# An astrophysical perspective of life. The growth of complexity

F. Sánchez  11affiliation: Instituto de Astrofísica de Canarias, Spain.
22affiliation: Department of Astrophysics, ULL, Spain. and E. Battaner
33affiliation: Instituto Carlos I de Física Teórica y Computacional, Spain.
Francisco Sánchez: Instituto de Astrofísica de Canarias, Vía Láctea s/n, 38205
La Laguna, Tenerife, Spain & Department of Astrophysics, University of La
Laguna, Avda. Astrofísico Francisco Sánchez s/n, 38206 Tenerife, Spain
(fsm@iac.es). Eduardo Battaner: Instituto Carlos I de Física Teórica y
Computacional, Universidad de Granada, Spain (battaner@ugr.es).

###### Abstract

The existence of life is one of the most fundamental problems of
astrophyisics. The intriguing existence of progressively complex and
apparently improbable living beings should be a general tendency of life in
the Universe. We are looking for general physical laws governing the growth of
complexity in any astrophysical environment. We posit the existence of a vital
scalar field. This scalar is sensitive to the gradient of the inverse of
specific entropy, such that its distribution tends to very high values in the
interior of living beings. Besides the classical mutations, vital field driven
mutations only produce decrements of entropy. The field equations give rise to
the existence of vital waves. This theory is able to deal with both the origin
of life and the evolution of life. We show that the growth of complexity is
accelerated by the vital field.

La existencia de vida es uno de los problemas fundamentales de la astrofísica.
La existencia intrigante de vivientes progresivamente complejos y
aparentemente improbables debería ser una tendencia general de la vida en el
Universo. Buscamos leyes físicas generales que gobiernen la emergencia de
complejidad en cualquier sistema astrofísico. Proponemos la existencia de un
campo vital escalar. Este escalar es sensible al gradiente del inverso de
entropía específica, de forma que su distribución tiende a adquirir valores
muy altos en el interior de los vivientes. Además de las mutaciones clásicas,
hay otras mutaciones inducidas por el campo vital que sólo producen
decrementos de entropía. Las ecuaciones del campo dan lugar a ondas vitales.
Mostramos cómo el campo vital acelera el crecimiento de complejidad. Esta
teoría puede aplicarse tanto el origen de la vida como a su evolución.

Hydrodynamics Astrobiology Cosmology

##  0.1 Introduction

It is a fact that our Universe contains life, even if observations are still
limited to only one planet. The origin and evolution of life in the Universe
requires a general insight much beyond the precise knowledge that biology has
reached in the particular case of living beings on Earth.

On the other hand, it is well known that, in the particular case of
terrestrial life, a large impulse pushes living beings to perpetuate and
benefit from any favourable circustance that arises and to progress. They
tenaciously adapt themselves to the medium, evolving and even modifying this
medium to improve their own evolution. Life efficiently uses any resource and
any kind of energy independently of its origin. This well-known and
fascinating behaviour makes it logical and necessary to undertake the general
problem of life in the evolution of the Universe as a fundamental
astrophysical problem.

In his celebrated book _What is life?_ Schrodinger (1944) wrote, ”From all
that we have learnt about the structure of living matter, we must be prepared
to find it working in a manner that cannot be reduced to the ordinary laws of
physics”. This opinion of Schrödinger’s inspired the present work, which aims
to tackle essentially biological problems from a different viewpoint based on
new physical laws. The complexity inherent in life and its evolution beckon us
to try out new emergent models built from hitherto untried perspectives.

We are not proposing an alternative to classical genetics but reinforce it
with new physical laws. This theory is neither teleological nor is based on
anthropic principle.

Many works have dealt with the physical interpretation of life, entropy,
complexity and evolution (some representative classic examples could be:
Prigogine, 1981; Prigogine & Stangers, 1984; Maturana & Varela, 1981; McShea &
Brandon, 2010; Kauffman, 2000; Margulis & Sagan, 1997; Schneider & Sagan,
2005; Krakauer, 2011; Grassberger, 1986; Day, 2012; Gould, 1996; Davies, 2019;
Adamsky et al., 2020; Zeravcic & Brenner, 2014) but their basic principles and
scopes are very different to the one presented here. There are many genetic
and astrobiological topics dealing with the evolution of life in the Universe,
but here we restrict ourselves to the entropic problem. This paper aims to
provide a fresh insight from a purely physics-based point of view.

We propose general physical laws for any kind of life in the Universe,
terrestrial life just being a particular case. We need, however, to make a
small number of general assumptions that are evident in terrestrial genetics;
namely, that the existence of a molecular code similar or dissimilar to our
DNA, that this code must be highly stable to subsist against replication and
thermal noise, and that this stability is not completely perfect, so that it
can be on rare occasions violated, thereby permitting mutations. The mechanism
of natural selection should also be at work.

We introduce a vital field characterized by vital density and velocity. The
vital field permeates the Universe and flows in space and time. We first
define what this vital field _is_ and later consider what this vital field
_does_.

The properties of the vital field are defined by two proposed equations
bearing formal similarity with the basic equations of the Eulerian Fluid
Mechanics. From these equations, we deduce the existence of vital waves. We
then propose that vital density has at least two effects on the evolution of
species. First, it induces another type of mutation characterized by
decrements of specific entropy (called VID mutations, Vital Intensity Driven
mutations). Second, vital waves enable a mechanism that speeds up the
dissemination of mutations.

The novel hypothesis proposed here may render some evolution processes more
effective. With both VID mutations and standard mutations, natural selection
continues to be the mechanism that decides the progress of evolution. We could
even suggest that the new hypothesis provides support to the traditional
theory of evolution and can therefore be considered as complementary to it. We
are not at all proposing an alternative to the standard theory of evolution.
We do not exclude standard genetical concepts; instead we simply add new ones,
albeit based on new physics.

##  0.2 Mathematical derivations

###  0.2.1 Vital field equations

The physical magnitude characterizing the complexity referred to here is
specific entropy (entropy per unit mass, or entropy per baryon), $s$. For
living beings we also use the thermodynamic concept of entropy, $S$, or rather
its statistical mechanical formulation according to the interpretation of
Boltzmann. We shall also speak of the entropy of DNA, or any other equivalent
code, on the basis of the relation between entropy and information by taking
into account the information contained microscopically in the genome. This
entropy is based on the thermodynamic weight (number of microstates of a
macrostate). Since the two should be intimately interrelated, there is no risk
of misinterpretation.

The scalar physical magnitude that defines the vital field is vital density,
$\rho(x,y,z,t)$, defined for all space and time. The characteristics of the
vital field are determined by two relations: the equation of vital flux and
the continuity equation.

We define the vital flux by the product $\rho\vec{v}$, where $\vec{v}$ is the
characteristic fluid velocity of the spatial variations of vital density (not
the velocity of propagation of the vitons, as we shall see later).

We write the equation of vital flux, i.e. using a hydrodynamic approach, as
follows:

${{\partial}\over{\partial t}}\left(\rho\vec{v}\right)+k_{1}\nabla\rho-
k_{2}\nabla{1\over s}=0$ (1)

where $k_{1}$ and $k_{2}$ are constants.

In order to interpret this equation, let us first suppose the simple case of a
stationary state:

$\nabla\left(k_{1}\rho-{k_{2}\over s}\right)=0$ (2)

The simplest way to integrate this equation is by ignoring the sign of the
gradient. We then obtain an equation that we can work with:

$\rho\propto{1\over s}$ (3)

In other words, only for stationary states, those points corresponding to the
interior of a living being with very low entropy will have high vital density;
outside, the density will be very low. The density will not be homogeneous
inside because the constituent organs will not possess the same complexity,
i.e. the same entropy.

In the general case around and within a living being, without considering
steady state conditions, Eq. 1 tells us that the vital flux varies, owing to
two gradients of opposing tendencies. The gradient of the inverse of the
specific entropy points from the exterior to the interior and will tend to
generate a vital flux towards the interior of the living being. The gradient
of density also points inwards, but because of its negative sign, generates a
vital flux outwards.

If the density distribution tends to have very high values inside, such values
cannot be attained instantaneously. If, for example, the living being is
displaced, the density distribution ’follows’, although with a slight delay
that is inappreciable if the velocity of the living being is very much less
than the speed of light. For that to occur it is necessary that the constants
$k_{1}$ and $k_{2}$ be very high (close to $c^{2}$). In the case of the
displacement of a living being, both gradients act jointly to make the
distribution of vital density and the inverse of the specific entropy
coincide. This implies that, in practice, the density distribution seems to be
in step with, or frozen into, the body of the living being.

Vital density does not represent life complexity, complexity being represented
by low specific entropy. But low entropy attracts the density as shown by Eq.
1 and this effect takes place over very short characteristic times. Steady
state conditions can be quickly reached and then the distribution of both the
density and the inverse of specific entropy may coincide.

The causes of variability in vital flux correspond to rapid processes with
very short characteristic times, in contradistinction to long-duration
processes (i.e. those that operate on evolutionary time scales), which we
shall discuss further on.

We call the second of the vital field equations the equation of continuity. In
fluid mechanics the equation of continuity establishes the conservation of
mass; here, in a similar way, the integrated density, is conserved throughout
space from -$\infty$ to +$\infty$. We therefore have:

$\iiint_{-\infty}^{+\infty}\rho d\tau=constant$ (4)

where d$\tau$ is the element of volume. The triple integral in this formula
means that the total integrated density of the vital field in the whole
Universe remains constant in time.

The equation of continuity tells us that if, in an element of volume, the
divergence of the flux is positive (i.e. if more flux enters than leaves the
volume element), the density will diminish:

${{\partial\rho}\over{\partial t}}=-\nabla\cdot\left(\rho\vec{v}\right)$ (5)

This equation establishes the conservation of interated density.
Interpretation of the equation of continuity of density brings us to the
physics that we wish to highlight. During its lifetime a living being is a
stable system. The only great changes occur at the foetal stage, infancy, and
death. Throughout the rest of its life the organism remains extraordinarily
constant. If its alimentation is of a chemical nature, the process of
ingestation introduces entropy. As the system is in equilibrium, this increase
of entropy must be eliminated through the expulsion of mass in the form of
faeces and urine, in the case of terrestrial warm-blooded animals the process
of perspiration results in heat exchange with the environment through the
latent heat of vaporization, and so on. A living organism may be thought of as
a stable heat engine (or a cyclic engine with a period of the order of one
day). A stable living organism is a great ejector of entropy into its
immediate surroundings. The ejection of entropy to the surroundings should
even be inherent to the very concept and definition of life.

For any given instant of time in the Universe the density distribution
associated with a living emitity will be very high (even though non-uniformly
distributed in the interior of the organized living entity) and very low in
its immediate surroundings, asymptotically approaching the value associated
with lifelessness towards infinity. The mean value would correspond to the
lifeless value; the high value of density in the interior of the organism
would compensate for the minimum value in its immediate surroundings through
the ejection of entropy.

That is to say, a living being is not simply a highly dissipative system that
consumes large quantities of energy in order to preserve its complexity, but
also, and this must be emphasized, it ejects a great quantity of entropy below
the lifeless density into its immediate surroundings. The density distribution
is illustrated in Fig. 1. A living organism absorbs energy from and
_entropizes_ its surroundings.

Figure 1: Vital density field distribution for a living being and its
environment.

To measure the density, we can benefit from Eq. 3 only when steady state
conditions are satisfied. This can be achieved inside a living being or in an
inhospitable medium, being nearly zero in an infertile medium. Inside a living
being, the value of the densities would be as difficult to estimate as the
specific entropy. The units of density would be the inverse of specific
entropy, or entropy per baryon (i.e. K/erg, for example), or would be non-
dimensional when the so-called system of geometrized units, with c, G and
Boltzmann-s constant equal to unity, is adopted, as is usual in cosmology and
other astrophysical studies. We consider life to be an astrophysical issue.

###  0.2.2 Vital field waves and vitons

We define vitons as particles associated with vital waves in accordance with
the interpretation of the wave-particle duality. We demonstrate, with the two
basic vital field equations, the existence of waves in the vital density
field. We now consider the external medium surrounding living organisms. Using
the method of linear perturbations and introducing small changes, magnitudes
susceptible to variation when traversed by the wave are transformed in the
following way:

$\begin{array}[]{l}\rho\rightarrow\rho+\rho^{\prime}\\\
\vec{v}\rightarrow\vec{v}+\vec{v^{\prime}}\end{array}$

Unprimed magnitudes correspond to the unperturbed medium, and primed
magnitudes are the perturbed magnitudes when traversed by the waves.

We introduce the following simplifications: (a) $\rho$ has no gradients and no
temporal variations; (b) the perturbationless velocity is zero; (c) the
perturbations are very small, so that terms implying a product of two
perturbed terms are negligible; and (d) there are no specific entropy
variations far removed from living beings, so that the term involving s does
not give rise to further complications. The waves would exist even without
these simplifying assumptions, but their mathematical treatment and physical
behaviour would be less simple. In particular, the propagation of vitons
inside living beings would be not so simple, where gradients of specific
entropy due to different organs would render the wave equation more realistic.

From the equation of continuity we get

${{\partial\rho^{\prime}}\over{\partial t}}+\rho\nabla\cdot\vec{v^{\prime}}=0$
(6)

and the equation of motion gives

$\rho{{\partial\vec{v^{\prime}}}\over{\partial t}}+k_{1}\nabla\rho^{\prime}=0$
(7)

We derive the first with respect to time and find the divergence in the
second, and, since the spatial and temporal derivatives are independent and
therefore commute, we obtain directly:

${{\partial^{2}\rho^{\prime}}\over{\partial
t^{2}}}=k_{1}\nabla^{2}\rho^{\prime}$ (8)

which is the wave equation. Now, $k_{1}$ is the wave velocity squared; that is
to say, the velocity of the vitons. We may suppose that $k_{1}=c^{2}$, where
$c$ is the speed of light, since these waves can propagate in vacuo. The wave
velocity (or the velocity of the vitons) could be equal to the velocity of
light or close to it. In the geometrized units, that are the custom in
cosmology, with $c=1$, $k_{1}=1$ also.

We could likewise derive a wave equation in terms of velocity. The propagation
of the waves is accompanied by small fluctuations in velocity,
$\vec{v^{\prime}}$. It is to be noticed that this velocity is not the wave
velocity, equal to $k_{1}^{1/2}$. In a fluid also, the perturbed velocities by
the passage of the wave are not equal to the sound speed.

Inside a living being, this wave equation can be produced in the following
way. A mutation can be produced at any given moment, giving rise to an abrupt
variation in entropy, which is equivalent to a sharp change in vital density.
A density jump in a medium with the capacity to support waves generates the
wave. In other words, mutations can create a wave and the emission of vitons.
The amplitude of this wave, being a purely geometric factor, diminishes as the
inverse square of distance. Vitons always travel at the speed of light or at
least at a very high speed. There is some similarity between (energy-photons)
and (entropy-vitons).

As in the case of normal mutations, VID mutations can be produced by auto-
replication or induced by external agents, X-rays, and other means. We cannot
provide examples of VID mutations, as their effects are noticeable only over
long time scales (consider, for example, the huge range of complexity of
mammals).

##  0.3 Chemistry of life

The origin, evolution and fate of living beings are the result of the
interrelation of both, physical and chemical processes. In this paper we focus
on the physical ones but it is clear that chemistry, more precisely,
biochemistry, is of the greatest importance to understand life, both
terrestrial and extraterrestrial. The knowledge of biochemistry is at present
incomplete but this is due to the great complexity of the problem rather than
the chemistry fundaments themselves require to be revised. Biochemistry of
cosmic life has been reviewed in several papers. The reader is addressed to
two excellent books by Schulze-Makuch & Irwin (2018) and Trigo-Rodríguez
(2010), with references therein. These authors not only examine the life
chemistry in the Earth, but also speculate on alternative cosmic niches and
study biochemistry under a cosmic perspective.

It is evident that life can develop at least when physical-chemical conditions
are available. A suitable habitat requires chemical heterogeneity. In Earth,
life macronutrients are C, H, O, N, K, Ca, P, Mg and S to form prebiotic
monomeric inorganic compounds such as H2, N2, CO2, NH3, HCN …Of great
importance was the emergence of photosynthesis, but this is a complicated
mechanism that could not be at work in the primitive forms of life. Early
chemolithotrophic life was important using redox metabolic reactions, in
particular the oxidation of hydrogen to water, with 2.6 eV per reaction,
coupled with the reduction of CO2 (metanogenesis, 1.4 eV per reaction), which
is as much energetic than light (about 2 eV per photon depending on
wavelength). Early cometary and meteoritic bombardment also delivered water
and organic compounds thus enriching primordial chemistry as first proposed by
Oro (1961). Carbonaceous chondrites are rich in organic molecules, as
witnesses of the protoplanetary disk chemistry.

Cyanobacteria and other chemoautotrophic organisms dramatically produced
oxygen changing the atmosphere, rendering oxidation conditions. Life went from
anaerobic to aerobic. This fact favoured the carbon chemistry with its ability
to form complex stable molecules, starting with methane and other alkanes,
reaching amino acids leading to proteins through connections by peptide bonds.
Also, cycloalkane and polycyclic aromatic hydrocarbons (already present in the
interstellar medium, in particular in molecular clouds) produced rings and
later three-dimensional macromolecules with a non-racemic left-handed
chirality. The carbon polymeric chemistry is able to build an almost unlimited
range of molecules with strong covalent bonds, providing a very rich
repertoire of metabolic reactions. This change of chemistry illustrates that
not only life is adapted to the atmosphere but also that the atmosphere is
adapted to life.

Life on Earth began with microscopic anaerobic prokaryotic cells, starting
with LUCA (Last Universal Common Ancestor). Two billions later the emergence
of eukaryotic cells took place as a fusion of two or more prokariotes
(Margulis & Sagan, 1995), with sizes grater by an order of magnitude. The
emergence of multicellular life is a poorly understood step. About 541
millions years ago the Cambrian explosion was the origin of bilaterian and
homeothermic phyla. This homeothermia was an important step as metabolic
reactions adapted to narrow to optimal temperature ranges. On the other hand,
homeothermia provided an efficient way to heat the environment thus expelling
entropy for keaping its disequilibrium state.

We cannot present a brief overview of the very wide branch of cosmochemistry.
We limit ourselves to brief comments on biochemistry topics closer to the
physics proposed in this paper: a) the necessity of a solvent, b) the required
boundary and c) the replication mechanism.

(a) The necessity of a solvent. Life is fundamentally a liquid based process.
Terrestrial life uses water as a solvent especially efficient due to its high
dipole moment. Atoms and molecules can move around nearly free to undergo
chemical reactions. There is a perfect compatibility between carbon polimeric
chemistry and water as a solvent. However, life was perhaps not originated in
oceans or sees but in interfaces liquid-solid like in puddles alternating
desiccation and rehydration, or porous media like pumice or even ice in
contact with liquid water. A solid phase could preserve initial and later
decreases of entropy.

(b) The required boundary. The existence of boundaries of living beings is of
great importance to isolate the regions of disequilibrium inside from the
environment closer to thermodynamic equilibrium, to isolate a region with very
low entropy and a medium with very large entropy. Biomembranes preserve the
high free energy state of the system from dissipation, encapsulating and
confining a high concentration of interacting solutes and macromolecules. But
isolation cannot be absolute. The semi-permeable boundaries must provide a
selective transport of nutrients and waste. For this difficult task
anphiphylic lipids (fatty acids, phospholipids) are essential as are
hidrophylic toward the aqueous environment and hydrophobic toward the cell
interior. The emergence of lipid protomembrane boundaries was a great jump in
the early evolution of life.

(c) The replication mechanism. At present, in Earthś life, the information is
stored in the one-dimensional form of a linear code (DNA) translated into the
there-dimensional structure of proteins but, to reach this complex, precise,
high fidelity mechanism, many previous insufficiently known steps were to be
surmounted. There is at present a great difficulty to identify the first
replicators. Kauffman (1995, 2000) proposed that the origin of life and the
transmission of hereditary information was not due to a single molecule but to
a network of interacting catalytic molecules. Probably there was a pre-RNA
world followed by an RNA world before reaching the DNA-RNA world in which RNA
is now an intermediate in the flow of information and responsible of the
production of catalytic and autocatalytic encymes. There must be a
transmission of information from parental to descendants or offs-springs to
assure an autonomous maintenance of phyla, a great biochemical challenge at
early epochs. These molecular codes permit the evaluation of the entropic
content of living individuals.

Much work must still be done to understand the establishment of terrestrial
life chemistry, and still much more work to speculate about extraterrestrial
life adapted to other energy sources and other habitats, but we do not suggest
that new chemistry fundaments are needed. On the other hand, it is here
suggested that complementary physics as introduced in this paper, should
favour the evolution to reach the chemical complexity of life, rendering some
difficult steps more plausible.

##  0.4 Results

###  0.4.1 Mutations driven by vital density

Standard mutations, which occur completely at random, may give rise to both
positive and negative variations in entropy in the molecular code. We add
another kind of mutation driven by the vital density field and characterized
solely by a diminution in entropy. We call these _VID mutations_ (Vital Driven
mutations). The existence of VID mutations is one of our fundamental
hypotheses. Both types are random, coexist, are subject to natural selection,
and drive the evolution of a given species. Although standard mutations might
well dominate, the introduced kind are also essential because they enable us
to better understand the progressive appearance of complexity through time, a
problem that we are using to test our model.

The simplest mathematical way to express the effect of VID mutation is to
suppose that

$ds=-k_{3}\rho dt$ (9)

In other words, the temporal diminution in specific entropy is proportional to
the vital density. Although the mutations are quantized in nature, after long
periods of time a continuous description may be permitted, as expressed in Eq.
9.

The real evolution of species is the result of the joint action of standard
and VID mutations, together with multiple interruptions, bifurcations, etc.
The graph of specific entropy as a function of time is extremely complex, as
testified by the history of evolution. In order to analyse the isolated effect
of VID mutations we plot the curve of the variation of entropy against time by
considering the ideal effect where VID mutations are alone responsible for
this variation. By isolating the tendency of VID mutations in this way we may
insert the result into a global interpretation involving all evolutionary
effects.

If we combine Eqs. 9 and 3,

$ds=-k_{3}{k_{1}\over k_{2}}{1\over s}dt=-k_{4}{1\over s}dt$ (10)

This differential equation integrates easily to produce:

${s\over s_{0}}=\sqrt{1-{{2k_{4}}\over{s_{0}^{2}}}t}$ (11)

We avoid a constant of integration by choosing a suitable time origin. The
constant $s_{0}$ represents the value of specific entropy at infinity,
approaching the lifeless level. The value of $k_{4}$ must be very small for
the effects of VID mutations to be applicable on evolutionary time scales.
After a time interval of, say, 106 yr, $k_{4}$ would be of the order of
3$\times$10-14 s-1. We calculate the value of the lifeless specific entropy
per baryon to be of the order of $s_{0}=5\times 10^{-15}ergK^{-1}$. The
estimation of this value is given in an appendix. Fig. 2 shows the variation
curve.

Figure 2: Variation of specific entropy with time due to the isolated effect
of VID mutations.

We see from Eq. 11 that for low times the specific entropy decreases slowly
until approximately the characteristic time $s_{0}^{2}k_{4}^{-1}$. At larger
times the decrease becomes faster until reaching its minimum value, which is
zero as the entropy is positive. The value of zero corresponds to DNA as a
pure crystal, equivalent to zero temperature. It is therefore an ideal limit.

On small time scales the action of VID mutations will not be very great; when,
however, it has attained a significant level of complexity, it could
predominate. The specific entropy would eventually correspond to what we may
term the _zero-entropy species_ , beyond which there could be neither greater
complexity nor further evolution. This ideal state would not, however, be
unchangeable, depending on ecological or environmental conditions, on the
action of normal mutation, etc. It is to be emphasized that this plot
considers only the isolated effect of VID mutations.

The zero-entropy species is not the end of evolution, because normal mutations
are always at work and can reverse the trend. Other evolution paths can then
be explored. This ideal end will be reached only if only VID mutations were
present, which is by no means assumed. In the real world other effects such as
bifurcations and extinctions could take place. On the other hand, there are
many paths to reducing entropy; there would be many end points. There would be
not just one zero-entropy species, but many, and there is no way of telling
which one of them would reach the final state chosen by natural selection. The
most complex species need not be the best adapted ones, so that almost
perfect, but fragile and ailing, species might result that are doomed to
extinction from the start.

If we put $s=0$ in Eq. 11, we get:

$s_{0}^{2}=2k_{4}\tau$ (12)

where $\tau$ is the characteristic time for VID mutations to reach the
condition of zero-entropy species. Bearing in mind the estimated values of
$s_{0}$ (see Appendix) and assuming $\tau$ to be of order $10^{9}$ yr,
somewhat less than, but of the order of, a Hubble time, we have an estimation
of the value of $k_{4}$ of the order of $10^{-14}s^{-1}$.

It is tempting to identify the zero-entropy species as some kind of _superior_
species, even to the point of imagining _superior humans_ , but such a notion
is untenable, given that a zero-entropy species would be the result of its
evolutionary path, which cannot be determined beforehand, so a zero-entropy
shark would be just as conceivable as a zero-entropy human or a zero-entropy
extraterrestrial, for example. It is possible that the shark species has
already reached a state close to zero entropy, further evolutionary progress
being difficult, given its phylogenesis. There is, then, no anthropic
principle at work in our hypothesis.

###  0.4.2 Spreading of mutations

The great stability of many molecules, particularly DNA or any other
equivalent code, implies that they are in a state of relative minimum energy.
For a mutation to occur and a different state to be reached, a high potential
energy barrier, $\Delta E$, must be surmounted, and that barrier must be much
higher than the thermal energy, $kT$ ($k$ being Boltzmann’ss constant and $T$
the temperature), so the molecule does not easily change its state. This
condition, equivalent to a very low temperature, was used by Schrödinger, who
considered the gene to be an aperiodic solid, ideally at a temperature of
absolute zero. Aside from a leap in entropy, $\Delta S$, the mutation would
undergo, independently of $\Delta E$, a change in energy, $E$, from its
initial to its final state, corresponding to a photon emitted in the mutation
(see Fig. 3).

Figure 3: Energy-entropy diagram showing states before and after a VID
mutation.

The existence of vitons may clarify the manner in which a mutation (whether or
not favourable, with $\Delta S$ either positive or negative) originating in an
individual of a species can spread to other individuals of the same species to
the point of mutating the entire species. Mutation is an entropic leap in a
DNA molecule that creates a sharp increase in vital density, giving rise to
the emission of a viton. When a viton meets a DNA molecule of the same
individual, or of other individuals of the same species, it will provoke the
same entropic leap, $\Delta S$, that gave rise to it, so that the viton will
have propagated the mutation of the emitting DNA molecule to the receptor DNA
molecule. In its turn, the receptor will emit another viton of exactly the
same entropic leap, $\Delta S$, thereby producing a scattering effect in the
density waves. An analogy may be drawn with the processes of emission and
absorption of photons. This mechanism would be at work in DNA molecules as
well as in any other molecule used as information code.

For transmission to be effective the viton must carry information on both
$\Delta S$ and $\Delta E$, the energetic leap required for the mutation to
take place. The viton would not only announce that a mutation is about to be
produced but also would supply the energy to make it happen. No problem arises
if the mutational leap gains more energy than it loses, i.e. if $E$ is greater
than $\Delta E$ (see Fig. 3).

When the receptor organism in turn becomes an emitter, the mutation will be
transmitted to neighbouring beings belonging to the species. In this way a
mutation originating in an individual may become a mutation of the entire
species.

A quantum leap in entropy could correspond numerically to more than one
mutation. The receptor organism might undergo a different mutation from that
of the emitting organism. This error in transmission does not imply any
evolutionary disadvantage since, for progressive mutation to occur, diminution
in entropy is more important than precisely where the dislocation in the
initial DNA takes place.

Standard mutations and VID mutations are not excluding each other. However
they are independent as they are produced by different mechanisms. Standard
mutations are the results of random changes in the DNA molecule (excluding
those produced by X-rays…) VID mutations are induced by vital waves. Let us
clarify this point by finding other similar effects in classical systems. In a
hot metal, for instance, electrons may escape the metal as a result of thermal
noise. In addition, if the metal is exposed to UV radiation the electrons also
escape due to the photoelectric effect. The two mechanisms co-exist, both
contribute to the escape of electrons and are independent. The UV photons
induce another type of ejecting electrons.

##  0.5 Discussion

With regard to the problem of life origin and evolution there are three types
of contributions: an astrophysical phase concerning the production of rather
complex biomolecules and the provision of habitable environments, the
contribution of genetics to identifying how evolution proceeds, and the
contribution of astrophysics in dealing with entropy and the probabilities
involved in evolution. Our paper only deals with the third aspect. How
biomolecules are formed in space, in the proto-solar system and in atmospheres
is of course crucial but are beyond the scope of our paper. Here we restrict
ourselves to the entropic problem of life.

When and how the vital field is originated in the Universe? Our vital field is
present in all coins of the Universe, even with in general very low values of
density, and its integrated value is a constant in time. The origin itself
should be a cosmological problem that is far from the present preliminary
state of the theory.

The existence of a vital field in the Universe provides an effective scenario
for other forms of life in the Universe and sheds new light on a number of
phenomena that are still controversial in the physics of life and its
evolution. With this model we have been able to explain two of these
phenomena: the appearance of species of growing complexity in Earth’s history.
This growing complexity should be a basic property also in other environments
in the Universe. We have also explained the rapid dissemination of a mutation
in an individual of the species that culminates in the mutation of the entire
species.

This theory could be generalized to incorporate relativistic and quantum
aspects, but given the exploratory and novel focus of this study, we have
preferred to introduce it in a simple mathematical way.

We have considered the special case of living beings, but the vital field can
be present in any place in the Universe, even in liveless points far from
living entities. Similarly, vital waves and vitons may be present anywhere.

The theory proposed here makes reference to the genome, but is quite
independent of the specific mutation mechanism. When considering physical
laws, which are in principle applicable to the entire Universe, these ideas
could have general implications concerning the emergence and evolution of life
as a cosmological fact.

Mutations, both standard and those driven by the vital field, must accept the
verdict of natural selection. For standard mutations $S$ may be positive or
negative; for VID mutations $S$ will be negative. The $\left[s,t\right]$ curve
for VID mutations has a gentle slope for short times, but the slope becomes
more pronounced for long periods of time and low specific entropy. This
implies that, if the evolution reach a very low level of entropy, VID
mutations could become of greater importance. To put it succinctly, complexity
breeds complexity.

Isolated VID mutations would lead to ideal zero-entropy species, but VID
mutations, still with the restriction that $\Delta S<$ 0, are also aleatory.
Furthermore, such species are the result of their evolutionary path. For that
reason we say that our theory is not teleological. Neither does man possess
any special privilege in our model, since it is not based in any way on the
anthropic principle.

The concept of _zero-entropy species_ is an idealization and would correspond
to a minimum of entropy depending on the evolution path. A rabbit can be close
to a _zero-entropy rabbit_ , for example. In this model, there is no end point
or point toward which it evolution converges; on the contrary, there are as
many end points as there are ways of reducing entropy at every step. As stated
in Sect. 3, the final end point will be the result of natural selection.
Randomness here is limited but not eliminated. We just reduce the
possibilities for VID mutations and preserve standard mutations. The adaption
to changing environmental and ecological conditions is here considered as
fundamental, too.

The density of the vital field tends to reach a distribution with very high
values inside a system of very low specific entropy. A living being is
conceived here as a system of very low entropy animated by very high values of
the density of the vital fluid at the same place occupied by the system.
However, we have not focused this study on the concept of life itself, but
rather on its evolution.

We have not considered sexual reproduction because the theory could be
applicable to any form of life in the Universe for which we do not know the
reproduction mechanism. In any case we do not at all exclude the advantages of
sex to the rise of new species and biodiversity.

The densities in the living being contribute to an evolution that favours the
diminution of entropy and, consequently, an increase in density. There is an
intrinsic instability: low entropy produces increase of density (Eq. 1) and
density reduces the entropy (Eq. 9). The fist process is fast and the second
slow. Although this instability takes place on evolutionary time scales, it is
present in every phase.

It is even present at the origin of life. In effect, inert matter can undergo
natural fluctuations from thermodynamic equilibrium; these lead to
fluctuations in the distribution of specific entropy, which imply vital
density fluctuations, which in turn produce a prezoic type of heterogeneity
that, because of this instability, leads to the first prokaryotes. In terms of
this model, the generation of life would not need to be stimulated by
mechanisms different from those of evolution, although the generative process
itself would require one of its most significant leaps.

One could argue that entropy is a magnitude that was introduced with other
purposes in mind, and that the notion of a quantum of entropy does not fit
into the original conception of entropy. However, Schrödinger has already
spoken of a quantum theory of life. In physics it has often been the case that
magnitudes have successfully been given a much broader interpretation than
when they were first introduced. Also, the concept of energy, although both
familiar and abstract, only began to be appreciated in the nineteenth and
twentieth centuries, more or less at the same time in which the concept of
entropy was established. On the other hand, we only consider quanta of entropy
when dealing with mutational transitions.

##  0.6 Conclusions

The ideas presented here are not an alternative to the standard evolution
paradigm, but rather introduce novel physical concepts that speed up the
evolution of life, thus rendering more probable the growth of complexity.

There is an intrinsic instability: low entropy produces en increase of vital
density; vital density produces a decrease of entropy. The first mechanism is
very fast and the second very slow, with typical characteristic evolution
times. As a result of this instability, the distribution of both low entropy
and vital density tends to reach very high values in small places, i.e. in
living beings.

The higher the degree of complexity the faster becomes the decrement in
entropy; therefore the faster the increase of vital density, the faster the
increase of complexity. The net effect of the VID mutations introduced here is
that complexity breeds complexity.

The similarity between the Eulerian fluid and the vital fluid considered here,
enables the existence of vital waves and their corresponding particles
(vitons). The speed of vitons should be vey high, as great as the light speed
or slightly lower. This provides another way of accelerating evolution in
terms of the faster disemination of mutations among individuals of the same
species. Energy-photons and entropy-vitons are another parallel that could be
emphasized.

When we view the complexity of life from a new perspective, specific
properties and laws emerge that we have tried here to identify. We have seen
how simple physical propositions, together with straightforward and simple
mathematics, have sufficed to clarify some of the most controversial questions
in the evolution of life on Earth and may even be used to establish general
guidelines for understanding life in the Universe. The growth of complexity
could be an intrinsic characteristic of any kind of life in diverse
astrophysical environments.

###### Acknowledgements.

Acknowledgements: We are very grateful to an anonymous referee for his/her
valuable comments. We are also very grateful to Estrella Florido and Terry
Mahoney. Helpful discussions with our colleagues José I. Illana, Antonio
Mampaso, Manuel Masip, Evencio Mediavilla, Fernando Moreno, Enrique Pérez,
Ramón Román and Ignacio Trujillo have improved the paper.

## References

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##  .7 Estimation of the value of the maximum specific entropy, $s_{0}$

Let us consider Boltzmann’s equation relating to entropy and thermodynamic
weight (the number of microstates compatible with a macrostate):

$S=k\ln{W}$ (13)

We assume that the inert matter is hydrogen. In one gram there are $N_{a}$
hydrogen atoms, $N_{a}$ being Avogadro’s number (approximately $6\times
10^{23}$). We can exchange each atom by each atom and always obtain the same
macroscopic result. The value of $W$ in the inanimate universe will then be
($6\times 10^{23}$)!, where the symbol ’!’ means factorial. This seems
difficult to calculate but we can use Stirling’ formula for very large
numbers:

$\ln{n!}=n\ln{n}-n$ (14)

In our case

$\ln{6\times 10^{23}!}=54.7\times 6\times 10^{23}-6\times 10^{23}\simeq
3\times 10^{25}$ (15)

The entropy of one gram in the inanimate universe will be of the order of

$1.38\times 10^{-16}\times 3\times 10^{25}=4\times 10^{9}erg/K$ (16)

and the entropy per baryon:

$s_{0}=5\times 10^{-15}erg/K$ (17)

In cosmological calculations ’geometrical units’ are often used with $c$, $G$,
$k$ = 1; the specific entropy then becomes non-dimension and has the value
$s_{0}=30$. With this value in Eq. 12 and assuming a characteristic time of
evolution of $10^{9}$ years we obtain $k_{4}$ to be of the order of
$10^{-14}s^{-1}$.


# A Breakdown Free Numerical Algorithm for Inverting General Tridiagonal
Matrices

Moawwad El-Mikkawy  Abdelrahman Karawia  Mathematics Department, Faculty of
Science, Mansoura University, Mansoura, 35516, Egypt

###### Abstract

In the current paper the authors linked two methods in order to evaluate
general n-th order tridiagonal determinants. A breakdown free numerical
algorithm is developed for computing the inverse of any $n\times n$ general
nonsingular tridiagonal matrix without imposing any constrains. The algorithm
is suited for implementation using any computer language such as FORTRAN,
PYTHON, MATLAB, MAPLE, C, C++, MACSYMA, ALGOL, PASCAL and JAVA. Some
illustrative examples are presented.

###### keywords:

Tridiagonal matrix, Determinants, Matrix inverse, Numerical algorithm,
Hadamard product

###### MSC:

[2010] 15A09, 15A15, 39A06, 65F05, 65Y20, 68W30

††journal: Journal of LaTeX Templatesmytitlenotemytitlenotefootnotetext: Fully
documented templates are available in the elsarticle package on CTAN.

##  1 Introduction

Applications of matrices in real life include physics, geology, statistics,
graphics, economic, business, animation, encryption, games and construction.
Definition 1. A band matrix $A_{n}=(t_{ij})_{i,j=1}^{n}$ is called tridiagonal
matrix if it takes the form:

$A_{n}=[t_{ij}]_{i,j=1}^{n}=trid(b_{i},d_{i},a_{i})=\left[\begin{array}[]{cccccc}d_{1}&a_{1}&0&\cdots&\cdots&0\\\
b_{1}&d_{2}&a_{2}&\ddots&&\vdots\\\ 0&b_{2}&d_{3}&\ddots&0&\vdots\\\
\vdots&\ddots&\ddots&\ddots&\ddots&0\\\ \vdots&&0&\ddots&\ddots&a_{n-1}\\\
0&\cdots&\cdots&0&b_{n-1}&d_{n}\\\ \end{array}\right]$ (1)

in which $t_{ij}=0$ whenever $|i-j|>1$.
Band matrices of tridiagonal type frequently appear in scientific and
engineering areas (see for instance [1, 2, 3, 4, 5]). The need to obtain the
inverse of nonsingular tridiagonal matrices arises in such cases. This problem
has been investigated by many authors. The interested reader may refer to [6,
7, 8, 9, 10, 11] and the references therein.
Perhaps the first complete analysis for inverting nonsingular tridiagonal
matrices without imposing any constraints was considered in [8]. Nevertheless
the resulting numerical algorithm is not breakdown free. It breaks down if any
principal submatrix is singular. By introducing the symbolic computation
technology, we analyzed and established an algorithm which is breakdown free
[12]. The computational complexity of the algorithms given in [8, 12] is
$O(n^{2})$.
In the applied domain, the numerical approach is generally preferable [10].
This is in fact the motivation of the present paper. We are mainly concerned
with constructing efficient numerical algorithms for computing the determinant
and the inverse of a general n-th order tridiagonal matrix.
The present article is organized as follows. A cost-efficient numerical
algorithm for evaluating general n-th order tridiagonal determinants is
presented in Section 2. In Section 3, the main result is given. Some
illustrative examples are also introduced.

##  2 A Cost-efficient Numerical Algorithm for Evaluating General n-th Order
Tridiagonal Determinants

In this section, we focus on constructing an efficient numerical algorithm for
the determinant of $A_{n}$ in (1). The algorithm is mainly based on the
following facts:

  1. 1.

The reversal matrix $J_{n}$ defined by:

$J_{n}=\left[\begin{array}[]{cccccc}0&0&\cdots&0&0&1\\\ 0&0&\cdots&0&1&0\\\
0&0&\cdots&1&0&0\\\ \vdots&\vdots&\iddots&\vdots&\vdots&\vdots\\\
0&1&\cdots&\cdots&\cdots\\\ 1&0&\cdots&0&0&0\\\ \end{array}\right]$ (2)

satisfies:
(i) $det(J_{n})=\pm 1$, and
(ii) $J_{n}=J_{n}^{t}=J_{n}^{-1}$.

  2. 2.

Let $M=[m_{ij}]$ be an $n\times n$ matrix. The rotate of $M$, denoted $M^{R}$
is defined as: $M^{R}=J_{n}MJ_{n}$. If $M=M^{R}$, then $M$ is called
centrosymmetric matrix. It is true that $det(M)=det(M^{R})$.

To study tridiagonal matrices, it is advantageous to introduce two
n-dimensional vectors, $\mathbf{c}=[c_{1},c_{2},\cdots,c_{n}]^{t}$, and
$\mathbf{e}=[e_{1},e_{2},\cdots,e_{n}]^{t}$, and two (n+1)-dimensional
vectors, $\mathbf{f}=[f_{0},f_{1},f_{2},\cdots,f_{n}]^{t}$, and
$\mathbf{g}=[g_{1},g_{2},\cdots,g_{n},g_{n+1}]^{t}$. These vectors are given
by:

$c_{1}=d_{1},c_{i}=d_{i}-\frac{a_{i-1}b_{i-1}}{c_{i-1}},i=2,3,\cdots,n,$ (3)
$e_{n}=d_{n},e_{i}=d_{i}-\frac{a_{i}b_{i}}{e_{i+1}},i=n-1,n-2,\cdots,1,$ (4)
$f_{0}=1,f_{i}=\left|\begin{array}[]{cccccc}d_{1}&a_{1}&0&\cdots&\cdots&0\\\
b_{1}&d_{2}&a_{2}&\ddots&&\vdots\\\ 0&b_{2}&d_{3}&a_{3}&0&\vdots\\\
\vdots&\ddots&\ddots&\ddots&\ddots&0\\\ \vdots&&0&\ddots&\ddots&a_{i-1}\\\
0&\cdots&\cdots&0&b_{i-1}&d_{i}\\\ \end{array}\right|,i=1,2,3,\cdots,n,$ (5)

and

$g_{n+1}=1,g_{i}=\left|\begin{array}[]{cccccc}d_{i}&a_{i}&0&\cdots&\cdots&0\\\
b_{i}&d_{i+1}&a_{i+1}&\ddots&&\vdots\\\ 0&b_{i+1}&d_{i+2}&a_{i+2}&0&\vdots\\\
\vdots&\ddots&\ddots&\ddots&\ddots&0\\\ \vdots&&0&\ddots&\ddots&a_{n-1}\\\
0&\cdots&\cdots&0&b_{n-1}&d_{n}\\\ \end{array}\right|,i=n,n-1,n-2,\cdots,1.$
(6)

The four vectors are related by [13]:

$f_{i}=\prod_{r=1}^{i}c_{r}=c_{i}f_{i-1},i=1,2,\cdots,n,$ (7)

and

$g_{i}=\prod_{r=i}^{n}e_{r}=e_{i}g_{i+1},i=n,n-1,\cdots,1$ (8)

Consequently,

$f_{n}=det(A_{n})=g_{1}$ (9)

The principal minors $f_{i}$’s and $g_{i}$’s in (5) and (6) satisfy the second
order linear difference equations:

$f_{i}=d_{i}f_{i-1}-a_{i-1}b_{i-1}f_{i-2},\quad i=2,3,\cdots,n,$ (10)

with $f_{0}=1,f_{1}=d_{1}$.
and

$g_{i}=d_{i}g_{i+1}-a_{i}b_{i}g_{i+2},\quad i=n-1,n-2,\cdots,1.$ (11)

with $g_{n+1}=1,g_{n}=d_{n}$
respectively. Note that for each $i=1,2,\cdots,n$, $f_{i}$ and $g_{i}$ are
determinants of order $i$ and $n+1-i$, respectively. By using the syntax of
Maple language, we have:

$f_{i}=det(submatrix(A_{n},1..i,1..i))\quad\textit{and}\quad
g_{i}=det(submatrix(A_{n},i..n,i..n))$

It is worth mentioned that any of the following conditions is sufficient (but
not necessary) to ensure that the tridiagonal matrix $A_{n}$ in (1) is
nonsingular.

  * 1.

(i) All $c_{i}\neq 0,i=1,2,\cdots,n$.

  * 2.

(ii) All $e_{i}\neq 0,i=1,2,\cdots,n$.

For example, consider the two matrices $A$ and $B$ given by:

$A=\left[\begin{array}[]{ccc}1&1&0\\\ 1&1&2\\\ 0&2&3\\\
\end{array}\right],B=\left[\begin{array}[]{ccc}3&2&0\\\ -1&1&1\\\ 0&1&1\\\
\end{array}\right]$

For these matrices, we have
$det(A)=-4\neq 0$ and $det(B)=2\neq 0$ although $c_{2}=e_{2}=0$.
Also, each of the following conditions is necessary and sufficient to ensure
that a symmetric tridiagonal matrix of the form (1) is positive definite[12].

  * 1.

(i) All $c_{i}>0$ for all $i=1,2,\cdots,n$.

  * 2.

(ii) All $e_{i}>0$ for all $i=1,2,\cdots,n$.

The nonsingularity of a matrix should be checked before considering the
problem of computing its inverse. So, we will present two numerical algorithms
for evaluating the determinant of $A_{n}$.
For both Algorithm 1 and Algorithm 2, the authors successfully linked two
methods together in order to evaluate n-th order tridiagonal determinants.
Throughout this paper $\sum_{k+1}^{k}(.)$ is assumed equal to $0$ and
$\prod_{k+1}^{k}(.)$ is assumed equal to $1$. Also, $p_{i}=-a_{i}$ and
$q_{i}=-b_{i},i=1,2,\cdots,n-1$.

Algorithm 1 (To evaluate $f_{1},f_{2},\cdots,f_{n}=det(A_{n})$)

Input: $n$ and the components of the vectors $\mathbf{a},\mathbf{b}$, and
$\mathbf{d}$.
Output: $det(A_{n})$.
Step 1: Set $c_{1}=d_{1}$, $f_{1}=d_{1}$, and $m=1$.
Step 2: While $m\leq n-1$ and $c_{m}\neq 0$ do
$m=m+1$,
$c_{m}=d_{m}-a_{m-1}b_{m-1}/c_{m-1}$,
$f_{m}=c_{m}f_{m-1}$,
End do.
Step 3: For $k=m+1$ to $n$ do
$f_{k}=d_{k}f_{k-1}-a_{k-1}b_{k-1}f_{k-2}$
End do.
Step 4: Set $det(A_{n})=f_{n}$.

Algorithm 2 (To evaluate $g_{n},g_{n-1},\cdots,g_{1}=det(A_{n})$)

Input: $n$ and the components of the vectors $\mathbf{a},\mathbf{b}$, and
$\mathbf{d}$.
Output: $det(A_{n})$.
Step 1: Set $e_{n}=d_{n}$, $g_{n}=d_{n}$, and $m=n$.
Step 2: While $m\geq 2$ and $e_{m}\neq 0$ do
$m=m-1$,
$e_{m}=d_{m}-a_{m}b_{m}/e_{m+1}$,
$g_{m}=e_{m}g_{m+1}$,
End do.
Step 3: For $k=m-1$ step $-1$ to $1$ do
$g_{k}=d_{k}g_{k+1}-a_{k}b_{k}g_{k+2}$
End do.
Step 4: Set $det(A_{n})=g_{1}$.

The Algorithm 1 and Algorithm 2 will be referred to as DETGTRIF and DETGTRIG
respectively.

##  3 The Main Result

This section is mainly devoted for developing a new efficient numerical
algorithm for computing the inverse of general nonsingular tridiagonal
matrices of the form (1).
Lemma 1: For $k=1,2,\cdots,n$, we have

$f_{k}g_{k+1}-d_{k}f_{k-1}g_{k+1}+f_{k-1}g_{k}=f_{n}$ (12)

Proof: for $k=1$, the left-hand side of (12) gives
$L.H.S=f_{1}g_{2}-d_{1}f_{0}g_{2}+f_{0}g_{1}=(d_{1})(g_{2})-(d_{1})(1)(g_{2})+(1)g_{1}=g_{1}=f_{n}=R.H.S$
Thus formula (12) holds for $k=1$.
We next assume that formula (12) holds for $k=1,2,\cdots,i-1$. That is

$f_{i-1}g_{i}-d_{i-1}f_{i-2}g_{i}+f_{i-2}g_{i-1}=f_{n}$ (13)

We want to prove that:

$f_{i}g_{i+1}-d_{i}f_{i-1}g_{i+1}+f_{i-1}g_{i}=f_{n}$ (14)

Consider the left-hand side of (13)
$f_{i-1}g_{i}-d_{i-1}f_{i-2}g_{i}+f_{i-2}g_{i-1}\\\ \hskip
56.9055pt=f_{i-1}g_{i}-f_{i-2}(g_{i-1}+a_{i-1}b_{i-1}g_{i+1})+f_{i-2}g_{i-1}\\\
\hskip 56.9055pt=f_{i-1}g_{i}-a_{i-1}b_{i-1}f_{i-2}g_{i+1}\\\ \hskip
56.9055pt=c_{i-1}f_{i-2}e_{i}g_{i+1}-a_{i-1}b_{i-1}f_{i-2}g_{i+1}\\\ \hskip
56.9055pt=f_{i-2}g_{i+1}(c_{i-1}e_{i}-a_{i-1}b_{i-1})\\\ \hskip
56.9055pt=c_{i-1}f_{i-2}g_{i+1}(e_{i}-\frac{a_{i-1}b_{i-1}}{c_{i-1}})\\\
\hskip 56.9055pt=f_{i-1}g_{i+1}(e_{i}-d_{i}+c_{i})$
Hence, we may write the formula (13) in the form:

$(c_{i}-d_{i}+e_{i})f_{i-1}g_{i+1}=f_{n}$ (15)

Expanding the left-hand side of (15), yields

$(c_{i}f_{i-1})g_{i+1}-d_{i}f_{i-1}g_{i+1}+f_{i-1}(e_{i}g_{i+1})=f_{n}$

Therefore, by using (7) and (8), we get

$f_{i}g_{i+1}-d_{i}f_{i-1}g_{i+1}+f_{i-1}g_{i}=f_{n}$

Consequently, the formula (15) is valid. This completes the proof.
Armed with the above result, we may now formulate the following result.
Theorem 1[8]: Let $A_{n}$ be a nonsingular tridiagonal matrix of the form (1).
Then $B_{n}=A_{n}^{-1}=[\alpha_{ij}]_{1\leq i,j\leq n}$ is given by

$\alpha_{ij}=\left\\{\begin{array}[]{lr}\frac{1}{c_{i}-d_{i}+e_{i}}&\textit{if}\quad
i=j\\\ -y_{i}\alpha_{i+1,j}&\textit{if}\quad i<j\\\
-v_{i-1}\alpha_{i-1,j}&\textit{if}\quad i>j\end{array}\right.\\\ $ (16)

where $y_{i}=\frac{a_{i}}{c_{i}},$ and
$v_{i}=\frac{b_{i}}{e_{i+1}},i=1,2,\cdots,n-1$.
Repeated applications of (16) yields

$\alpha_{ij}=\left\\{\begin{array}[]{lr}\frac{1}{c_{i}-d_{i}+e_{i}}&\textit{if}\quad
i=j\\\ (\prod_{r=i}^{j-1}\frac{p_{r}}{c_{r}})\alpha_{jj}&\textit{if}\quad
i<j\\\ (\prod_{r=j}^{i-1}\frac{q_{r}}{e_{r+1}})\alpha_{jj}&\textit{if}\quad
i>j\end{array}\right.\\\ $ (17)

Theorem 2. Consider the matrix $A_{n}$ in (1). If $det(A_{n})\neq 0$ then the
inverse of the matrix $A_{n}$ is given by $A_{n}^{-1}=[\alpha_{ij}]_{1\leq
i,j\leq n}$ with $\alpha_{ij}$ given by:

$\alpha_{ij}=\left\\{\begin{array}[]{lr}f_{i-1}g_{i+1}/f_{n}&\textit{if}\quad
i=j\\\ (f_{i-1}g_{j+1}/f_{n})(\prod_{r=i}^{j-1}p_{r})&\textit{if}\quad i<j\\\
(f_{j-1}g_{i+1}/f_{n})(\prod_{r=j}^{i-1}q_{r})&\textit{if}\quad
i>j\end{array}\right.$ (18)

Proof: By using (7) and (8), we see that
$\alpha_{ii}=\frac{1}{c_{i}-d_{i}+e_{i}}=\frac{1}{\frac{f_{i}}{f_{i-1}}-d_{i}+\frac{g_{i}}{g_{i+1}}}=\frac{f_{i-1}g_{i+1}}{f_{i}g_{i+1}-d_{i}f_{i-1}g_{i+1}+f_{i-1}g_{i}}=\frac{f_{i-1}g_{i+1}}{f_{n}}$
by using Lemma 1.
Now, if $i<j$, then
$\alpha_{ij}=(\prod_{r=i}^{j-1}\frac{p_{r}}{c_{r}})\alpha_{jj}=\frac{\prod_{r=i}^{j-1}p_{r}}{\prod_{r=i}^{j-1}c_{r}}\frac{f_{j-1}g_{j+1}}{f_{n}}=\frac{\prod_{r=i}^{j-1}p_{r}}{\prod_{r=1}^{i-1}c_{r}\prod_{r=i}^{j-1}c_{r}}\frac{f_{j-1}g_{j+1}f_{i-1}}{f_{n}}\\\
\hskip 14.22636pt=\frac{f_{i-1}g_{j+1}}{f_{n}}\prod_{r=i}^{j-1}p_{r}$ having
used (7).
Similarly, we can prove that:
For $i>j$, we have

$\alpha_{ij}=\frac{f_{j-1}g_{i+1}}{f_{n}}\prod_{r=j}^{i-1}q_{r}$

This completes the proof.
Following [7], we see that

$\alpha_{ij}=[g_{j+1}(\sum_{k=1}^{j-1}\delta_{ik}f_{k-1}\prod_{r=k}^{j-1}p_{r})+\delta_{ij}f_{j-1}g_{j+1}+f_{j-1}(\sum_{j+1}^{n}\delta_{ik}g_{k+1}\prod_{r=j}^{k-1}q_{r})]/f_{n}$
(19)

is satisfied (see also [2]).
where $\delta_{ij}$ is the Kronecker delta symbol which is equal to $1$ if
$i=j$ and $0$ if $i\neq j$. It should be noticed that we may obtain (18) from
(19).
At this point, we can introduce the new numerical algorithm for inverting any
general nonsingular tridiagonal matrix $A_{n}$ of the form (1).

Algorithm 3 (To find the $n\times n$ inverse matrix of a general tridiagonal
matrix
$A_{n}$ of the form (1)).

Input: The components $a_{i},b_{i},i=1,2,\cdots,n-1$ and
$d_{i},i=1,2,\cdots,n$.
Output: The inverse matrix$A_{n}^{-1}=[\alpha_{ij}]_{1\leq i,j\leq n}$.
Step 1: Compute $f_{1},f_{2},\cdots,f_{n}$ using the DETGTRIF algorithm. If
$f_{n}=0$ then print”Singular matrix” and stop.
Step 2: Compute $g_{n+1},g_{n},\cdots,g_{1}$ using the DETGTRIG algorithm.
Step 3: Calculate the elements of the inverse matrix
$\alpha_{ii},i=1,2,3,\cdots,n$,
$\alpha_{ij},i=1,2,3,\cdots,n-1,j=i+1,i+2,\cdots,n$,
and $\alpha_{ij},j=1,2,3,\cdots,n-1,i=j+1,j+2,\cdots,n$ by using (18).

The Algorithm 3 will be referred to as HINVGTRI algorithm.
Using the HINVGTRI algorithm, a procedure has been written in Maple and can be
obtained from the authors upon request.
Remark 1. If $a_{k}=0,1\leq k\leq n-1$, then
$\alpha_{ij}=0,i=1,2,\cdots,k;j=k+1,k+2,\cdots,n$. Similarly, If
$b_{k}=0,1\leq k\leq n-1$, then
$\alpha_{ij}=0,j=1,2,\cdots,k;i=k+1,k+2,\cdots,n$.
Remark 2. If $f_{k}=0,1\leq k\leq n-1$, then
$\alpha_{k+1,j}=\alpha_{j,k+1}=0,j=k+1,k+2,\cdots,n$. Similarly, If
$g_{k}=0,2\leq k\leq n$, then
$\alpha_{i,k-1}=\alpha_{k-1,i}=0,i=1,2,\cdots,k-1$.
Remark 3. The inverse matrix $B_{n}$ of the matrix $A_{n}$ of the form in (1)
can be written as the Hadamard product, also called element-wise product.

$B_{n}=R\circ S$ (20)

where $R=(r_{ij})_{1\leq i,j\leq n}$ and $S=(s_{ij})_{1\leq i,j\leq n}$ are
given by:

$r_{ij}=r_{ji}=\left\\{\begin{array}[]{lr}f_{i-1}g_{i+1}/\Delta&\textit{if}\quad
i=j\\\ f_{i-1}g_{j+1}/\Delta&\textit{if}\quad i\neq j\end{array}\right.$ (21)
$s_{ij}=\left\\{\begin{array}[]{lr}1&\textit{if}\quad i=j\\\
\prod_{r=i}^{j-1}p_{r}&\textit{if}\quad i<j\\\
\prod_{r=j}^{i-1}q_{r}&\textit{if}\quad i>j\end{array}\right.$ (22)

and

$\alpha_{ij}=r_{ij}\times s_{ij},i,j=1,2,\cdots,n.$ (23)

Note that in formula (21), $\Delta=det(A_{n})$.
Remark 4. For centrosymmetric tridiagonal matrices, we have
$f_{i}=g_{n+1-i},i=0,1,2,\cdots,n$.

##  4 Illustrative Examples

In this section, We are going to present some illustrative examples.
Example 1. Consider $A_{4}=\left[\begin{array}[]{cccc}2&-1&0&0\\\ -2&2&1&0\\\
0&1&2&3\\\ 0&0&-1&-3\\\ \end{array}\right]$
Find $A_{4}^{-1}$.
Solution: We have
$a=[-1,1,3]^{t}$, $b=[-2,1,-1)]^{t}$, and $d=[2,2,2,-3]^{t}$ by using the
DETGTRIF algorithm, we get $det(A_{4})=0$. Therefore, $A_{4}^{-1}$ does not
exist.
Example 2. Consider the $n\times n$ symmetric tridiagonal matrix $A_{n}$ given
by [14]:
$A_{n}=\left[\begin{array}[]{ccccc}-1&1&&&\\\ 1&-2&1&&\\\
\ddots&\ddots&\ddots&&\\\ &&1&-2&1\\\ &&&1&-\frac{(n-1)}{n}\\\
\end{array}\right]$
Find $A_{n}^{-1}$.
Solution: We have
$a=b=[1,1,\cdots,1]^{t}$ and $d=[-1,-2,-2,\cdots,-2,-\frac{(n-1)}{n}]^{t}$,
Step 1: $c_{i}=-1,i=1,2,\cdots,n-1$ and $c_{n}=\frac{1}{n}$.
$e_{i}=-(\frac{i-1}{i}),i=n,n-1,\cdots,2$
and $e_{1}=1$. $f_{i}=(-1)^{i},i=1,2,\cdots,n-1$ and
$f_{n}=det(A_{n})=\frac{(-1)^{n-1}}{n}$.
$g_{i}=(-1)^{n+1-i}(\frac{i-1}{n}),i=n,n-1,\cdots,2$ and
$g_{1}=det(A_{n})=\frac{(-1)^{n-1}}{n}$.
Step 2: When $i=j$ then $\alpha_{ii}=f_{i-1}g_{i+1}/det(A_{n})=i$
Step 3: When $i<j$ then $\alpha_{ij}=(-1)^{j-i}f_{i-1}g_{j+1}/det(A_{n})=j$
Step 4: When $i>j$ and since $A_{n}^{-1}$ is symmetric then $\alpha_{ij}=i$.
So $A_{n}^{-1}=[max(i,j)]_{1\leq i,j\leq n}$
Example 3. Compute $A^{-1}=\left[\begin{array}[]{cccc}25&-9&0&0\\\
-9&13&-4&0\\\ 0&-4&5&-1\\\ 0&0&-1&1\\\ \end{array}\right]^{-1}$
Solution: We have
$a=b=[-9,-4,-1]^{t}$ and $d=[25,13,5,-1]^{t}$.
$f=[1,25,244,820,576]^{t}$,and $g=[576,36,4,1,1]$.
Consequently, $\Delta=576$,
and
$A^{-1}=\frac{1}{576}\left[\begin{array}[]{cccc}36&36&36&36\\\
36&100&100&100\\\ 36&100&244&244\\\ 36&100&244&820\\\ \end{array}\right]$
Example 4. Consider the centrosymmetric matrix
$A_{n}=\left[\begin{array}[]{ccccc}2&-1&&&\\\ -1&2&-1&&\\\
\ddots&\ddots&\ddots&&\\\ &&-1&2&-1\\\ &&&-1&2\\\ \end{array}\right]$
Find $A^{-1}$.
Solution: We have $a=b=[-1,-1,\cdots,-1]^{t}$ and $d=[2,2,\cdots,2]^{t}$.
Since $A_{n}$ is centrosymmetric matrix, then
$f_{i}=i+1,g_{i}=n+2-i,i=1,2,\cdots,n$.
$A^{-1}=X\circ R,\quad x_{ij}=1for1\leq i,j\leq n$,
Thus
$A^{-1}=R,r_{ij}=r_{ji}=\frac{1}{n+1}f_{i-1}g_{j+1}=\frac{1}{n+1}(i)(n+1-j)=\frac{i(n+1-j)}{n+1}$.

Example 5. Consider $A_{5}=\left[\begin{array}[]{ccccc}1&-1&0&0&0\\\
-1&3&-1&0&0\\\ 0&-1&1&-1&0\\\ 0&0&-1&1&-1\\\ 0&0&0&-1&1\\\ \end{array}\right]$
Find $A_{5}^{-1}$.
Solution: We have $a=b=[-1,-1,-1,-1]^{t}$ and $d=[1,3,1,1,1]^{t}$.
Thus, $f=[1,1,2,1,-1,-2]^{t}$, $g=[-2,-3,-1,0,1,1]^{t}$ and

$A^{-1}=\left[\begin{array}[]{ccccc}\frac{3}{2}&\frac{1}{2}&0&-\frac{1}{2}&-\frac{1}{2}\\\
\frac{1}{2}&\frac{1}{2}&0&-\frac{1}{2}&-\frac{1}{2}\\\ 0&0&0&-1&-1\\\
-\frac{1}{2}&-\frac{1}{2}&-1&-1&-\frac{1}{2}\\\
-\frac{1}{2}&-\frac{1}{2}&1&-1&\frac{1}{2}\\\ \end{array}\right]$.

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