2}(1+\log d_{\mathbb{K}})(1+\log B)A_{1}\cdots A_{t}\right),$

where $B\geq\max\\{|b_{1}|,\ldots,|b_{t}|\\},$ and
$A_{i}\geq\max\\{d_{\mathbb{K}}h(\eta_{i}),|\log\eta_{i}|,0.16\\},$ for all
$i=1,\ldots,t.$

We cite the following lemma from [7], which is a version of the reduction
method based on the Baker-Davenport lemma [3], and we use it to reduce some
upper bounds on the variables. Recall that, for a real number $\theta,$ we put
$||\theta||=\min\\{|\theta-n|:n\in\mathbb{N}\\},$ the distance from $\theta$
to the nearest integer.

###### Lemma 4

Let $M$ be a positive integer, $p/q$ be a convergent of the continued fraction
of the irrational $\gamma$ such that $q>6M$, and let $A,B,\mu$ be some real
numbers with $A>0$ and $B>1$. If $\epsilon:=||\mu q||-M||\gamma q||>0$, then
there is no solution to the inequality

$0<|u\gamma-v+\mu|<AB^{-w},$

in positive integers $u,v$ and $w$ with

$u\leq M\quad\text{and}\quad w\geq\frac{\log(Aq/\epsilon)}{\log B}.$

##  3 Proof of Theorem 1 and 2

Proof of Theorem 1: Assume that the equation (1) holds. We will need the
relations among the variables $n,m,k$ and $d$ through this section. Note that,
we may write the number of digits of $L_{k}$ as
$d=\lfloor\log_{10}L_{k}\rfloor+1$ where $\lfloor\theta\rfloor$ is the floor
function of $\theta$ , that is the greatest integer less than or equal to
$\theta.$ Thus,

$\displaystyle d=\lfloor\log_{10}L_{k}\rfloor+1\leq 1+\log_{10}L_{k}$
$\displaystyle\leq
1+\log_{10}{(2\alpha^{k})}=1+{k}\log_{10}{\alpha}+\log_{10}{2}$

and

$d=\lfloor\log_{10}L_{k}\rfloor+1>\log_{10}L_{k}\geq\log_{10}{\alpha^{k-1}}\geq{(k-1)}\log_{10}{\alpha}.$

From the above relations we may get more explicit bounds for $d$ as

$\frac{k-1}{5}<d<\frac{k+6}{4},$ (5)

by using the fact that $(1/5)<\log_{10}{\alpha}\cong 0.208...<(1/4)$ and
$\log_{10}{2}<0.31.$

In particular,

$L_{k}=10^{\log_{10}L_{k}}<10^{d}<10^{1+\log_{10}L_{k}}<10L_{k}.$

From the last inequality together with (1) we write

$\alpha^{n-2}\leq F_{n}=10^{d}F_{m}+L_{k}\leq
10L_{k}F_{m}+L_{k}<11F_{m}L_{k}<22\alpha^{m+k-1}\leq\alpha^{m+k+6}$

and

$\alpha^{n-1}\geq
F_{n}=F_{m}10^{d}+L_{k}>F_{m}L_{k}+L_{k}>F_{m}L_{k}\geq\alpha^{m+k-3}.$

Hence, we have that

$m+k-2<n<m+k+8.$ (6)

Before further calculations, we wrote a short computer program to search the
variables $n,m$ and $k$ satisfying (1) in the range $0\leq m,k<200$ and we
found only the Fibonacci numbers given in Theorem 1. So from now on we may
assume that $\max\\{m,k\\}\geq 200.$

Note that, we may take $n-k\geq 4.$ Indeed, using the well-known fact
$L_{k}=F_{k+1}+F_{k-1}$, see for example [10], we may write equation (1) as

$F_{n}=10^{d}F_{m}+F_{k+1}+F_{k-1}.$

Then, clearly $n\neq k$ and $n\neq k+1.$ If $m=0,$ then the case $F_{n}=L_{k}$
is possible only for $L_{k}\in\\{1,2,3\\},$ that is $\max\\{m,k\\}<3,$ a
contradiction. So $m\neq 0$ and hence from the inequality

$F_{n}=10^{d}F_{m}+L_{k}\geq 10^{d}+L_{k}>2L_{k}=2(F_{k+1}+F_{k-1}),$

we see that the cases $n=k+2$ and $n=k+3$ are not also possible. So we get
that

$n-k\geq 4.$ (7)

Using Binet formula for Fibonacci and Lucas sequences, we rewrite equation (1)
as

$\dfrac{\alpha^{n}-\beta^{n}}{\sqrt{5}}=\dfrac{\alpha^{m}}{\sqrt{5}}10^{d}-\dfrac{\beta^{m}}{\sqrt{5}}10^{d}+L_{k},$
$\dfrac{\alpha^{n}}{\sqrt{5}}-\dfrac{10^{d}\alpha^{m}}{\sqrt{5}}=\dfrac{\beta^{n}}{\sqrt{5}}-\dfrac{10^{d}\beta^{m}}{\sqrt{5}}+L_{k}.$

Multiplying both sides of the above equation by ${\sqrt{5}}/{\alpha^{n}},$ and
taking absolute values of the both sides of it, we get that

$\left|1-\dfrac{10^{d}}{\alpha^{n-m}}\right|\leq\dfrac{|\beta^{n}|}{\alpha^{n}}+\dfrac{10^{d}|\beta^{m}|}{\alpha^{n}}+\dfrac{L_{k}\sqrt{5}}{\alpha^{n}}$
$\qquad\qquad\qquad\leq\dfrac{1}{\alpha^{2n}}+\dfrac{10L_{k}}{\alpha^{n+m}}+\dfrac{2\alpha^{k}\sqrt{5}}{\alpha^{n}}$
$\qquad\qquad\qquad<\dfrac{1}{\alpha^{2n}}+\dfrac{20\alpha^{k}}{\alpha^{n+m}}+\dfrac{2\sqrt{5}}{\alpha^{n-k}}.$

Since $20<\alpha^{7},$ we find that

$\Lambda_{1}:=\left|1-\dfrac{10^{d}}{\alpha^{n-m}}\right|<\dfrac{3}{\alpha^{n-k-7}}.$
(8)

Let $(\eta_{1},b_{1})=(10,d)$ and $(\eta_{2},b_{2})=(\alpha,-(n-m)),$ where
$\eta_{1},\eta_{2}\in\mathbb{K}=\mathbb{Q}(\sqrt{5}).$ We take
$d_{\mathbb{K}}=2$, to be the degree of the real number field $\mathbb{K}.$
Since $h(\eta_{1})=\log{10},$ $h(\eta_{2})=\dfrac{1}{2}\log{\alpha},$ we take
$A_{1}=2\log{10},$ $A_{2}=\log{\alpha}.$ Suppose that $n-m<d.$ Then from the
two relations (5) and (6), we get that

$k-2<n-m<d<(k+6)/4,$

which implies $k\leq 4.$ Since $L_{4}=7,$ we have $d=1.$ Hence $n=m,$ that is
$F_{n}=10F_{n}+L_{k},$ which is clearly false . So we take

$B:=\max\\{|b_{i}|\\}=\max\\{d,n-m\\}=n-m.$

In (8), $\Lambda_{1}\neq 0.$ Indeed if $\Lambda_{1}=0,$ then we get that
$\alpha^{n-m}=10^{d}\in\mathbb{Q}$, which is possible only for $n=m$ and we
know that this is not the case. So $\Lambda_{1}\neq 0.$ Now we apply Theorem 3
to $\Lambda_{1}$ and we get that

$\log{(\Lambda_{1})}>-1.4\cdot 30^{5}\cdot 2^{4.5}\cdot 2^{2}(1+\log
2)(1+\log{(n-m)})\cdot 2\log{10}\cdot\log\alpha.$

On the other hand, taking the logarithm of both sides of (8), we get also

$\log{(\Lambda_{1})}<\log 3-(n-k-7)\log\alpha.$

Combining last two inequalities, we get that

$n-k-7<2.41\cdot 10^{10}\cdot(1+\log{(n-m)}).$ (9)

We will turn (9) later. Now we rewrite (1) as

$\dfrac{\alpha^{n}}{\sqrt{5}}-\alpha^{k}-10^{d}F_{m}=\dfrac{\beta^{n}}{\sqrt{5}}+\beta^{k},$
$\dfrac{\alpha^{n}}{\sqrt{5}}\left(1-\alpha^{k-n}\sqrt{5}\right)-10^{d}F_{m}=\dfrac{\beta^{n}}{\sqrt{5}}+\beta^{k}.$

Note that, by (7), $1-\alpha^{k-n}\sqrt{5}>0$ and
$1<\dfrac{1}{1-\alpha^{k-n}\sqrt{5}}<2$ for $n-k\geq 4.$ So, we may divide,
and than take the absolute value of both sides of the last equality to get
that

$\left|1-\dfrac{F_{m}10^{d}\sqrt{5}}{{\alpha^{n}}\left(1-\alpha^{k-n}\sqrt{5}\right)}\right|<\left(\dfrac{1}{1-\alpha^{k-n}\sqrt{5}}\right)\left(\dfrac{|\beta^{n}|}{\alpha^{n}}+\dfrac{|\beta^{k}|\sqrt{5}}{\alpha^{n}}\right)$
$\qquad\qquad<2\left(\dfrac{1}{\alpha^{2n}}+\dfrac{\sqrt{5}}{\alpha^{n+k}}\right).$

So

$\left|\Lambda_{2}\right|:=\left|1-\dfrac{10^{d}F_{m}\sqrt{5}}{{\alpha^{n}}\left(1-\alpha^{k-n}\sqrt{5}\right)}\right|<\dfrac{7}{\alpha^{2k}}.$
(10)

Let $(\eta_{1},b_{1})=(10,d)$, $(\eta_{2},b_{2})=(\alpha,-n)$ and
$(\eta_{3},b_{3})=\left(\dfrac{F_{m}\sqrt{5}}{\left(1-\alpha^{k-n}\sqrt{5}\right)},1\right)$.
Again $\eta_{1},\eta_{2}$ and $\eta_{3}$ are all belongs to the real quadratic
number field $\mathbb{K}=\mathbb{Q}(\sqrt{5}).$ So we take $d_{\mathbb{K}}=2$,
to be the degree of $\mathbb{K},$ $h(\eta_{1})=\log{10}$,
$h(\eta_{2})=(1/2)\log(\alpha)$ and for $h(\eta_{3}),$ we need the properties
of logarithmic height given in the Preliminaries, so that

$\displaystyle
h(\eta_{3})=h\left(\dfrac{F_{m}\sqrt{5}}{1-\alpha^{k-n}\sqrt{5}}\right)$
$\displaystyle\leq h(F_{m})+h(\sqrt{5})+h(1-\alpha^{k-n}\sqrt{5})$
$\displaystyle\leq\log(F_{m})+h(\sqrt{5})+h(\alpha^{k-n}\sqrt{5})+\log 2$
$\displaystyle\leq(m-1)\log(\alpha)+\dfrac{|k-n|}{2}\log\alpha+1.$

Since $m-1<n-k+1$ from (6), we write
$h(\eta_{3})<1+\dfrac{3(n-k)+2}{2}\log(\alpha).$

So we take

$A_{1}=2\log{10},\quad A_{2}=\log\alpha\quad\text{and}\quad
A_{3}=2+{(3(n-k)+2)}\log(\alpha).$

By (5), if $n<d<(k+6)/4,$ then we find that $4n-6<k$ and hence $m+3n<8$ from
(6), a contradiction since $\max\\{m,k\\}\geq 200.$ So

$B:=\max\\{|b_{i}|\\}=\max\\{d,n,1\\}=n.$

We show that $\Lambda_{2}\neq 0.$ For this purpose, assume that
$\Lambda_{2}=0.$ Then we get that

$\alpha^{n}-\alpha^{k}\sqrt{5}=10^{d}F_{m}\sqrt{5},$

Conjugating this expression in $\mathbb{K}$ we obtain

$\beta^{n}+\beta^{k}\sqrt{5}=-10^{d}F_{m}\sqrt{5}.$

From these last two equality, we find that

$10\sqrt{5}\leq 10^{d}F_{m}\sqrt{5}=|\beta^{n}+\beta^{k}\sqrt{5}|\leq
2\max\\{\beta^{n},\beta^{k}\sqrt{5}\\}\leq 2\sqrt{5},$

a contradiction. So we conclude that $\Lambda_{2}\neq 0.$ Thus, we apply
Theorem 3 to $\Lambda_{2}$ given in (10) and we get that

$\log{(\Lambda_{2})}>-1.4\cdot 30^{6}\cdot 3^{4.5}\cdot 2^{2}(1+\log 2)(1+\log
n)2\log{10}\log\alpha\cdot[2+{(3(n-k)+2)}\log(\alpha)].$ (11)

On the other hand, from (10), we know that

$\log{(\Lambda_{2})}<\log{7}-2k\log\alpha.$ (12)

Combining (11) and (12) we get that

$k<2.24\cdot 10^{12}\cdot(1+\log n)[2+(3(n-k)+2)\log(\alpha)].$ (13)

Now, we focus on two inequalities (13) and (9) to get a bound for $n$ by
examining the cases $k\leq m$ and $m<k$ separately. First, assume that $k\leq
m.$ Then $n-m\leq n-k$ and therefore, from (9), we find

$n-k-7<2.41\cdot 10^{10}\cdot(1+\log{(n-k)})$

which means that $n-k<8\cdot 10^{11}.$ So it follows that

$n<1.7\cdot 10^{12},$

since from the left side of (6), we know that $m-2<n-k$ and from the right
side of (6), $n<2m+8.$

Let $m\leq k.$ Then $n<2k+8.$ From (9), in particular we have that

$n-k-7<2.41\cdot 10^{10}\cdot(1+\log{n}).$ (14)

Substituting (14) into (13) and using the fact that $(n-8)/2<k,$ we find that

$n<7\cdot 10^{26}.$ (15)

So whether $m\leq k$ or not, in either case, we have that $n<7\cdot 10^{26}.$
Now, we reduce this upper bound on $n.$

###### Lemma 5

If the equation (1) holds, then $m\leq 150.$ In particular, if $k\leq m,$ then
the equation (1) has only solution for $F_{n}\in\\{1,2,3,13,21,34\\}.$

Proof. Suppose that $m>150.$ Let $\Gamma_{1}:=d\log{10}-(n-m)\log\alpha.$
Since $141<m-9<n-k-7,$ from (8)

$\left|\Lambda_{1}\right|:=\left|\exp({\Gamma_{1}})-1\right|<\dfrac{3}{\alpha^{n-k-7}}<\dfrac{1}{2}.$

Recall that, when $v<\dfrac{1}{2},$ the inequality

$\left|\exp(u)-1\right|<v\quad\text{implies that}\quad\left|u\right|<2v.$

So, it follows that $\left|\Gamma_{1}\right|<\dfrac{6}{\alpha^{n-k-7}}.$
Moreover, $\Gamma_{1}\neq 0,$ since $\Lambda_{1}.$ Thus, we write

$0<\left|\dfrac{d}{n-m}-\dfrac{\log\alpha}{\log
10}\right|<\dfrac{6}{(n-m)\alpha^{n-k-7}\log 10}.$ (16)

Note that, we have

$\dfrac{6}{(n-m)\alpha^{n-k-7}\log 10}<\dfrac{1}{2(n-m)^{2}},$

for otherwise

$\dfrac{6}{(n-m)\alpha^{n-k-7}\log 10}\geq\dfrac{1}{2(n-m)^{2}}$

implies that

$n>n-m>\dfrac{\alpha^{n-k-7}\log 10}{12}>\dfrac{\alpha^{141}\log
10}{12}>5.6\cdot 10^{28},$

which is a contradiction because of (15). Then we have that

$0<\left|\dfrac{\log\alpha}{\log
10}-\dfrac{d}{n-m}\right|<\dfrac{1}{2(n-m)^{2}},$

which implies that $\dfrac{d}{n-m}$ is a convergent of continued fractions of
$\dfrac{\log\alpha}{\log 10}$, say $\dfrac{d}{n-m}=\dfrac{p_{i}}{q_{i}}.$
Since $\gcd(p_{i},q_{i})=1$, we have $q_{i}\leq n-m\leq n<7\cdot 10^{26}.$ Let
$[a_{1},a_{2},a_{3},a_{4},a_{5},\ldots]=[0,4,1,3,1,\ldots]$ be the continued
fraction expansion of $\log\alpha/\log 10$. With the help of Maple, we find
that $i<54$ and $\max\\{a_{i}\\}=a_{37}=106$ for $i=1,2,\ldots,54.$ So from
the well known property of continued fractions, see for example [9, Theorem
1.1.(iv)], we get that

$\dfrac{1}{108(n-m)^{2}}\leq\dfrac{1}{(a_{i}+2)(n-m)^{2}}<\left|\dfrac{\log\alpha}{\log
10}-\dfrac{d}{n-m}\right|<\dfrac{6}{(n-m)\alpha^{n-k-7}\log 10}$

which means

$n>n-m>\dfrac{\alpha^{n-k-7}\log 10}{6\cdot 108}>10^{27}.$

But this is also a contradiction, since $n<7\cdot 10^{26}.$ Therefore, we
conclude that $m\leq 150.$

By Lemma 5, from now on, we assume that $m\leq k$ and hence $n<2k+8.$
Moreover, we write also $n-k<m+8\leq 158.$ By substituting this upper bound
for $n-k$ into the (13), we get a better bound for $n$ as

$n-8<2k<2\cdot 2.24\cdot 10^{12}\cdot(1+\log n)(476\log(\alpha)+2).$ (17)

So from (17), it follows that

$n<4.5\cdot 10^{16}.$

Let

$\Gamma_{2}:=d\log{10}-n\log\alpha+\log\left({\dfrac{F_{m}\sqrt{5}}{1-\alpha^{k-n}\sqrt{5}}}\right)$

so that
$\left|\Lambda_{2}\right|:=\left|\exp{(\Gamma_{2})}-1\right|<\dfrac{7}{\alpha^{2k}}.$
Then $\left|\Gamma_{2}\right|<\dfrac{14}{\alpha^{2k}},$ since
$\dfrac{1}{\alpha^{2k}}<\dfrac{1}{2}.$ So, from (3),

$0<\left|d\dfrac{\log{10}}{\log\alpha}-n+\dfrac{\log\left({\dfrac{F_{m}\sqrt{5}}{1-\alpha^{k-n}\sqrt{5}}}\right)}{\log\alpha}\right|<\dfrac{14}{\alpha^{2k}\log\alpha}.$

Now, we take

$M:=4.5\cdot
10^{16}>n>d\qquad\text{and}\qquad\tau:=\dfrac{\log{10}}{\log\alpha}.$

Then, in the continued fraction expansion of irrational $\tau$, we take
$q_{60},$ the denominator of the $60th$ convergent of $\tau,$ which exceeds
$6M.$ Now, with the help of Maple, we calculate

$\epsilon_{m,n-k}:=||\mu_{m,n-k}q_{60}||-M||\tau q_{60}||$

for each $1\leq m\leq 150$ and $4\leq n-k<m+8,$ where

$\mu_{m,n-k}:=\dfrac{\log\left({\dfrac{F_{m}\sqrt{5}}{1-\alpha^{k-n}\sqrt{5}}}\right)}{\log\alpha},\qquad
q_{60}=2568762252997982327345614176552$

and we see that

$0.00034<\epsilon_{50,20}\leq\epsilon_{m,n-k},\qquad\text{for all}\quad
m,n-k.$

Let $A:=\dfrac{14}{\log\alpha},$ $B:=\alpha$ and $\omega:=2k$ . Thus from
Lemma 4, we find that

$2k<\dfrac{\log{\left(Aq_{60}/{0.00034}\right)}}{\log B}<170.$

So we get that $k<85$ which is a contradiction because of bound of $k.$ Thus,
we conclude that the numbers 1, 2, 3, 13, 21 and 34 are the only Fibonacci
numbers which are expressible as a concatenation of a Fibonacci and a Lucas
number.

Proof of Theorem 2: Assume that the equation (2) holds. As
$d=\lfloor\log_{10}F_{k}\rfloor+1$ is the number of digits of $F_{k},$ we have
that

$d=\lfloor\log_{10}F_{k}\rfloor+1\leq 1+\log_{10}F_{k}\leq
1+\log_{10}{\alpha^{k-1}}=1+{(k-1)}\log_{10}{\alpha}<\frac{k+3}{4}$

and

$d=\lfloor\log_{10}F_{k}\rfloor+1>\log_{10}F_{k}\geq\log_{10}{\alpha^{k-2}}={(k-2)}\log_{10}{\alpha}>\frac{k-2}{5}$

since $(1/5)<\log_{10}{\alpha}\cong 0.208...<(1/4).$ Hence, we have that

$\frac{k-2}{5}<d<\frac{k+3}{4},$ (18)

and

$F_{k}\leq
10^{\lfloor\log_{10}F_{k}\rfloor}<10^{d}<10^{1+\lfloor\log_{10}F_{k}\rfloor}\leq
10F_{k}.$

From the last inequality together with (2), we can find a range of $n$
depending on $m$ and $k.$ More precisely,

$\alpha^{n-1}\leq F_{n}=L_{m}10^{d}+F_{k}\leq
L_{m}10F_{k}+F_{k}<11L_{m}F_{k}<22\alpha^{m+k-1}<\alpha^{m+k+7}$

and

$\alpha^{n-2}\geq F_{n}=L_{m}10^{d}+F_{k}\geq
L_{m}F_{k}+F_{k}>L_{m}F_{k}>\alpha^{m+k-3}.$

Hence

$m+k-1<n<m+k+7.$ (19)

When $k,m\leq 200$, by a quick computation, we see that the only solutions of
(2) are those given in Theorem 2. So from now on, we assume that
$\max\\{m,k\\}>200.$ Using Binet formula for Fibonacci and Lucas sequences, we
rewrite equation (2) as

$\dfrac{\alpha^{n}}{\sqrt{5}}-\dfrac{\beta^{n}}{\sqrt{5}}=\left({\alpha^{m}+\beta^{m}}\right)10^{d}+F_{k},$
$\dfrac{\alpha^{n}}{\sqrt{5}}-\alpha^{m}10^{d}=\dfrac{\beta^{n}}{\sqrt{5}}+\beta^{m}10^{d}+F_{k},$

Now, we multiply both sides of the above equation by
${\sqrt{5}}/{\alpha^{n}},$ and take the absolute value of both sides of it.
Thus, we get

$\displaystyle\left|1-\dfrac{10^{d}}{\alpha^{n-m}/\sqrt{5}}\right|$
$\displaystyle\leq\dfrac{\left|\beta\right|^{n}}{\alpha^{n}}+\dfrac{\left|\beta\right|^{m}10^{d}\sqrt{5}}{\alpha^{n}}+\dfrac{F_{k}\sqrt{5}}{\alpha^{n}}$
$\displaystyle<\dfrac{1}{\alpha^{2n}}+\dfrac{10F_{k}\sqrt{5}}{\alpha^{n+m}}+\dfrac{\alpha^{k-1}\alpha^{2}}{\alpha^{n}}$
$\displaystyle<\dfrac{1}{\alpha^{2n}}+\dfrac{\alpha^{5}\alpha^{k-1}\alpha^{2}}{\alpha^{n+m}}+\dfrac{1}{\alpha^{n-k-1}}$
$\displaystyle<\dfrac{3}{\alpha^{n-k-6}}.$

So, we have that

$\Lambda_{3}:=\left|1-\dfrac{10^{d}\sqrt{5}}{\alpha^{n-m}}\right|<\dfrac{3}{\alpha^{n-k-6}}.$
(20)

Now, we may apply Theorem 3 to the left side of the inequality (20) with
$(\eta_{1},b_{1})=(10,d)$, $(\eta_{2},b_{2})=(\alpha,-(n-m))$ and
$(\eta_{3},b_{3})=(\sqrt{5},1)$. Since $\eta_{1},$ $\eta_{2}$ and $\eta_{3}$
belong to the real quadratic number field $\mathbb{K}=\mathbb{Q}(\sqrt{5})$,
we take $d_{\mathbb{K}}=2$, to be the degree of the number field $\mathbb{K}.$
Since $h(\eta_{1})=\log{10},$ $h(\eta_{2})=\dfrac{1}{2}\log{\alpha},$ and
$h(\eta_{3})=\log\sqrt{5}$ we take $A_{1}=2\log{10},$ $A_{2}=\log{\alpha}$ and
$A_{3}=\log 5.$ Assume that $\Lambda_{3}=0.$ Then, we get that
$\alpha^{n-m}=10^{d}\sqrt{5},$ that is, $\alpha^{2(n-m)}=5\cdot
10^{2d}\in\mathbb{Q},$ but this is false for $n-m\neq 0$ and one can see that
the case $n-m=0$ is not possible from (2). So $\Lambda_{3}\neq 0.$

Now, we claim that $\max\\{d,n-m\\}=n-m.$ Indeed, if $n-m<d$ then from the
inequalities (18) and (19), we get that $k-1<n-m<d<(k+3)/4.$ Thus $k<3,$ which
means that $d=1$ and hence $n-m<1,$ that is $n=m.$ But from the identity
$L_{i}=F_{i-1}+F_{i+1},$ we see that the case $n=m$ is not possible. So

$B:=\max\\{|b_{i}|\\}=\max\\{d,n-m,1\\}=n-m.$

Now, we are ready to apply Theorem 3 to $\Lambda_{3}$ given in (20), and we
get that

$\log{(\Lambda_{3})}>-1.4\cdot 30^{6}\cdot 3^{4.5}\cdot 2^{2}(1+\log
2)(1+\log{(n-m)})\cdot 2\log{10}\cdot\log\alpha\cdot\log 5.$

Combining this inequality with the one directly obtained from (20), as
$\log(\Lambda_{3})<\log 3-(n-k-6)\log\alpha,$ we get that

$n-k-6<7.19\cdot 10^{12}(1+\log{(n-m)}).$ (21)

Now, we rewrite (2) as

$\dfrac{\alpha^{n}}{\sqrt{5}}-L_{m}10^{d}-\dfrac{\alpha^{k}}{\sqrt{5}}=\dfrac{\beta^{n}}{\sqrt{5}}+\dfrac{\beta^{k}}{\sqrt{5}},$
$\dfrac{\alpha^{n}}{\sqrt{5}}\left(1-\alpha^{k-n}\right)-L_{m}10^{d}=\dfrac{\beta^{n}}{\sqrt{5}}+\dfrac{\beta^{k}}{\sqrt{5}}.$

After dividing both sides of the last equation by
$\dfrac{\alpha^{n}}{\sqrt{5}}\left(1-\alpha^{k-n}\right),$ and taking the
absolute value of both sides of it, we get that

$\left|1-\dfrac{L_{m}10^{d}\sqrt{5}}{\alpha^{n}(1-\alpha^{k-n})}\right|\leq\left(\dfrac{1}{1-\alpha^{k-n}}\right)\left(\dfrac{|\beta^{n}|}{\alpha^{n}}+\dfrac{|\beta^{k}|}{\alpha^{n}}\right).$

Taking into account $1<\dfrac{1}{1-\alpha^{k-n}}<2$ for $n-k\geq 2,$ we get
that

$\left|\Lambda_{4}\right|:=\left|1-\dfrac{L_{m}10^{d}\sqrt{5}}{\alpha^{n}\left(1-\alpha^{k-n}\right)}\right|<\dfrac{4}{\alpha^{2k}}.$
(22)

Let $(\eta_{1},b_{1})=(10,d)$, $(\eta_{2},b_{2})=(\alpha,-n)$ and
$(\eta_{3},b_{3})=\left(\dfrac{L_{m}\sqrt{5}}{1-\alpha^{k-n}},1\right).$ All
of $\eta_{1},\eta_{2}$ and $\eta_{3}$ belong to real quadratic number field
$\mathbb{K}=\mathbb{Q}(\sqrt{5})$, which has degree $d_{\mathbb{K}}=2.$ Since
$h(\eta_{1})=\log{10},$ $h(\eta_{2})=(1/2)\log(\alpha)$ and

$h(\eta_{3})\leq
h(L_{m})+h(\sqrt{5})+h(1-\alpha^{k-n})\leq\log(L_{m})+\log(\sqrt{5})+h(\alpha^{k-n})+\log
2$ $<\log(2\alpha^{m})+\log(\sqrt{5})+\dfrac{|k-n|}{2}\log\alpha+\log 2.$

So

$h(\eta_{3})<\dfrac{3(n-k)+2}{2}\log(\alpha)+\log(4\sqrt{5})$

where we used the fact that $m<n-k+1$ from (19). So we take $A_{1}=2\log{10},$
$A_{2}=\log\alpha$ and $A_{3}=(3(n-k)+2)\log(\alpha)+\log(80).$ Clearly,
$B:=\max\\{|b_{i}|\\}=\max\\{d,n,1\\}=n.$ Also $\Lambda_{4}\neq 0.$ To show
this fact assume that $\Lambda_{4}=0.$ Then, we get that
$\alpha^{n}-\alpha^{k}=10^{d}L_{m}\sqrt{5}.$ Conjugating in $\mathbb{K},$ we
find $\beta^{n}-\beta^{k}=-10^{d}L_{m}\sqrt{5}.$ Adding side by side the last
two equalities we obtain that $L_{n}-L_{k}=0,$ that is $n=k,$ a contradiction.
So $\Lambda_{4}\neq 0.$ Thus, we apply Theorem 3 to $\Lambda_{4}$ given in
(22), and we get that

$\log{(\Lambda_{4}|)}>-1.4\cdot 30^{6}\cdot 3^{4.5}\cdot 2^{2}(1+\log
2)(1+\log n)A_{3}2\log{10}\log\alpha.$ (23)

On the other hand, from (22), taking into account that
$\log{(\Lambda_{4})}<\log 4-2k\log\alpha,$ we get that

$k<2.24\cdot 10^{12}\cdot(1+\log n)[(3(n-k)+2)\log(\alpha)+\log(80)].$ (24)

Now, we use the two inequalities (21) and (24) to get an initial bound on the
variable $n$. To do this, first assume that $k\leq m.$ Then, $n-m\leq n-k$ and
hence, from (21), we write

$n-k-6<7.19\cdot 10^{12}(1+\log{(n-k)})$

which implies that $n-k<3\cdot 10^{14}.$ So it follows that

$n<10^{15},$

since from (19), we know that $n<2m+4$ and $m<n-k+1.$

Let $m\leq k.$ From (21), in particular, we have that

$n-k-6<7.19\cdot 10^{12}(1+\log{n}).$

Substituting (3) into (24) and using the fact that $n<2k+7,$ we find that

$n<2.5\cdot 10^{29}.$ (25)

So, whether $m\leq k$ or not, we have that $n<2.5\cdot 10^{29}.$

###### Lemma 6

If the equation (2) holds then $m\leq 168.$ In particular, if $k\leq m,$ then
the equation has the solution only for $F_{n}\in\\{13,21\\}.$

Proof. Suppose that $m>168.$ Let

$\Gamma_{3}:=d\log{10}-(n-m)\log\alpha+\log\sqrt{5}.$ (26)

Since $161<m-7<n-k-6,$
$\left|\Lambda_{3}\right|:=\left|\exp{(\Gamma_{3})}-1\right|<\dfrac{3}{\alpha^{n-k-6}}<\dfrac{1}{2}.$
Hence, we get that $\left|\Gamma_{3}\right|<\dfrac{6}{\alpha^{n-k-6}}.$ Thus
from (26), we write

$0<\left|d\dfrac{\log{10}}{\log\alpha}-(n-m)+\dfrac{\log\sqrt{5}}{\log\alpha}\right|<\dfrac{6}{\alpha^{n-k-6}\log\alpha}.$

Let $M:=2.5\cdot 10^{29}>n>d$ and $\tau=\dfrac{\log{10}}{\log\alpha}.$ Then,
in the continued fraction expansion of irrational $\tau$, we see that
$q_{60},$ the denominator of the $60th$ convergent of $\tau,$ exceeds $6M.$
With the help of Maple, we calculate

$\epsilon:=||\mu q_{60}||-M||\tau q_{60}||$

where $\mu=\dfrac{\log\sqrt{5}}{\log\alpha}$ and we find that
$0.017775<\epsilon.$ Let $A:=\dfrac{6}{\log\alpha},$ $B:=\alpha$ and
$\omega:=n-k-6.$ From Lemma 4, we find that

$n-k-6<\dfrac{\log{\left(Aq_{60}/\epsilon\right)}}{\log B}<160.$

But this contradicts the fact that $161<m-7<n-k-6.$ So, we conclude that
$m\leq 168.$

By Lemma 6, from now on, we deal with only the case $m<k.$ Since $n-k<m+7\leq
175,$ by substituting this upper bound for $n-k$ into (24), we get that

$n<2k+7<2\cdot 2.24\cdot 10^{12}\cdot(1+\log n)[527\log(\alpha)+\log(80)]+7.$
(27)

So from (27), it follows that

$n<4.6\cdot 10^{16}.$

Let

$\Gamma_{4}:=d\log{10}-n\log\alpha+\log{\left(\dfrac{L_{m}\sqrt{5}}{1-\alpha^{k-n}}\right)}.$
(28)

So

$\left|\Lambda_{4}\right|:=\left|\exp{\Gamma_{4}}-1\right|<\dfrac{4}{\alpha^{2k}}.$

Then, $\left|\Gamma_{4}\right|<\dfrac{8}{\alpha^{2k}},$ since
$\dfrac{1}{\alpha^{2k}}<\dfrac{1}{2}.$ Thus, from (28),

$0<\left|\dfrac{\Gamma_{4}}{\log\alpha}\right|<\dfrac{}{\alpha^{2k}\log\alpha}.$
(29)

Now, we take $M:=4.6\cdot 10^{16}>n>d$ and
$\tau:=\dfrac{\log{10}}{\log\alpha}$ which is irrational. Then, in the
continued fraction expansion of $\tau$, we take $q_{91},$ the denominator of
the $91th$ convergent of $\tau,$ which exceeds $6M.$ Now, with the help of
Maple we calculate

$\epsilon_{m,n-k}:=||\mu_{m,n-k}q_{91}||-M||\tau q_{91}||$

for each $0\leq m\leq 168$ and $3\leq n-k<m+7,$ where

$\mu_{m,n-k}:=\dfrac{\log{\left(\dfrac{L_{m}\sqrt{5}}{1-\alpha^{k-n}}\right)}}{\log\alpha}$

except for $n-k=4$ and $n-k=8.$

For these two values $\epsilon_{m,4}<0$ and $\epsilon_{m,8}<0.$ To overcome
this problem, we use the periodicity of Fibonacci sequences. With an easy
computation, one can see that there is no positive integer $t$ in the range
$1\leq t\leq 20$ which satisfies neither $F_{t}\equiv F_{t+4}\pmod{5}$ nor
$F_{t}\equiv F_{t+8}\pmod{5}.$ Since the period of the Fibonacci sequence
modulo 5 is 20 [10, Theorem 35.6.], we conclude that there is no positive
integer which satisfies either of these two congruences. So, from (2), we take
$n-k\neq 4$ and $n-k\neq 8.$

Let $A:=\dfrac{8}{\log\alpha},$ $B:=\alpha$ and $\omega:=2k.$ Thus, from Lemma
4 we get that, the inequality (29) has solutions only for

$2k\leq\dfrac{\log{\left(Aq_{91}/{0.000109}\right)}}{\log B}\leq 233.$

So, we get that $m<k<117,$ which is a contradiction because of bound of $k.$
Thus we conclude that the equation (2) has no solution when
$F_{n}\not\in\\{13,21\\}.$ This completes the proof.

Discussion : In this paper, we aimed to search the results of mixed
concatenations of the numbers belonging to the different sequences in the
Fibonacci and Lucas particular case and we found that there are only a handful
of Fibonacci numbers that can be written as a mixed concatenation of a
Fibonacci and a Lucas numbers. The results of this paper together with those
of [1, 4] bring the question to the mind that could there exist a
nondegenerate _binary_ recurrence sequence which contains infinitely many
terms which are expressible as a mixed concatenation of, say, a Fibonacci and
a Lucas numbers? Let us define a sequence $u_{n}=10F_{n}+1.$ Then, clearly,
all terms of this sequence are a concatenation of a Fibonacci and $1.$ So, in
the above question, we can not omit the expression ”nondegenerate _binary_ ”.
As a final remark, it is also worth noting that just using some properties and
identities of Fibonacci and Lucas numbers, we could get only a few partial
results so we used the method in this paper, which is the effective
combination of Baker’s method together with the reduction method.

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